aa r X i v : . [ m a t h . OA ] M a r Noncommutative Geometry of QuantizedCoverings
Petr R. Ivankov* e-mail: * [email protected] 26, 2020 ontents C ∗ -inductive limits of nonunital algebras . . . . . . . . . . . . . . . . 151.3 Inclusions of some C ∗ algebras . . . . . . . . . . . . . . . . . . . . . . 161.4 Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 C ∗ -algebras . . . . . . . . . . . . . . . . 863.3.2 Universal coverings of operator spaces . . . . . . . . . . . . . 873.4 Induced representations . . . . . . . . . . . . . . . . . . . . . . . . . . 873.5 Coverings of spectral triples . . . . . . . . . . . . . . . . . . . . . . . . 913.5.1 Constructive approach . . . . . . . . . . . . . . . . . . . . . . . 923.5.2 Axiomatic approach . . . . . . . . . . . . . . . . . . . . . . . . 94 C ∗ -algebras 97 C ∗ -algebras . . . . . . . . . . . . . . . . . 1364.7 Continuous structures and sheaves . . . . . . . . . . . . . . . . . . . . 1484.8 Lifts of continuous functions . . . . . . . . . . . . . . . . . . . . . . . 1544.9 Geometrical lifts of spectral triples . . . . . . . . . . . . . . . . . . . . 1644.9.1 Finite-fold lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.9.2 Infinite lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.10 Finite-fold coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.10.1 Coverings of C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . 1684.10.2 Induced representation . . . . . . . . . . . . . . . . . . . . . . 1794.10.3 Coverings of spectral triples . . . . . . . . . . . . . . . . . . . . 1824.10.4 Unoriented spectral triples . . . . . . . . . . . . . . . . . . . . 1884.11 Infinite coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.11.1 Inverse limits of coverings in topology . . . . . . . . . . . . . 1904.11.2 Algebraic construction in brief . . . . . . . . . . . . . . . . . . 2004.11.3 Coverings of C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . 2014.11.4 Universal coverings and fundamental groups . . . . . . . . . 2144.11.5 Induced representation . . . . . . . . . . . . . . . . . . . . . . 2164.11.6 Coverings of spectral triples . . . . . . . . . . . . . . . . . . . . 2204.12 Supplementary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 2294.12.1 Families of operator spaces . . . . . . . . . . . . . . . . . . . . 2304.12.2 Operator spaces and coverings . . . . . . . . . . . . . . . . . . 2354.12.3 Some operator systems . . . . . . . . . . . . . . . . . . . . . . 239 C ∗ -algebras induced by coverings . . . . . . . . . . . . . . . . 2455.2.2 Geometrical lifts of spectral triples . . . . . . . . . . . . . . . . 2455.3 Finite-fold coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465.3.1 Coverings of C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . 2464.3.2 Coverings of operator spaces . . . . . . . . . . . . . . . . . . . 2645.3.3 Induced representation . . . . . . . . . . . . . . . . . . . . . . 2655.3.4 Coverings of spectral triples . . . . . . . . . . . . . . . . . . . . 2695.3.5 Unoriented spectral triples . . . . . . . . . . . . . . . . . . . . 2765.4 Infinite coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2775.4.1 Coverings of C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . 2775.4.2 Coverings of operator spaces . . . . . . . . . . . . . . . . . . . 2965.4.3 Universal coverings and fundamental groups . . . . . . . . . 2995.4.4 Induced representation . . . . . . . . . . . . . . . . . . . . . . 3015.4.5 Coverings of spectral triples . . . . . . . . . . . . . . . . . . . . 305 C ∗ -Hilbert modules over commutative C ∗ -algebras 309 C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . 3317.2.2 Coverings and irreducible representations . . . . . . . . . . . 3437.3 Infinite coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 C ∗ -algebra . . . . . . . . . . . . . . . . . . . . . . 3708.2.2 Induced representation . . . . . . . . . . . . . . . . . . . . . . 3798.2.3 Coverings of spectral triples . . . . . . . . . . . . . . . . . . . . 3828.3 Infinite coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3868.3.1 Equivariant representation . . . . . . . . . . . . . . . . . . . . 3878.3.2 Inverse noncommutative limit . . . . . . . . . . . . . . . . . . 3908.3.3 Covering of spectral triple . . . . . . . . . . . . . . . . . . . . . 4115 Isospectral deformations and their coverings 415
10 The double covering of the quantum group SO q ( ) C ∗ -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 43510.3 SO q ( ) as an unoriented spectral triple . . . . . . . . . . . . . . . . . 436 Appendices 441A Topology 443
A.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443A.2 Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447A.2.1 Unique path lifting . . . . . . . . . . . . . . . . . . . . . . . . . 449A.2.2 Regular and universal coverings . . . . . . . . . . . . . . . . . 449A.3 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
B Algebra 453
B.1 Algebraic Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . 453B.2 Finite Galois coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . 453B.3 Profinite and residually finite groups . . . . . . . . . . . . . . . . . . 454
C Functional analysis 457
C.1 Weak topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457C.2 Fourier transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
D Operator algebras 461
D.1 C ∗ -algebras and von Neumann algebras . . . . . . . . . . . . . . . . . 461D.2 States and representations . . . . . . . . . . . . . . . . . . . . . . . . . 465D.2.1 GNS construction . . . . . . . . . . . . . . . . . . . . . . . . . . 465D.2.2 Irreducible representations . . . . . . . . . . . . . . . . . . . . 466D.3 Inductive limits of C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . 470D.4 Hilbert modules and compact operators . . . . . . . . . . . . . . . . . 471D.5 Hermitian modules and functors . . . . . . . . . . . . . . . . . . . . . 474D.6 Strong Morita equivalence for C ∗ -algebras . . . . . . . . . . . . . . . 4756.7 Operator spaces and algebras . . . . . . . . . . . . . . . . . . . . . . . 476D.8 C ∗ -algebras of type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479D.8.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479D.8.2 Fields of operators . . . . . . . . . . . . . . . . . . . . . . . . . 482D.8.3 C ∗ -algebras as cross sections and their multiplies . . . . . . . 483 E Spectral triples 485
E.1 Definition of spectral triples . . . . . . . . . . . . . . . . . . . . . . . . 485E.2 Representations of smooth algebras . . . . . . . . . . . . . . . . . . . 487E.3 Noncommutative differential forms . . . . . . . . . . . . . . . . . . . 488E.3.1 Noncommutative connections and curvatures . . . . . . . . . 489E.3.2 Connection and curvature . . . . . . . . . . . . . . . . . . . . . 490E.4 Commutative spectral triples . . . . . . . . . . . . . . . . . . . . . . . 491E.4.1 Spin c manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 491E.4.2 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . 493E.5 Coverings of Riemannian manifolds . . . . . . . . . . . . . . . . . . . 494E.6 Finite spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495E.7 Product of spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . 496 F Noncommutative torus and Moyal plane 497
F.1 Noncommutative torus T n Θ . . . . . . . . . . . . . . . . . . . . . . . . 497F.1.1 Definition of noncommutative torus T n Θ . . . . . . . . . . . . . 497F.1.2 Geometry of noncommutative tori . . . . . . . . . . . . . . . . 499F.2 Moyal plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 G Foliations and operator algebras 509
G.1 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509G.2 Operator algebras of foliations . . . . . . . . . . . . . . . . . . . . . . 515G.2.1 Restriction of foliation . . . . . . . . . . . . . . . . . . . . . . . 516G.2.2 Lifts of foliations . . . . . . . . . . . . . . . . . . . . . . . . . . 517
H Miscellany 521
H.1 Pre-order category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521H.2 Flat connections in the differential geometry . . . . . . . . . . . . . . 521H.3 Quantum SU ( ) and SO ( ) . . . . . . . . . . . . . . . . . . . . . . . . 524H.4 Presheaves and Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 527H.5 Isospectral deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 5287 ntroduction Gelfand-Na˘imark theorem [2] states the correspondence between locally com-pact Hausdorff topological spaces and commutative C ∗ -algebras. So a noncommu-tative C ∗ -algebra can be regarded as a noncommutative generalization of a topo-logical space. Further development of noncommutative geometry gives general-izations of following classical geometric an topological notions.Table 1: Mapping between classical and noncommutative geometryClassical notion Noncommutative generalizationTopological space C ∗ -algebraMeasure space von Neumann algebraRiemannian manifold Spectral tripleTopological K -theory K -theory of C ∗ -algebrasHomology and cohomology Noncommutative homology and cohomologyIn this book we continue development of the noncommutative geometry, thisbook contains generalizations of following notions.9able 2: Mapping between geometry of topological coverings and noncommuta-tive ones Classical notion Noncommutative generalizationCovering Noncommutative coveringTheorem about covering Theorem about coveringof Riemannian manifold of spectral tripleFundamental group of a space π ( X ) Fundamental group of a C ∗ -algebra π ( A ) Flat connections given by Noncommutative flat connectionscoverings given by noncommutative coveringsUnoriented spin c -manifolds Unoriented spectral triplesThere is a set of theories of noncommutative coverings (e.g. [16,68]). In contrarythe presented here theory gives results which are (almost) equivalent to the clas-sical topological theory. In particular covering spaces of commutative spaces arealso commutative. This fact yields pure algebraic definition of the fundamentalgroup (cf. the Theorem 4.11.39 and the Corollary 4.11.40).The Chapter 1 contains preliminary results. The material of this chapter can beread as needed.The Chapter 2 contains the construction of noncommutative of finite-old cov-erings of C ∗ -algebras and operator spaces. Sections 2.1 -2.3 are basic and neededfor the further reading of this book. Other sections are written for those who areinterested in following applications of this theory: • Coverings and strong Morita equivalence. • Noncommutative path lifting. 10
Coverings of spectral triples. • Finite noncommutative coverings and flat connections. • Unoriented spectral triples.The Chapter 3 is devoted to noncommutative infinite coverings of C ∗ -algebrasand operator spaces. Sections 3.1 - 3.4 are basic. The Section 3.5.1 is interesting forthose who are interested in coverings of spectral triples.The Chapter 4 contains applications of described in Chapters 2 and 3 to com-mutative coverings. It is proven the one to one correspondence between geometryof topological coverings and "noncommutative ones" presented in the Table 2.The Chapter 5 is devoted to noncommutative coverings of C ∗ -algebras withcontinuous trace. It is proven that the theory of noncommutative coverings of C ∗ -algebras with continuous trace contains all ingredients of right row of the Table2. In the Chapter 6 the coverings of spaces of vector fields are being discussed.Sections of vector fields of Hilbert spaces have the natural structure of operatorspaces. Application of the results of the Chapters 2 and 3 yield finite-fold andinfinite noncommutative coverings of these spaces.In the Chapter 7 we consider noncommutative finite-fold and infinite coveringsof foliations.The Chapter 8 is devoted to noncommutative coverings of noncommutative tori.The Chapter 9 is devoted to noncommutative coverings of isospectral deforma-tions. We consider "noncommutative finite-fold coverings" only. The presented inSections 2.7, 2.8 and 2.9 of coverings of spectral triples is applied to isospectraldeformations.The Chapter 10 is devoted to the two-listed covering of quantum SO ( ) byquantum SU ( ) ; 112 hapter 1 Preliminaries
This research is based on the noncommutative generalizations of both topologi-cal spaces and coverings. Following two theorems describe these generalizations.
Theorem 1.1.1. [2] (Gelfand-Na˘ımark). Let A be a commutative C ∗ -algebra and let X be the spectrum of A. There is the natural ∗ -isomorphism γ : A → C ( X ) . Theorem 1.1.2. [56] Suppose X and Y are compact Hausdorff connected spaces andp : Y → X is a continuous surjection. If C ( Y ) is a projective finitely generated Hilbertmodule over C ( X ) with respect to the action ( f ξ )( y ) = f ( y ) ξ ( p ( y )) , f ∈ C ( Y ) , ξ ∈ C ( X ) , then p is a finite-fold covering. Following table contains special symbols.13ymbol Meaningˆ A or A ∧ Spectrum of a C ∗ - algebra A with the hull-kernel topology(or Jacobson topology) A + Cone of positive elements of C ∗ - algebra, i.e. A + = { a ∈ A | a ≥ } A G Algebra of G - invariants, i.e. A G = { a ∈ A | ga = a , ∀ g ∈ G } Aut ( A ) Group of * - automorphisms of C ∗ - algebra AA ′′ Enveloping von Neumann algebra of AB ( H ) Algebra of bounded operators on a Hilbert space H C (resp. R ) Field of complex (resp. real) numbers C ( X ) C ∗ - algebra of continuous complex valuedfunctions on a compact space X C ( X ) C ∗ - algebra of continuous complex valued functions on a locallycompact topological space X equal to 0 at infinity C c ( X ) Algebra of continuous complex valued functions on atopological space X with compact support C b ( X ) C ∗ - algebra of bounded continuous complex valuedfunctions on a locally compact topological space X G (cid:16) e X | X (cid:17)
Group of covering transformations of covering e X → X [70] H Hilbert space K = K ( H ) C ∗ - algebra of compact operators on the separable Hilbert space H K ( A ) Pedersen ideal of C ∗ -algebra AC ∗ - lim −→ C ∗ -inductive limitlim ←− Inverse limit M ( A ) A multiplier algebra of C ∗ -algebra A M n ( A ) The n × n matrix algebra over C ∗ -algebra A N A set of positive integer numbers N A set of nonnegative integer numbers supp a Support of a ∈ C b ( X ) rep x or rep Ax An irreducible representation A → B ( H ) which correspondsto a point x ∈ ˆ A of spectrum of A (cf. (D.2.3)). Z Ring of integers Z n Ring of integers modulo nk ∈ Z n An element in Z n represented by k ∈ Z X \ A Difference of sets X \ A = { x ∈ X | x / ∈ A }| X | Cardinal number of a finite set X [ x ] The range projection of element x of a von Neumann algebra. f | A ′ Restriction of a map f : A → B to A ′ ⊂ A , i.e. f | A ′ : A ′ → B .2 C ∗ -inductive limits of nonunital algebras Definition 1.2.1.
We say that a C ∗ -algebra A is connected if the only central projec-tions of M ( A ) are 0 and 1. Let A be a C ∗ -algebra. Denote by A ∼ the unital C ∗ -algebra given by A ∼ def = ( A if A is unital A + if A is not unital (1.2.1)where A + is the minimal unitization of A (cf. Definition D.1.8). Definition 1.2.3.
An injective *-homomorphism φ : A → B is said to be unital if itis unital in the sense of the Definition D.3.1 or can be uniquely extended up to theunital (in sense of the Definition D.3.1) of *-homomorphism φ ∼ : A ∼ → B ∼ where A + and B + are minimal unitizations of A and B (cf. Definition D.1.8). Remark 1.2.4.
The Definition 1.2.3 is a generalization of the Definition D.3.1.
Remark 1.2.5.
Any unital *-homomorphism φ : A → B can be uniquely extendedup to the unital *-homomorphism M ( φ ) : M ( A ) → M ( B ) of multipliers. Let { A λ } λ ∈ Λ be a family of C ∗ -algebras where Λ denotes an directed set (cf.Definition A.1.3). Suppose that for every µ , ν with µ ≤ ν , there exists the uniqueunital (in sense of the Definition 1.2.3) injective *-homomorphism f µν : A µ ֒ → A ν satisfying f µν = f µλ ◦ f λν where µ < λ < ν . If C ∗ -algebras A λ are not unitalthen there are natural unital (in sense of the Definition D.3.1) unique injective *-homomorphisms f ∼ µν : A ∼ µ → A ∼ ν of minimal unitiazations. From the TheoremD.3.4 it turns out that C ∗ -inductive limit C ∗ - lim −→ Λ A ∼ λ of (cid:8) A ∼ λ (cid:9) . Definition 1.2.7.
In the situation of 1.2.6 consider injective *-homomorphisms A λ ֒ → C ∗ - lim −→ Λ A ∼ λ of (cid:8) A ∼ λ (cid:9) as inclusions A λ ⊂ C ∗ - lim −→ Λ A ∼ λ of (cid:8) A ∼ λ (cid:9) . The C ∗ -norm completion of the union ∪ λ ∈ Λ A λ ⊂ C ∗ - lim −→ Λ A ∼ λ is said to be the C ∗ - inductivelimit of { A λ } . It is denoted by C ∗ - lim −→ λ ∈ Λ A λ or C ∗ - lim −→ A λ . Remark 1.2.8.
The Definition 1.2.7 is a generalization of the Definition D.3.3. Thereis the evident generalizations of the Theorem D.3.4 and the Proposition D.3.6.
Lemma 1.2.9.
If A + is the minimal unitization of a nonunital C ∗ -algebra A and Ω A , Ω A + are state spaces of A and A + respectively then one has Ω A = (cid:8) τ + ∈ Ω A + | τ + ( ⊕ ) =
0; where 0 ⊕ ∈ A ⊕ C ∼ = A + (cid:9) . (1.2.2)15 roof. Any state τ : A → C induces the state τ + : A + → C given by τ + ( a ⊕ λ ) = τ ( a ) for any a ⊕ λ ∈ A ⊕ C ∼ = A + . Clearly τ + satisfies to (1.2.2). Corollary 1.2.10.
If a C ∗ -algebra A is a C ∗ -inductive limit (in sense of the Definition1.2.7) of A λ ( λ ∈ Λ ), the state space Ω of A is homeomorphic to the projective limit of thestate spaces Ω λ of A λ .Proof. Let us consider unital injective *-homomorphisms f + µν : A + µ → A + ν (in thesense of the Definition D.3.1) of the minimal unitiazations and let b A + be a C ∗ -inductive limit (in sense of the Definition 1.2.7) of (cid:8) A + λ (cid:9) . From the Theorem D.3.7it follows that the state space Ω b A + is the projective limit Ω + λ of A + λ . From theLemma 1.2.9 it turns that Ω b A ⊂ Ω b A + and Ω λ ⊂ Ω + λ and for any λ ∈ Λ . Moreoverevery state b τ ∈ Ω b A is mapped onto τ λ ∈ Ω λ . It follows that Ω is homeomorphicto the projective limit of the state spaces Ω λ . C ∗ algebras Definition 1.3.1.
Suppose that there is a C ∗ -algebra D and a Hilbert D -module X D with D -valued product h· , ·i D : X D × X D → D . Suppose that there is a C ∗ -subalgebra A of D and a C ∗ -Hilbert A -module X A with product h· , ·i A . An inclu-sion X A ⊂ X D is said to be exact if following conditions hold h ξ , η i A = h ξ , η i D ∀ ξ , η ∈ X A , (1.3.1) ∀ η ∈ X D \ X A ∃ ξ ∈ X A h ξ , η i D / ∈ A (1.3.2) Definition 1.3.2.
Let us consider the situation of the Definition (1.3.1). Denote by L ( X D ) the space of D -linear adjointable maps (cf. Definition D.4.7). L ∈ L D ( X D ) is said to be A - continuous if LX A ⊂ X A . We write L ∈ L A ( X A ) .Following lemma is a consequence of the above definition. Lemma 1.3.3.
In the situation of the definition (1.3.1) L ∈ L D ( X D ) is A-continuous ifand only if h ξ , L η i D ∈ A ∀ ξ , η ∈ X A (1.3.3) Corollary 1.3.4.
In the situation of the definition (1.3.1) a self-adjoint L = L ∗ ∈ L D ( X D ) is A-continuous if and only if h ξ , L ξ i D ∈ A ∀ ξ ∈ X A (1.3.4) Proof.
Follows from the Lemma 1.3.3 and the polarization equality (D.4.3).16 orollary 1.3.5.
Let { ξ α ∈ X A } α ∈ A be a family such that the C -linear span of { ξ α } isdense in X A . A self-adjoint L : X D → X D is A-continuous if and only if ∀ α ∈ A h ξ α , L ξ α i D ∈ A . (1.3.5) Lemma 1.3.6.
Let a ∈ B ( H ) + be a positive operator, and let G ε : R → R be a boundedcontinuous map given by G ε ( x ) = x ≤ x ε < x ≤ ε x > ε ε >
0. (1.3.6)
If p def = G ε ( a ) then for any positive b ∈ B ( H ) + such that b ≤ a one has k b − pbp k ≤ ε k a k . (1.3.7) Proof. If p ∈ B ( H ) is the spectral projection of a (cf. Definition D.2.29) on the set ( − ∞ , ε ) then one has the direct sum H = p H ⊕ ( − p ) H such that ax = apx ∀ x ∈ ( − p ) H , k ay k ≤ ε k y k ∀ y ∈ p H Denote by a ′ def = a , b ′ def = b , H ′ = ( − p ) H , H ′′ = p H . According to ourconstruction one has a ′ x = a ′ px , ∀ x ∈ H ′ ; (cid:13)(cid:13) a ′ y (cid:13)(cid:13) ≤ ε k y k , ∀ y ∈ H ′′ .Suppose that y ∈ H ′′ is such that k b ′ y k > ε k y k . If p y is the projector along y then following inequality (cid:13)(cid:13) b ′ y (cid:13)(cid:13) = (cid:13)(cid:13) b ′ p y y (cid:13)(cid:13) = (cid:13)(cid:13) b ′ p y (cid:13)(cid:13) k y k = q(cid:13)(cid:13) p y b ′ b ′ p y (cid:13)(cid:13) k y k = q(cid:13)(cid:13) p y bp y (cid:13)(cid:13) k y k << q(cid:13)(cid:13) p y ap y (cid:13)(cid:13) k y k = q(cid:13)(cid:13) p y a ′ a ′ p y (cid:13)(cid:13) k y k = (cid:13)(cid:13) a ′ p y (cid:13)(cid:13) k y k = (cid:13)(cid:13) a ′ p y y (cid:13)(cid:13) ≤ ε k y k yields a contradiction. If x ∈ H ′ and y ∈ H ′′ then k x + y k = k x k + k y k andfollowing condition holds. ( x + y ) ( b − pbp ) ( x + y ) = y ( b − pbp ε ) y ++ x ( b − pbp ) y + y ( b − pbp ) x (1.3.8)17aking into account k yby k = (cid:13)(cid:13) yb ′ b ′ y (cid:13)(cid:13) ≤ ε (cid:13)(cid:13) b ′ (cid:13)(cid:13) k y k ≤ ε k a k k y k ≤ ε k a k k x + y k , k ypbpy k ≤ k yby k ≤ ε k a k k x + y k , k xby k = k ybx k = (cid:13)(cid:13) xb ′ b ′ y (cid:13)(cid:13) ≤ ε (cid:13)(cid:13) b ′ (cid:13)(cid:13) k x k k y k ≤ ε k a k k x k k y k ≤≤ ε k a k k x k + k y k = ε k a k k x + y k k xpbpy k = k ypbpx k ≤ ε k a k k x + y k k ( x + y ) ( b − pbp ) ( x + y ) k ≤ ε k a k k x + y k ,hence k b − pbp k ≤ ε k a k . Lemma 1.3.7. (i) If B ⊂ K ( X A ) is a hereditary subalgebra and X BA the norm closureof X A B then there is the natural isomorphismB ∼ = K (cid:16) X BA (cid:17) (1.3.9) (ii) If X A is a Hilbert module, and Y A ⊂ X A is a submodule then Y A is the normcompletion of X A K ( Y A ) . Moreover K ( Y A ) ⊂ K ( X A ) is a hereditary subalbegra.Proof. (i) The poof has two parts (a) B ⊂ K (cid:0) X BA (cid:1) and (b) K (cid:0) X BA (cid:1) ⊂ B .(a) B ⊂ K (cid:0) X BA (cid:1) . Let b ∈ B + be a positive element. If ε > a ∈ ∑ nj = ξ j ih η j ∈ K ( X A ) such that k a − b k < ε /2. If G ε isgiven by (1.3.6) and p = G ε / ( k a k ) ( b ) ∈ B then from the Lemma 1.3.6 it turnsout k b − pbp k ≤ ε k p k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = ξ j p ih η j p − pbp (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p n ∑ j = ξ j ih η j − b ! p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = ξ j ih η j − b (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = ξ j p ih η j p − b (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = ε .Taking into account ∑ nj = ξ j p ih η j p ∈ K (cid:0) X BA (cid:1) we conclude b ∈ K (cid:0) X BA (cid:1) .(b) K (cid:0) X BA (cid:1) ⊂ B . If b ∈ K (cid:0) X BA (cid:1) and ε > b ′ ∈ ∑ nj = ξ j b ′ j ih η j b ′′ j ∈K (cid:0) X BA (cid:1) such that k b − b ′ k < ε . Since B ⊂ K ( X A ) is a hereditary b ′ j , b ′′ j ∈ B one has ξ j b ′ j ih η j b ′′ j = b ′∗ j (cid:0) ξ j ih η j (cid:1) b ′′ j ∈ B . Since K (cid:0) X BA (cid:1) is norm closed onehas b ∈ B .(ii) Every a ∈ K ( Y A ) is represented by a C ∗ -norm convergent series a = ∞ ∑ j = ξ j ih η j ξ j , η j ∈ Y A so for any ξ ∈ X A ξ a = ∞ ∑ j = (cid:10) ξ j , ξ (cid:11) η j .Since any summand of the above series lies in Y A and the series is norm convergentone has ξ a ∈ Y A , equivalently X A K ( Y A ) ⊂ Y A . If ξ ∈ Y A then ξ ih ξ ∈ K ( Y A ) . If G ε is given by (1.3.6) then p = G ε / k ξ k ( ξ ih ξ ) ∈ K ( Y A ) and k ξ − ξ p k < ε , so ξ liesin the norm completion of X A K ( Y A ) . If b ′ b ′′ ∈ K ( Y A ) then b ′ = ∞ ∑ j = ξ ′ j ih η ′ j , b ′′ = ∞ ∑ k = ξ ′′ k ih η ′′ k and for any a ∈ K ( X A ) one has b ′ ab ′′ = ∞ ∑ j = k = ξ ′ j ih η ′ j a , ξ ′′ k ih η ′′ k Every summand of the above sum lies in K ( Y A ) and the sum is C ∗ -norm con-vergent, so b ′ ab ′′ ∈ K ( Y A ) . From the Lemma D.1.17 it follows that K ( X ′ A ) is ahereditary subalbebra of K ( X A ) . 19 emma 1.3.8. Let B ⊂ K ( X A ) be a hereditary subalgebra, and let and X BA the normclosure of X A B. For any positive b ∈ B + and any ξ ∈ X A there is η ∈ X BA such that ≤ b ′ ≤ b ⇒ (cid:13)(cid:13)(cid:10) ξ b ′ , ξ (cid:11) A − (cid:10) η b ′ , η (cid:11) A (cid:13)(cid:13) < ε . Proof. f G ε is given by (1.3.6) and p = G ε / ( k b kk ξ k ) ( b ) ∈ B then from the Lemma1.3.6 it turns out (cid:13)(cid:13) b ′ − pb ′ p (cid:13)(cid:13) ≤ ε k ξ k < ε k ξ k .If η = ξ p then η ∈ X BA and one has (cid:13)(cid:13)(cid:10) ξ b ′ , ξ (cid:11) A − (cid:10) η b ′ , η (cid:11) A (cid:13)(cid:13) < ε . Lemma 1.3.9.
Let X A be a Hilbert A-module, let Y A be a be a Hilbert A-module withthe inclusion ι : Y A ֒ → X A and the projection p : X A ֒ → Y A such that p ◦ ι = Id Y A .Following conditions hold(i) There is the natural inclusion ι L : L ( Y A ) ֒ → L ( X A ) such that ι L ( L ( Y A )) is ahereditary subalgebra of L ( X A ) .(ii) The inclusion induces the natural inclusion ι K : K ( Y A ) ֒ → K ( X A ) such that ι K ( K ( Y A )) is a hereditary algebra of both L ( X A ) and K ( X A ) . Moreover one has ι K ( K ( Y A )) = L ( Y A ) ∩ K ( X A ) . (1.3.10) (iii) ι ( Y A ) is the norm completion of X A ι K ( K ( Y A )) .Proof. Let us define the inclusion ι L : L ( Y A ) ֒ → L ( X A ) ; a ( ξ ι A ap A ξ ) ∀ ξ ∈ X A . (1.3.11)(i) If a ∈ L ( X A ) and b , b ′ ∈ ι ( L ( X B )) then one has bab ′ ξ = ι bpa ι b ′ p ξ = ι (cid:0) bpa ι b ′ (cid:1) p ξ , (1.3.12)and taking into account (1.3.11) we conclude bab ′ ∈ L (cid:16) X C ( X ) (cid:17) . From the LemmaD.1.17 it turns out that ι L ( L ( Y A )) is a hereditary subalgebra of L ( X A ) . We write Y A ⊂ X A and L ( Y A ) ⊂ L ( X A ) instead of ι : Y A ֒ → X A respectively ι L : L ( Y A ) ֒ → ( X A ) .(ii) Similarly to (1.3.11) define the inclusion ι K : K ( Y A ) ֒ → K ( X A ) ; a ( ξ ι A ap A ξ ) ∀ ξ ∈ X A . (1.3.13)From (1.3.12) it turns out that ι K : K ( Y A ) is a hereditary algebra of K ( X A ) . Oth-erwise K ( X A ) is a hereditary algebra of L ( X A ) , hence ι K ( K ( Y A )) is a hereditaryalgebra of L ( X A ) . The inclusion K ( Y A ) ⊂ L ( Y A ) ∩ K ( X A ) is already proven.Suppose a ∈ L ( Y A ) ∩ K ( X A ) , we can suppose that a is positive. For any ε thereare ξ j , η j ∈ X A such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − n ∑ j = ξ j ih η j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε G ε is given by (1.3.6) and p ′ = G ε / ( k a k ) ( ι ( b )) ∈ ι ( K ( Y A )) then from (1.3.7) itfollows that (cid:13)(cid:13) a − p ′ ap ′ (cid:13)(cid:13) < ε k p ′ k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ′ a − n ∑ j = ξ j ih η j ! p ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ′ ap ′ − n ∑ j = ξ j p ′ ih η j p ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − n ∑ j = ξ j p ′ ih η j p ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε .and taking into account ξ j p ′ , η j p ′ ∈ ι ( Y A ) one has a ∈ ι K ( K ( Y A )) .(iii) Consider a compact operator a = ∞ ∑ j = ξ j ih η j ∈ K ( Y A ) (1.3.15)where ξ j , η j ∈ Y A . If ξ ∈ X A then ξ ι (cid:0) ξ j ih η j (cid:1) = ι (cid:0) ξ j (cid:1) (cid:10) ξ , ι (cid:0) η j (cid:1)(cid:11) A ∈ Y A . Takinginto account that (1.3.15) is norm convergent one has ξι ( a ) ∈ Y A , hence the normcompletion of X A ι K ( K ( Y A )) is a submodule of Y A . Conversely let η ∈ Y A and let b = η ih η ∈ K ( Y A ) . If G ε is given by (1.3.6) and p ′ = G δ / ( k η k ) ( ι ( b )) ∈ ι ( K ( Y A )) ,then one has ι ( η ) p ′ ∈ X A ι K ( K ( Y A )) , (cid:13)(cid:13) ι ( η ) − ι ( η ) p ′ (cid:13)(cid:13) ≤ δ .However δ is arbitrary real number, it turns out ι ( η ) lies in the closure of X A ι K ( K ( Y A )) .21 .4 Miscellany Lemma 1.4.1.
Let B is a C ∗ -subalgebra of a separable C ∗ -algebra A containing an approx-imate unit for A. Suppose that A is separable and suppose there are a , ..., a n ∈ M ( A ) such that M ( A ) = a M ( B ) + ... + a n M ( B ) . (1.4.1) Let p : A → A ′ be a surjective morphism of C ∗ -algebras, and let e p : M ( A ) → M ( A ′ ) isits extension given by the Theorem D.1.34. Then M ( A ′ ) is a finitely generated e p ( M ( B )) -module.Proof. If a ′ ∈ M ( A ′ ) then from the Theorem D.1.34 it turns out that there is a ∈ M ( A ) such that a ′ = e p ( a ) . Otherwise from there are b , ..., b n ∈ M ( B ) such that a = a b + ... + a n b n . It turns out a ′ = e p ( a ) e p ( b ) + ... + e p ( a n ) e p ( b n ) ,i.e. M ( A ′ ) is an e p ( M ( B )) -module, generated by e p ( a ) , ..., e p ( a n ) ∈ M ( A ′ ) .Let X be a C ∗ -Hilbert A -module. We also write X A instead X and h x , y i X A instead h x , y i A , the meaning depends on context. If ℓ ( A ) is the standard Hilbert A -module (cf. Definition D.4.11) There is the natural *-isomorphism K (cid:0) ℓ ( A ) (cid:1) ∼ = A ⊗ K (1.4.2)Any element a ∈ K (cid:0) ℓ ( A ) (cid:1) corresponds to an infinite matrix a . . . a j ,1 . . .... . . . ... . . . a j ,1 . . . a j , j . . .... ... ... . . . ∈ K (cid:0) ℓ ( A ) (cid:1) . (1.4.3)The free finitely generated A -module A n is also C ∗ -Hilbert A module with theproduct D(cid:8) a j (cid:9) j = n , (cid:8) b j (cid:9) j = n E A n = n ∑ j = a ∗ j b j . (1.4.4)The algebra of compact operators is given by K ( A n ) = M n ( A ) (1.4.5)22 emark 1.4.2. Similarly to (1.4.3) and (1.4.5) any compact operator on the separa-ble Hilbert space can be represented by an infinite or a finite matrix x . . . x j ,1 . . .... . . . ... . . . x j ,1 . . . x j , j . . .... ... ... . . . ∈ K (cid:0) L ( N ) (cid:1) ,or x . . . x n ,1 ... . . . ... x n ,1 . . . x n , n ∈ K (cid:0) L ( {
1, ..., n } ) (cid:1) = M n ( C ) (1.4.6) Definition 1.4.3.
Let us consider an algebra of finite or infinite matrices given by(1.4.6). For every j , k ∈ N a matrix given by (1.4.6) is said to be j , k - elementary if x pr = δ pj δ rk .Denote by e jk the j , k -elementary matrix. We say that x = ( x , ..., x n ) ∈ C n or x = ( x , x , ... ) ∈ ℓ ( N ) is j - elementary if x k = δ jk . Denote by e j the j -elementaryvector. Lemma 1.4.4.
If we consider the situation of the Lemma D.4.16 then { p n } n ∈ N is theapproximate unit for K (cid:0) ℓ ( A ) (cid:1) .Proof. Follows from the Definition D.1.13 and the Lemma D.4.16.
Consider the situation of the Lemma D.4.16. If both A and B are positiveoperators such that A ≥ B then one has k A − Ap n k = k A ( − p n ) k = q k ( − p n ) A ( − p n ) k ≥≥ q k ( − p n ) B ( − p n ) k = k B − Bp n k so we conclude A ≥ B ⇒ k A − Ap n k ≥ k B − Bp n k (1.4.7) Lemma 1.4.6.
If K : ℓ ( A ) → ℓ ( A ) is a compact operator (in the sense of Mishchenko)then one has lim n → ∞ k p n K p n − K k =
0. (1.4.8)23 roof.
From (ii) of the Lemma D.4.16 it follows that lim n → ∞ k K p n − K k =
0, hencefrom k p n k = n → ∞ k p n ( K p n − K ) k = lim n → ∞ k p n K p n − p n K k = n → ∞ k p n K − K k = n → ∞ k p n K p n − K k = Lemma 1.4.7.
Let H ′ and H ′′ be Hilbert spaces. Ifa def = (cid:18) b cc ∗ d (cid:19) ∈ B (cid:0) H ′ ⊕ H ′′ (cid:1) + is a positive bounded operator then one has (cid:13)(cid:13)(cid:13)(cid:13) a − (cid:18) b
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k d k + q k d k k b k ≤ k d k + q k d k k a k . (1.4.9) Proof. If H def = H ′ ⊕ H ′′ , x ′ ∈ H ′ , y ′′ ∈ H ′′ , x def = ( x ′ , 0 ) ∈ H ′ ⊕ H ′′ and y def = ( y ′′ ) ∈H ′ ⊕ H ′′ then from then from the Cauchy–Schwarz inequality it follows that | ( x , ay ) H | ≤ q ( x , ax ) H q ( y , ay ) H = q ( x ′ , bx ′ ) H ′ q ( y ′′ , dy ′′ ) H ′′ . (1.4.10)On the other hand from k c k = k c ∗ k = sup k x ′ k = k y ′′ k = (cid:12)(cid:12)(cid:0) x ′ , cy ′′ (cid:1) H ′ (cid:12)(cid:12) = sup k x ′ k = k y ′′ k = (cid:12)(cid:12)(cid:0)(cid:0) x ′ , 0 (cid:1) , a (cid:0) y ′′ (cid:1)(cid:1) H (cid:12)(cid:12) ,and taking into account (1.4.10) one has k c k ≤ p k b k p k d k . Otherwise (cid:13)(cid:13)(cid:13)(cid:13) a − (cid:18) b
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k d k + k c k ≤ k d k + q k d k k b k .From k b k ≤ k a k it follows that k d k + p k d k k b k ≤ k d k + p k d k k a k . Lemma 1.4.8.
Let A, B , B be C ∗ -algebras. If φ : B ⊕ B → A is a surjective *-homomorphism then A = φ ( B ⊕ { } ) ⊕ φ ( { } ⊕ B ) .Proof. If A = φ ( B ⊕ { } ) ⊕ φ ( { } ⊕ B ) then there is a nonzero b ∈ φ ( B ⊕ ) ∩ φ ( { } ⊕ B ) . If both b ∈ B and b ∈ B are such that φ ( b ⊕ ) = φ ( ⊕ b ) = b then φ ( b ⊕ ) φ ( ⊕ b ∗ ) = bb ∗ . Otherwise from ( b ⊕ ) ( ⊕ b ∗ ) = = φ (( b ⊕ ) ( ⊕ b ∗ )) = φ ( b ⊕ ) φ ( ⊕ b ∗ ) = bb ∗ , hence k bb ∗ k = k b k =
0. However we supposed that b is not zero. The contradiction proves this lemma.24 emark 1.4.9. The Lemma 1.4.8 is an algebraical analog of that any connectedcomponent of the topological space is mapped into a connected component.
Corollary 1.4.10.
Let A, B , B be C ∗ -algebras. If φ : B ⊕ B → A is a surjective*-homomorphism and A is not a direct sum of more than one nontrivial algebras then φ ( B ⊕ { } ) = { } or φ ( { } ⊕ B ) = { } . hapter 2 Noncommutative finite-foldcoverings
Here the noncommutative generalization of the Theorem 1.1.2 is being consid-ered.
Definition 2.1.1.
Let π : A ֒ → e A be an injective unital *-homomorphism of con-nected C ∗ -algebras. Let G be group of *-automorphisms of e A such that A ∼ = e A G def = n a ∈ e A | a = ga ; ∀ g ∈ G o . We say that the triple (cid:16) A , e A , G (cid:17) and/or the quadruple (cid:16) A , e A , G , π (cid:17) and/or *-homomorphism π : A ֒ → e A is a noncommutative finite-foldquasi-covering . We write G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) def = G . (2.1.1) Lemma 2.1.2. If (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-covering then thereis an approximate unit of e A which is contained in A.Proof.
From the Theorem D.1.14 there is an approximate unit { e u λ } λ ∈ Λ of e A . Forany g ∈ G and e x ∈ e A one haslim λ k x − x ( g e u λ ) k = lim λ (cid:13)(cid:13)(cid:13) g (cid:16) g − x − (cid:16) g − x (cid:17) e u λ (cid:17)(cid:13)(cid:13)(cid:13) = { g e u λ } λ ∈ Λ is an approximate unit of e A . If v λ = | G | ∑ g e u λ ∈ A , then, ofcourse, { v λ } λ ∈ Λ is an approximate unit of e A , i.e. { v λ } is an approximate unit of e A contained in A . 27 emma 2.1.3. If (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-covering then e A isa C ∗ -Hilbert A-module with respect to the following A-valued product h a , b i A = ∑ g ∈ G g ( a ∗ b ) . (2.1.2) Proof.
Direct calculation shows that the product (2.1.2) satisfies to conditions (a)-(d) of the Definition D.4.3, Let us prove that e A is closed with respect to the norm(D.4.1). The norm k·k H of A -Hilbert pre-module e A A is given by k e a k H = vuut(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G g ( e a ∗ e a ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C where k·k C is the C ∗ -norm. From the above equation and taking into account k e a k C = p k e a ∗ e a k C it turns out k e a k C ≤ k e a k H , k e a k H ≤ q | G | k e a k C .The algebra e A is closed with respect to C ∗ -norm, hence a C ∗ -Hilbert A -premodule e A A is closed with respect to the C ∗ -Hilbert A -premodule norm. Definition 2.1.4. If (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-coveringthen product given by (2.1.2) is said to be the Hilbert product associated with thenoncommutative finite-fold quasi-covering (cid:16) A , e A , G , π (cid:17) . Definition 2.1.5.
Let π : A ֒ → e A be an injective unital *-homomorphism of con-nected C ∗ -algebras such that following conditions hold:(a) If Aut (cid:16) e A (cid:17) is a group of *-automorphisms of e A then the group G = n g ∈ Aut (cid:16) e A (cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ A o is finite.(b) A ∼ = e A G def = n a ∈ e A (cid:12)(cid:12)(cid:12) a = ga ; ∀ g ∈ G o .We say that the triple (cid:16) A , e A , G (cid:17) and/or the quadruple (cid:16) A , e A , G , π (cid:17) and/or *-homomorphism π : A ֒ → e A is a noncommutative finite-fold pre-covering .28 emark 2.1.6. Any noncommutative finite-fold pre-covering is a noncommutativefinite-fold quasi-covering.
Remark 2.1.7.
Indeed conditions (a), (b) of the Definition 2.1.5 state an isomor-phism between ordered sets of groups and C ∗ -algebras and it is an analog thefundamental theorem of Galois theory. (cf. [44] Chapter Six, § 6 for details). Let A ⊂ e A be an inclusion of C ∗ -algebras such that e A is a finitely generatedleft A -module, i.e. ∃ e a , ... e a n e A = A e a + ..., A e a n .From e A = e A ∗ it turns out that e A = e a ∗ A + ..., e a ∗ n A ,hence one has e A is a finilely generated left A -module ⇔⇔ e A is a finilely generated right A -module. (2.1.3) Definition 2.1.9.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold pre-covering(resp. quasi-covering). Suppose both A and e A are unital. We say that (cid:16) A , e A , G , π (cid:17) is an unital noncommutative finite-fold covering (resp. unital noncommutative finite-foldquasi-covering ) if e A is a finitely generated C ∗ -Hilbert left and/or right A -modulewith respect to product given by (2.1.2). Remark 2.1.10.
Above definition is motivated by the Theorem 1.1.2.
Remark 2.1.11.
From the Equation 2.1.3 it follows that the usage of left A -modulesin the Definition 2.1.9 is equivalent to the usage of right ones. Remark 2.1.12.
From the Lemma D.4.13 it turns out that e A is a projective A -module. Definition 2.1.13.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold pre-covering(resp. quasi-covering) such that following conditions hold:(a) There are unitizations A ֒ → B , e A ֒ → e B .(b) There is an unital noncommutative finite-fold quasi-covering (cid:16) B , e B , G , e π (cid:17) such that π = e π | A (or, equivalently A ∼ = e A ∩ B ) and the action G × e A → e A is induced by G × e B → e B . 29e say that the triple (cid:16) A , e A , G (cid:17) and/or the quadruple (cid:16) A , e A , G , π (cid:17) and/or *-homomorphism π : A ֒ → e A is a noncommutative finite-fold covering with unitization ,(resp. noncommutative finite-fold quasi-covering with unitization ). Remark 2.1.14.
If we consider the situation of the Definition 2.1.13 then since bothinclusions A ֒ → B , e A ֒ → e B correspond to essential ideals then one has n g ∈ Aut (cid:16) e A (cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ A o ∼ = n g ∈ Aut (cid:16) e B (cid:17) (cid:12)(cid:12)(cid:12) gb = b ; ∀ b ∈ B o ,it turns out that (cid:16) A , e A , G , π (cid:17) is a pre-covering ⇒ (cid:16) B , e B , G , e π (cid:17) is a pre-covering . (2.1.4) Remark 2.1.15.
Any noncommutative finite-fold covering with unitization is anoncommutative finite-fold quasi-covering with unitization.
Lemma 2.1.16. If (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-covering then:(i) There is the natural inclusion ϕ : M ( A ) ֒ → M (cid:16) e A (cid:17) of C ∗ -algebras.(ii) If an action of G on M (cid:16) e A (cid:17) is induced by the action of G on e A then there is thenatural *-isomorphism ψ : M (cid:16) e A (cid:17) G ∼ = M ( A ) (2.1.5) Proof. (i) From the Lemma 2.1.2 and the Proposition D.1.15 it turns out that π extends to the inclusion M ( A ) ⊂ M (cid:16) e A (cid:17) . (2.1.6)(ii) For any a ∈ M ( A ) one has g ϕ ( a ) = ϕ ( a ) ∀ g ∈ G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) ,hence ϕ ( M ( A )) ⊂ M (cid:16) e A (cid:17) G . On the other hand if a ∈ M (cid:16) e A (cid:17) G and b ∈ e A G then ab ∈ e A is such that g ( ab ) = ( ga ) ( gb ) = ab ∈ e A G . From A = e A G it follows thatthere is the following map ψ : M (cid:16) e A (cid:17) G → M ( A ) . (2.1.7)Clearly ψ ◦ ϕ = Id M ( A ) and since ϕ is injective we conclude that both ϕ and ψ areisomorphisms. 30 efinition 2.1.17. A (cid:16) A , e A , G , π (cid:17) noncommutative finite-fold pre-covering is saidto be noncommutative finite-fold covering if there is an increasing net { u λ } λ ∈ Λ ⊂ M ( A ) + of positive elements such that(a) There is the limit β - lim λ ∈ Λ u λ = M ( A ) in the strict topology of M ( A ) (cf. Definition D.1.12).(b) If for all λ ∈ Λ both A λ and e A λ are C ∗ -norm completions of u λ Au λ and u λ e Au λ respectively then for every λ ∈ Λ a quadruple (cid:16) A λ , e A λ , G , π | A λ : A λ ֒ → e A λ (cid:17) is a noncommutative finite-fold covering with unitization. The action G × e A λ → e A λ , is the restriction on e A λ ⊂ e A of the action G × e A → e A . Remark 2.1.18.
The Definition 2.1.17 implicitly uses the given by the Lemma 2.1.16inclusion M ( A ) ⊂ M (cid:16) e A (cid:17) . Remark 2.1.19. If { u λ } λ ∈ Λ = n M ( A ) o then the Definition 2.1.17 yields a noncom-mutative finite-fold covering with unitization, i.e. every noncommutative finite-fold covering with unitization is a noncommutative finite-fold covering. Remark 2.1.20.
From the Lemma D.1.17 it turn out that for all λ ∈ Λ both A λ and e A λ are hereditary subalbebras of A and e A respectively. Remark 2.1.21.
The Definition 2.1.17 is motivated by the Theorem 4.10.8.
Definition 2.1.22.
The group G is said to be the finite covering transformation group (of (cid:16) A , e A , G , π (cid:17) ) and we use the following notation G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) def = G . (2.1.8) Lemma 2.1.23.
In the situation of the Definition 2.1.17 the union ∪ λ ∈ Λ e A λ is dense in e A.Proof. If ε > e a ∈ e A there is λ ∈ Λ such that k e a − u λ e au λ k < ε (2.1.9)From u λ e au λ ∈ e A λ one concludes that ∪ λ ∈ Λ e A λ is dense in e A . Remark 2.1.24.
There are alternative theories of noncommutative coverings (e.g.[16]), however I do not know the theory which gives a good definition of thefundamental group (cf. Remark 4.10.9). 31 .2 Basic constructions
Definition 2.2.1.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold quasi-covering.If M (cid:16) e A (cid:17) is the multiplier algebra of e A then there is the natural action of G on M (cid:16) e A (cid:17) such that for any e a ∈ M (cid:16) e A (cid:17) , e b ∈ e A and g ∈ G a following conditionholds ( g e a ) e b = g (cid:16)e a (cid:16) g − e b (cid:17)(cid:17) , e b ( g e a ) = g (cid:16)(cid:16) g − e b (cid:17) e a (cid:17) , (2.2.1)We say that action of G on M (cid:16) e A (cid:17) is induced by the action of G on e A . Lemma 2.2.2.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold quasi-covering. Suppose e B ⊂ M (cid:16) e A (cid:17) is such that G e B = e B and B is a unitization of A such that B = e B G . IfG = n g ∈ Aut (cid:16) e A (cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ A oe G = n g ∈ Aut (cid:16) e B (cid:17) (cid:12)(cid:12)(cid:12) gb = b ; ∀ b ∈ B o then there is the natural injective homomorphism of groupsG ֒ → e G . Proof. If b ∈ B then b = | G | ∑ g ∈ G gb . Otherwise from e B ⊂ M (cid:16) e A (cid:17) it turns out thatit is a directed set { e a λ } ∈ e A such that there is the limit b = β -lim e a λ , so one has b = | G | ∑ g ∈ G g β - lim e a λ = | G | β - lim ∑ g ∈ G g e a λ (2.2.2)For any g ∈ G from g ∑ g ′ ∈ G g ′ e a λ = ∑ g ∈ G g ′ e a λ , and (2.2.2) one concludes gb = b ,i.e. g defines element e g ∈ e G . From e A ⊂ e B it turns out that the map g e g isinjective. Corollary 2.2.3. If (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-covering and bothA and e A are connected (cf. Definition 1.2.1) nonunital C ∗ -algebras then the inclusion π : A ֒ → e A is unital in sense of the Definition 1.2.3. roof. The given by the Lemma 2.1.16 *-homomorphism ϕ : M ( A ) ֒ → M (cid:16) e A (cid:17) is injective, it follows that ϕ (cid:16) M ( A ) (cid:17) =
0. Otherwise g M ( e A ) = M ( e A ) for any g ∈ G , it follows that 1 M ( e A ) ∈ ϕ ( M ( A )) . Since e A is connected 1 M ( e A ) is theunique central idempotent of M (cid:16) e A (cid:17) On the other hand A is connected, so 1 M ( A ) is the unique central idempotent of M ( A ) . It follows that ϕ (cid:16) M ( A ) (cid:17) = M ( e A ) , ϕ ( A + ) = e A + . Lemma 2.2.4.
If P is a finitely generated Hilbert A-module then there is a finite subset ne a , ..., e a n , e b , ..., e b n o ⊂ P such that n ∑ j = e b j ih e a j = End ( P ) A . Proof. If P is generated by e a j , ..., e a n ∈ P as a right A -module and S ∈ End ( P ) A isgiven by S = n ∑ j = e a j ih e a j (2.2.3)then S is self-adjoint. From the Corollary 1.1.25 of [41] it turns out that S is strictlypositive. Otherwise P is a finitely generated right A -module, so from the Exercise15.O of [76] it follows that S is invertible. Let R ⊂ R + be a spectrum S of andsuppose that φ : R + → R + is given by z z . If T = φ ( S ) then TS = End ( P ) A so if e b j = T e a j then n ∑ j = T e a j ih e a j = n ∑ j = e b j ih e a j = End ( P ) A . (2.2.4) Remark 2.2.5.
From the Lemma 2.2.4 it follows that if (cid:16) A , e A , G , π (cid:17) is an unitalnoncommutative finite-fold covering (cf. Definition 2.1.9) then there is a finitesubset ne a , ..., e a n , e b , ..., e b n o ⊂ e A e a = n ∑ j = e b j ∑ g ∈ G g (cid:16)e a ∗ j e a (cid:17)! ; ∀ e a ∈ e B (2.2.5) Remark 2.2.6.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold covering withunitization. If (cid:16) B , e B , G , φ (cid:17) is an unital noncommutative finite-fold covering whichsatisfies to conditions (a), (b) of the Definition 2.1.13 then there is a finite subset ne a , ..., e a n , e b , ..., e b n o ⊂ e A such that e b = n ∑ j = e b j ∑ g ∈ G g (cid:16)e a ∗ j e b (cid:17)! ; ∀ e b ∈ e B ,(cf.(2.2.5)). In particular if e a ∈ e A then e a = n ∑ j = e b j ∑ g ∈ G g (cid:16)e a ∗ j e a (cid:17)! and taking into account e a ∗ j e a ∈ e A , ∑ g ∈ G g (cid:16)e a ∗ j e a (cid:17) ∈ e A G = A one has e A = e b A + ... + e b n A (2.2.6)From (2.2.6) it follows that e A = A e b ∗ + ... + A e b ∗ n (2.2.7) Lemma 2.2.7.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold quasi-covering withunitization. If A is a separable C ∗ -algebra then e A is a separable C ∗ -algebra.Proof. Follows from (2.2.6).
Lemma 2.2.8.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold quasi-covering with uni-tization. Let (cid:16) B , e B , G , e π (cid:17) be an unital noncommutative finite-fold quasi-covering whichsatisfies to conditions (a), (b) of the Definition 2.1.13. Then there is ne b , ..., e b n o ⊂ e B suchthat M (cid:16) e A (cid:17) = e b M ( A ) + ... + e b n M ( A ) . (2.2.8)34 roof. It is known that for any e a ∈ M (cid:16) e A (cid:17) there is a directed set { e a λ } ∈ e A suchthat there is the limit e a = β -lim e a λ with respect to the strict topology (cf. DefinitionD.1.12). From the Remark 2.2.5 turns out that there is finite subset ne a , ..., e a n , e b , ..., e b n o ⊂ e B such that e b = n ∑ j = e b j ∑ g ∈ G g (cid:16)e a ∗ j e a (cid:17)! ; ∀ e b ∈ e B ,so one has e a λ = n ∑ j = e b j ∑ g ∈ G g (cid:16)e a ∗ j e a λ (cid:17)! .From the above equation it turns out e a = β - lim e a λ = β - lim n ∑ j = e b j ∑ g ∈ G g (cid:16)e a ∗ j e a λ (cid:17)! = n ∑ j = e b j ∑ g ∈ G g (cid:16)e a ∗ j β - lim e a λ (cid:17)! == n ∑ j = e b j ∑ g ∈ G g (cid:16)e a ∗ j e a (cid:17)! = n ∑ j = e b j a j ; where a j = ∑ g ∈ G g (cid:16)e a ∗ j e a (cid:17) ∈ M ( A ) .The element e a ∈ M (cid:16) e A (cid:17) is arbitrary, so one has M (cid:16) e A (cid:17) = e b M ( A ) + ... + e b n M ( A ) . Remark 2.2.9.
Any e b ∈ e B is a multiplier of e A , in particular e b , ..., e b n ∈ M (cid:16) e A (cid:17) , sofrom 2.2.8 it follows that M (cid:16) e A (cid:17) is a finitely generated M ( A ) -module. Corollary 2.2.10.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold quasi-covering withunitization (cf. Definition 2.1.13).(i) There is an unital noncommutative finite-fold quasi-covering (cid:16) M ( A ) , M (cid:16) e A (cid:17) , G , φ (cid:17) .(ii) Moreover if (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold covering with unitization(cf. Definition 2.1.13) then (cid:16) M ( A ) , M (cid:16) e A (cid:17) , G , φ (cid:17) is an unital noncommutativefinite-fold covering (cf. Definition 2.1.9). roof. From the Lemma 2.2.8 it turns out that M (cid:16) e A (cid:17) is a finitely generated M ( A ) -module.(i) The morphism φ : M ( A ) → M (cid:16) e A (cid:17) is injective, hence from the Lemma 2.1.16it turns out that M ( A ) = M (cid:16) e A (cid:17) G .(ii) Both A and e A are essential ideals of M ( A ) and M (cid:16) e A (cid:17) respectively, so one has n g ∈ Aut (cid:16) M (cid:16) e A (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ M ( A ) o == n g ∈ Aut (cid:16) e A (cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ A o = G ,hence the group n g ∈ Aut (cid:16) M (cid:16) e A (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ M ( A ) o is finite. Corollary 2.2.11.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold quasi-covering withunitization. Suppose both A and e A are separable and φ : M ( A ) ֒ → M (cid:16) e A (cid:17) is theinjective *-homomorphism given by the Lemma 2.1.16. Let p : e A → e A ′ be a surjectivemorphism of C ∗ -algebras which commutes with G; that is:g ◦ p = p ◦ g ; ∀ G . (2.2.9) Then the following conditions hold:(i) There is the natural action of G on e A ′ ;(ii) If A ′ = p ( A ) then M (cid:16) e A ′ (cid:17) is a finitely generated Hilbert M ( A ′ ) -module.Proof. From the Corollary 2.2.10 it turns out that there is an unital noncommutativefinite-fold covering (cid:16) M ( A ) , M (cid:16) e A (cid:17) , G , φ (cid:17) .(i) For any e a ′ ∈ e A ′ there is e a ∈ e A such that p ( e a ) = e a ′ , moreover if e a , e a ∈ e A aresuch that p ( e a ) = p ( e a ) = e a ′ then from (2.2.9) it turns out p ( g e a ) = p ( g e a ) forany g ∈ G . From this fact one can correctly define g e a ′ def = p ( g e a ) .(ii) From the Remark 2.2.9 it turns out that there are e a , ..., e a n ∈ M (cid:16) e A (cid:17) such that M (cid:16) e A (cid:17) = e a M ( A ) + ... + e a n M ( A ) .From the Lemma 2.2.7 it follows that e A is a separable C ∗ -algebra. From the Theo-rem D.1.34 p extends to the surjective morphism p : M (cid:16) e A (cid:17) → M (cid:16) e A ′ (cid:17) . For any36 a ′ ∈ M (cid:16) e A ′ (cid:17) there is e a ∈ M (cid:16) e A (cid:17) such that e a ′ = p ( e a ) . Otherwise e a = e a a + ... + e a n a n ; where a , ..., a n ∈ M ( A ) ,hence one has e a ′ = e a ′ a ′ + ... + e a ′ n a ′ n ; where e a ′ = p ( e a ) , a ′ j = p (cid:0) a j (cid:1) ∈ M (cid:0) A ′ (cid:1) , e a ′ j = p (cid:0)e a j (cid:1) , for all j =
1, ..., n .Since e a ′ is arbitrary element of M (cid:16) e A ′ (cid:17) one has M (cid:16) e A ′ (cid:17) = e a ′ M (cid:0) A ′ (cid:1) + ... + e a ′ M (cid:0) A ′ (cid:1) ,i.e. M (cid:16) e A ′ (cid:17) is a finitely generated M ( A ′ ) -module. Any e a ′ , e b ′ ∈ M (cid:16) e A ′ (cid:17) can berepresented as strict limits e a ′ = β - lim e a ′ λ ; e a ′ λ ∈ e A ′ , e b ′ = β - lim e b ′ λ ; e b ′ λ ∈ e A ′ ,hence from ∑ g ∈ G g (cid:16)e a ′∗ λ e b ′ λ (cid:17) ∈ A ′ one can define M ( A ′ ) -valued product De a ′ , e b ′ E M ( A ′ ) = β - lim ∑ g ∈ G g (cid:16)e a ′∗ λ e b ′ λ (cid:17) . (2.2.10)and similarly to the Lemma 2.1.3 one can prove that M (cid:16) e A ′ (cid:17) is complete withrespect to the topology induced by the product (2.2.10), i.e. M (cid:16) e A ′ (cid:17) is a Hilbert M ( A ′ ) -module. If A ⊂ e A is an inclusion of C ∗ -algebras such that there is an approximateunit for e A containing in A then from the Proposition D.1.15 it turns out that theinclusion extends to an *-homomorphism M ( A ) → M (cid:16) e A (cid:17) . If p : e A → e A ′ issurjective morphism of C ∗ -algebras and { e u λ } is an approximate unit for e A then p ( { e u λ } ) is an approximate unit for e A ′ . From these circumstances it turns out ifapproximate unit for e A containing in A then there is approximate unit for e A ′ con-taining in A ′ = p ( A ) , hence the inclusion A ′ ⊂ e A ′ extends to a *-homomorphism M ( A ′ ) → M (cid:16) e A ′ (cid:17) . In particular from the proof of the Lemma 2.1.16 it turns outthat if inclusion A ⊂ e A is such there is (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-covering, then there is the natural *-homomorphism M ( A ′ ) → M (cid:16) e A ′ (cid:17) .37 emma 2.2.13. Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold quasi-covering withunitization (cf. Definitions 2.1.1, 2.1.9), such that A is separable. Let p : e A → e A ′ asurjective morphisms of C ∗ -algebras which commutes with G; that is:g ◦ p = p ◦ g ; ∀ g ∈ G . (2.2.11) If A ′ = p ( A ) and the natural *-homomorphism M ( A ′ ) → M (cid:16) e A ′ (cid:17) is injective, and e A ′ is connected then following conditions hold:(i) There is an unital noncommutative finite-fold quasi-covering (cid:16) M (cid:0) A ′ (cid:1) , M (cid:16) e A ′ (cid:17) , G , e π (cid:17) . (ii) There is a noncommutative finite-fold quasi-covering with unitization (cid:16) A ′ , e A ′ , G , e π | A ′ (cid:17) . Remark 2.2.14.
From the Lemma 2.2.7 it follows that e A is a separable C ∗ -algebra.From (2.2.11) it follows that there is correctly defined action G × e A ′ → e A ′ givenby gp ( e a ) = p ( g e a ) for any g ∈ G . This action follows from the equation (2.2.11). Proof. (of the Lemma 2.2.13) . Firstly we prove that both quadruples (cid:16) A ′ , e A ′ , G , e π | A ′ (cid:17) and (cid:16) M ( A ′ ) , M (cid:16) e A ′ (cid:17) , G , e π (cid:17) are quasi-coverings, i.e. both satisfy to the Definition2.1.1. If a ′ ∈ A ′ then there is a ∈ A such that a ′ = p ( a ) hence one has ga ′ = p ( ga ) = p ( a ) = a ′ , i.e. a ∈ e A ′ G . Conversely if a ′ ∈ e A ′ G then a ′ = | G | ∑ g ∈ G ga ′ and if a ′ = p ( e a ) then a ′ = | G | p ∑ g ∈ G g e a ! and taking into account ∑ g ∈ G g e a ∈ A one has a ∈ p ( A ) = A ′ . In result we con-clude that A ′ = e A ′ G . Otherwise M ( A ′ ) → M (cid:16) e A ′ (cid:17) is an injective *-homomorphism,so from the Lemma 2.1.16 it turns out M ( A ′ ) ∼ = M (cid:16) e A ′ (cid:17) G .(i) One should prove that M (cid:16) e A ′ (cid:17) is a finitely generated Hilbert module over38 ( A ′ ) . The application of the equation (2.1.2) and the Definition 2.1.4 providesthe structure of Hilbert M ( A ′ ) -module on M (cid:16) e A ′ (cid:17) . From the Remark 2.2.9 it turnsout that M (cid:16) e A (cid:17) is a finitely generated C ∗ -Hilbert module over M ( A ) . Since thereare e b , ..., e b n ∈ M (cid:16) e A (cid:17) such that M (cid:16) e A (cid:17) = e b M ( A ) + ... + e b n M ( A ) the algebra e A is separable. From the Theorem D.1.34 it turns out that there isthe extension of p up to surjective morphism p : M (cid:16) e A (cid:17) → M (cid:16) e A ′ (cid:17) . For any e a ′ ∈ M (cid:16) e A ′ (cid:17) there is e a ∈ M (cid:16) e A (cid:17) such that e a ′ = p ( e a ) . If e a = e b a + ... + e b n a n where a , ..., a n ∈ A then e a ′ = e b ′ a ′ + ... + e b ′ n a ′ n where a ′ , ..., a ′ n ∈ A ′ ; e b ′ j = p (cid:16)e b j (cid:17) ; a ′ j = p (cid:0) a j (cid:1) ; ∀ j =
1, ..., n .So one has M (cid:16) e A ′ (cid:17) = e b ′ M (cid:0) A ′ (cid:1) + ... + e b ′ n M (cid:0) A ′ (cid:1) i.e. M (cid:16) e A ′ (cid:17) is the finitely generated module over M ( A ′ ) . It turns out that (cid:16) M (cid:0) A ′ (cid:1) , M (cid:16) e A ′ (cid:17) , G , e π (cid:17) is an unital noncommutative finite-fold quasi-covering (cf. Definition 2.1.9).(ii) Both A ′ and e A ′ are essential ideals of M ( A ′ ) and M (cid:16) e A ′ (cid:17) respectively it turnsout that (cid:16) A ′ , e A ′ , G , e π | A ′ (cid:17) is a noncommutative finite-fold quasi-covering with uni-tization (cf. Definition 2.1.13) Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold pre-covering (cf. Defini-tion 2.1.5). Let B ⊂ A a connected C ∗ -subalgebra such that there is C ∗ -subalgebra e B ⊂ e A which satisfies to the following conditions • B is not a direct sum of more then one C ∗ -algebras. • e B ∪ g e B = { } for all nontrivial g ∈ G .39 π ( B ) ⊂ M g ∈ G g e B (2.2.12) • If p e B : L g ∈ G g e B → e B is the natural projection p e B ◦ π | B : B ≈ −→ e B is *-isomorprhism. Definition 2.2.16.
In the described in 2.2.15 situation we say that the Equation(2.2.12) a direct sum decomposition of (cid:16) A , e A , G , π (cid:17) . Lemma 2.2.17.
Let (cid:16) A , e A , G , π (cid:17) , (cid:16) A , e A , G , π (cid:17) be finite-fold noncommutative pre-coveings. Let B ⊂ A be a C ∗ subalgebras such there are e B ⊂ e A , e B ⊂ e A and directsummand decompositions (cf. Definition 2.2.16). π ( B ) ⊂ M g ∈ G g e B , π ( B ) ⊂ M g ∈ G g e B . (2.2.13) If there is a injective *- homomorphism π : e A ֒ → e A such that π = π ◦ π then therethe following conditions hold: • If p e : L g ∈ G g e B → e B is the natural projection then there is the unique g ′ ∈ G such that p e ◦ π (cid:16) g ′ e B (cid:17) = { } . • If e B ′ = g ′ e B then there is the surjecive homomorphism φ : G → G such that π (cid:16) g e B ′ (cid:17) ⊂ M g ∈ G φ ( g )= g g e B ′ Proof.
From the Corollary 1.4.8 it follows that there is the unique g ′ ∈ G such that p e ◦ π (cid:16) g ′ e B (cid:17) = { } .If g ∈ G is any element e B ′ = g ′ e B and p g : L g ′ ∈ G g ′ e B → g e B is the naturalprojection then from the Corollary 1.4.8 it follows that the unique g ∈ G suchthat p g ◦ π (cid:16) g e B ′ (cid:17) = { } . So there is the map φ : G → G , g = φ ( g ) ⇔ p g ◦ π (cid:16) g e B ′ (cid:17) = { } . (2.2.14)40enote by H = n h ∈ G | p h π (cid:16) e B ′ (cid:17) = { } o ⊂ G , and B ′ = ⊕ h ∈ H h B . Any g ∈ G yields the ∗ -isomorphism B ′ → g B ′ with the inverse isomorphism g − : gB ′ → B ′ . So for any h ∈ H one has g h g − ∈ H , i.e. H ⊂ G is a normalsubgroup. If g ∈ G and h ∈ H then one has p h π (cid:16) e B ′ (cid:17) = { } ⇒ p g h g π (cid:16) e B ′ (cid:17) = { } AND p g g π (cid:16) e B ′ (cid:17) = { }⇒ φ ( g h ) = φ ( g ) ,i.e. φ is equivalent to the surjective homomorphism of G onto its factorgroup G → G / H . Definition 2.3.1.
Let (cid:16) A , e A , G , π (cid:17) is a finite-fold noncommutative covering suchthat e A A def = e A is of the C ∗ -Hilbert A -module (cf. 2.1.3, 2.1.4). If ρ : A → B ( H ) isa representation and e ρ = e A − Ind A e A ρ : e A → B (cid:16) e H (cid:17) is given by (D.5.3), i.e. e ρ is theinduced representation (cf. Definition D.5.5) then we say that e ρ is induced by thepair (cid:16) ρ , (cid:16) A , e A , G , π (cid:17)(cid:17) . If X = e A ⊗ A H is the algebraic tensor product then according to (D.5.4) thereis a sesquilinear C -valued product ( · , · ) X on X given by (cid:16)e a ⊗ ξ , e b ⊗ η (cid:17) X = (cid:16) ξ , De a , e b E A η (cid:17) H (2.3.1)where ( · , · ) H means the Hilbert space product on H , and h· , ·i e A is given by (2.1.4).So X is a pre-Hilbert space. There is a natural map e A × (cid:16) e A ⊗ A H (cid:17) → e A ⊗ A H given by e A × (cid:16) e A ⊗ A H (cid:17) → e A ⊗ A H , (cid:16)e a , e b ⊗ ξ (cid:17) e a e b ⊗ ξ . (2.3.2)The space e H of the representation e ρ = e A − Ind A e A ρ : e A → B (cid:16) e H (cid:17) is the Hilbertnorm completion of the pre-Hilbert space X = e A ⊗ A H and the action of e A on e H is the completion of the action on e A ⊗ A H given by (2.3.2).41 emma 2.3.3. If A → B ( H ) is faithful then e ρ : e A → B (cid:16) e H (cid:17) is faithful.Proof. If e a ∈ e A is a nonzero element then a = h e a e a ∗ , e a e a ∗ i A = ∑ g ∈ G g ( e a ∗ e a e a e a ∗ ) ∈ A is a nonzero positive element. There is ξ ∈ H such that ( a ξ , ξ ) H >
0. However ( a ξ , ξ ) H = ( ξ , h e a e a ∗ , e a e a ∗ i A ξ ) H = ( e a e a ∗ ⊗ ξ , e a e a ∗ ⊗ ξ ) e H = (cid:16)e a e ξ , e a e ξ (cid:17) e H > e ξ = e a ∗ ⊗ ξ ∈ e A ⊗ A H ⊂ e H . Hence e a e ξ = Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold covering, let ρ : A → B ( H ) be a faithful non-degenerated representation, and let e ρ : e A → B (cid:16) e H (cid:17) is inducedby the pair (cid:16) ρ , (cid:16) A , e A , G , π (cid:17)(cid:17) . There is the natural action of G on e H induced bythe map g ( e a ⊗ ξ ) = ( g e a ) ⊗ ξ ; e a ∈ e A , g ∈ G , ξ ∈ H . (2.3.3)There is the natural orthogonal inclusion H ֒ → e H (2.3.4)induced by inclusions A ⊂ e A ; A ⊗ A H ⊂ e A ⊗ A H .If e A is an unital C ∗ -algebra then the inclusion (2.3.4) is given by ϕ : H ֒ → e H , ξ e A ⊗ ξ (2.3.5)where 1 e A ⊗ ξ ∈ e A ⊗ A H is regarded as element of e H . The inclusion (2.3.5) is notisometric. From (cid:10) e A , 1 e A (cid:11) = ∑ g ∈ G ( e A | A ) g e A = (cid:12)(cid:12)(cid:12) G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17)(cid:12)(cid:12)(cid:12) A it turns out ( ξ , η ) H = (cid:12)(cid:12)(cid:12) G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17)(cid:12)(cid:12)(cid:12) ( ϕ ( ξ ) , ϕ ( η )) e H ; ∀ ξ , η ∈ H (2.3.6)42ction of g ∈ G (cid:16) e A | A (cid:17) on e A can be defined by representation as g e a = g e ag − ,i.e. ( g e a ) ξ = g (cid:16)e a (cid:16) g − ξ (cid:17)(cid:17) ; ∀ ξ ∈ e H . Definition 2.3.5.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold quasi-covering.An let φ be a positive functional on A . Any positive functional ψ : e A → C is saidto be a lift of φ if k ψ k = k φ k and φ = ψ | A . We say that the lift ψ is G - invariant if ψ ( e a ) = ψ ( g e a ) for each g ∈ G and e a ∈ e A . Lemma 2.3.6. If (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-covering then thereis the one-to-one correspondence between positive functionals on A and their G-invariantlifts.Proof. For any positive functional φ : A → C there is a G -invariant lift ψ : e A → C given by e a | G | φ ∑ g ∈ G g e a ! . (2.3.7)If ψ ′ , ψ ′′ are two G -invariant lifts of φ then for any e a ∈ e A one has ψ ′ ( e a ) = | G | ∑ g ∈ G ψ ′ ( g e a ) = | G | φ ∑ g ∈ G g e a ! = | G | ∑ g ∈ G ψ ′′ ( g e a ) = ψ ′′ ( e a ) ,i.e. ψ ′ = ψ ′′ . Lemma 2.3.7. If (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-covering then fol-lowing conditions hold:(i) If e X = e A ∧ is the spectrum of e A then there is the natural action G × e X → e X ,(ii) If X = ˆ A is the spectrum of A then there is the natural surjective continuous mapp : e X → X ; rep e A e x rep e A e x (cid:12)(cid:12)(cid:12) A ∼ = rep Ap ( e x ) (2.3.8) (cf. Proposition D.2.25, and (D.2.3) ), which is G-invariant, i.e.p ◦ g = p ; ∀ g ∈ G , (2.3.9) (iii) The map p naturally induces the homeomorphism p G : e X / G ≈ −→ X . roof. (i) If e ρ : e A → B (cid:16) e H (cid:17) is an irreducible representation then for any g ∈ G there is the irreducible representation e ρ g : e A → B (cid:16) e H (cid:17) , e ρ g ( e a ) = e ρ ( g e a ) ; ∀ e a ∈ e A .The map e ρ e ρ g gives a homeomorphism of e X . Such homeomorphisms give anaction G × e X → e X , such that g e ρ = e ρ g for any g ∈ G .(ii) Let e τ : e A → C be a pure state on e A , and let τ def = e τ | A A → C be the restriction. If e a ∈ e A + is a positive element such that e τ ( e a ) = a = ∑ g ∈ G g e a ∈ A is a positiveelement such that e τ ( a ) =
0. It turns out that the restriction τ def = e τ | A : A → C is a state. Let p : A → C be a positive functional such that p < τ . From theHahn–Banach theorem it follows that there is a positive functional e p : e A → C such e p < e τ . Since e τ is there is t ∈ [
0, 1 ] such that e p = t e τ . One has the G invariant state e τ ⊕ : e A → C , e a | G | ∑ g ∈ G e τ If e p ⊕ : e A → C , e a | G | p ∑ g ∈ G e τ ! then from e p ⊕ = t e τ ⊕ , p = e p ⊕ | A and τ = e τ ⊕ | A it follows that p = t τ . So the state τ is pure, and the map e τ τ yields the given by (2.3.8) map. Let us prove that themap is continuous. From the Theorem D.2.17 any closed set U ⊂ X correspondsto a closed two-sided ideal I ⊂ A such that U = { ρ ∈ X | ρ ( I ) = { }} .The set p − ( U ) corresponds to the closed two-sided ideal e I I = \ e x ∈ e X rep e x ( I )= ker rep e x , (2.3.10)44o the preimage of U , is closed, it turns out that p is continuous. From the Propo-sition D.2.25 it turns out that p is a surjective map. From the condition (b) of theDefinition 2.1.5 it turns out that ga = a for any a ∈ A and g ∈ G , so one has ( g e ρ ) ( a ) = e ρ ( a ) for any e ρ ∈ e X . It follows that p ◦ g = p ; ∀ g ∈ G . From theLemma 2.3.6 it turns out that p is a bijective map.(iii) Any pure state φ : A → C uniquely defines the given by 2.3.7 invariant state e φ . Otherwise e φ uniquely depends on the G -orbit of any pure state e ψ : f → C suchthat e ψ (cid:12)(cid:12) A = φ . So the map p G : e X / G ≈ −→ X is bijective. Open subsets of e X / G correspond to open subsets e U ⊂ e X such that G e U = e U . Otherwise such opensubset corresponds to the closed two-sided G -invariant ideal e I , i.e. G e I = e I . If I def = A ∩ π − (cid:16)e I (cid:17) then e I is the generated by π ( I ) ideal. There are mutually in-verse maps between closed two sided ideals of A an closed two-sided G -invariantideals e A given by I the generated by π ( I ) ideal, e I A ∩ π − (cid:16)e I (cid:17) This maps establishes the one to one correspondence between open subsets of both e X / G and X , i.e. p G : e X / G ≈ −→ X is a homeomorphism. Remark 2.3.8.
From the equation (2.3.8) it turns out that there is the natural iso-morphism p e A e x : rep e A e x ( A ) ∼ = −→ rep Ap ( e x ) ( A ) , rep e A e x ( a ) rep Ap ( e x ) ( a ) ; ∀ a ∈ A . (2.3.11) Corollary 2.3.9.
Let (cid:16) A , e A , G , π (cid:17) be a pure noncommutative finite-fold quasi-covering(cf. Definition 2.1.13). If the spectra X and e X of both A and e A respectively then for anyn ∈ N the set X n = n x ∈ X | (cid:12)(cid:12)(cid:12) p − ( x ) (cid:12)(cid:12)(cid:12) ≥ n o is an open subset of X .Proof. If x ∈ X n then there is m ≥ n such that (cid:12)(cid:12) p − ( x ) (cid:12)(cid:12) = m . There is a subset e x ∈ p − ( x ) and the subset { g , ..., g m } such that ∀ j , k ∈ { g , ..., g m } j = k ⇒ g j e x = g k e x e X is Hausdorff there is an open neighborhood e U of e x such that ∀ x ∈ e U ∀ j , k ∈ { g , ..., g m } j = k ⇒ g j e x = g k e x .So one has ∀ e x ∈ [ g ∈ G g e U (cid:12)(cid:12)(cid:12) p − ( x ) (cid:12)(cid:12)(cid:12) ≥ m .From the Lemma 2.3.7 it turns out that there is an open subset U ⊂ X such that p − ( U ) = S g ∈ G g e U , so one has ∀ x ∈ U (cid:12)(cid:12)(cid:12) p − ( p ( x )) (cid:12)(cid:12)(cid:12) ≥ m . Suppose that If (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-coveringand A is separable. There is an isomorphism e A ≈ A L P of A -modules then any e a ∈ End (cid:16) e A (cid:17) A can be represented as a sum e a = α ( e a ) + β ( e a ) + γ ( e a ) + δ ( e a ) of A -module homomorphisms α : A → A , β : P → A , γ : A → P , δ : P → P . If M ( e a ) = (cid:18) α ( e a ) β ( e a ) γ ( e a ) δ ( e a ) (cid:19) (2.3.12)then M (cid:0)e a ′ e a ′′ (cid:1) = M (cid:0)e a ′ (cid:1) M (cid:0)e a ′′ (cid:1) . If e ρ : e A → B (cid:16) e H (cid:17) an irreducible representation then e H = ρ (cid:16) e A (cid:17) ξ . (2.3.13)If H def = e ρ ( α ( A )) ξ . (2.3.14)then from the Lemma 2.3.7 one has the irreducible representation ρ : A → B ( H ) ; ρ ( a ) (cid:0)e ρ (cid:0) a ′ (cid:1) ξ (cid:1) def = e ρ (cid:0) aa ′ (cid:1) ξ ∀ a , a ′ ∈ A . (2.3.15)46f p H : e H → e H is the projector onto H then ∀ e a ∈ e A e ρ ( α ( e a )) = p H e ρ ( e a ) p H , ρ ( β ( e a )) = p H e ρ ( e a ) ( − p H ) , e ρ ( γ ( e a )) = ( − p H ) e ρ ( e a ) p H , ρ ( δ ( e a )) = ( − p H ) e ρ ( e a ) ( − p H ) , e ρ ( e a ) = (cid:18) e ρ ( α ( e a )) e ρ ( β ( e a )) e ρ ( γ ( e a )) e ρ ( δ ( e a )) (cid:19) (2.3.16) Definition 2.3.12.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold quasi-covering.The minimal hereditary subalgebra of e A which contains π ( A ) , is said to be the reduced algebra of (cid:16) A , e A , G , π (cid:17) . The reduced algebra is denoted by e A red . Remark 2.3.13.
Clearly one has an inclusion A ⊂ e A red (2.3.17) Remark 2.3.14. If e A ′ is a hereditary subalgebra of e A which contains A then for all g e A ′ a hereditary subalgebra of e A which contains A . If follows that G e A red = e A red .Hence there is the natural action G × e A red → G × e A red (2.3.18)Moreover from e A G = A and A ⊂ e A red it follows that e A G red = A . (2.3.19) Remark 2.3.15.
For any e x ∈ e X one has rep e x (cid:16) e A red (cid:17) = rep e x ( A ) (2.3.20)Really if rep e x (cid:16) e A red (cid:17) $ rep e x ( A ) then e A ′ def = n e a ′ ∈ e A red (cid:12)(cid:12)(cid:12) rep e x (cid:0)e a ′ (cid:1) ∈ rep e x ( A ) o is a hereditary algebra of e A which contains π ( A ) and e A ′ $ e A red .47 emark 2.3.16. From the Lemma 2.3.7 and the Proposition D.2.20 it follows thatthe spectrum of e A red coincides with the spectrum of e A as a topological space. If e X is the spectrum of both e A red and e A then the inclusion G ֒ → Aut (cid:16) e A (cid:17) induces ahomomorphism h : G → Homeo (cid:16) e X (cid:17) .If G red def = n g ∈ Aut (cid:16) e A (cid:17) (cid:12)(cid:12)(cid:12) g e a red = e a red ; ∀ e a red ∈ e A red o ⊂ G (2.3.21)then from the coincidence of spectra of e A and e A red it follows that G red = ker (cid:16) G → Homeo (cid:16) e X (cid:17)(cid:17) . (2.3.22)In particular G red is normal subgroup of G and there is the inclusion G / G red ⊂ Homeo (cid:16) e X (cid:17) . (2.3.23)Moreover from the Lemma 2.3.7 it follows that there is a homeomorphism X ∼ = e X / ( G / G red ) . (2.3.24) The following result about algebraic Morita equivalence of Galois extensions isin fact rephrasing of described in [4, 15] constructions. Let (cid:16) A , e A , G (cid:17) be an unitalfinite-fold noncommutative covering. Denote by e A ⋊ G a crossed product , i.e. e A ⋊ G ia a C ∗ -algebra which coincides with a set of maps from G to e A as a set, andoperations on e A are given by ( a + b ) ( g ) = a ( g ) + b ( g ) , ( a · b ) ( g ) = ∑ g ′ ∈ G a (cid:0) g ′ (cid:1) (cid:16) g ′ (cid:16) b (cid:16) g ′− g (cid:17)(cid:17)(cid:17) ∀ a , b ∈ e A ⋊ G , ∀ g ∈ G , a ∗ ( g ) = (cid:16) a (cid:16) g − (cid:17)(cid:17) ∗ ; ∀ a ∈ e A ⋊ G , ∀ g ∈ G . (2.4.1)Let us construct a Morita context (cid:16) e A ⋊ G , A , e A ⋊ G e A A , A e A e A ⋊ G , ϕ , ψ (cid:17) A e A e A ⋊ G and e A ⋊ G e A A coincide with e A as C -spaces. Left and right actionof e A ⋊ G on e A is given by a e a = ∑ g ∈ G a ( g ) ( g e a ) , e aa = ∑ g ∈ G g − ( e aa ( g )) , ∀ a ∈ e A ⋊ G , ∀ e a ∈ e A .Left (resp. right) action of A on e A we define as left (resp. right) multiplication by A . Denote by ϕ : e A ⊗ A e A → e A ⋊ G , ψ : e A ⊗ e A ⋊ G e A → A the maps such that ϕ (cid:16)e a ⊗ e b (cid:17) ( g ) = e a (cid:16) g e b (cid:17) , ψ (cid:16)e a ⊗ e b (cid:17) = ∑ g ∈ G g (cid:16)e a e b (cid:17) , ∀ e a , e b ∈ e A , g ∈ G .From above equations it follows that ϕ (cid:16)e a ⊗ e b (cid:17) e c = ∑ g ∈ G (cid:16) ϕ (cid:16)e a ⊗ e b (cid:17) ( g ) (cid:17) g e c == ∑ g ∈G e a (cid:16) g e b (cid:17) ( g e c ) = e a ∑ g ∈ G g (cid:16)e b e c (cid:17) = e a ψ (cid:16)e b ⊗ e c (cid:17) , e a ϕ (cid:16)e b ⊗ e c (cid:17) = ∑ g ∈ G g − (cid:16)e a e bg e c (cid:17) == ∑ g ∈ G g − (cid:16)e a e b (cid:17)! e c = ∑ g ∈ G g (cid:16)e a e b (cid:17)! e c = ψ (cid:16)e a ⊗ e b (cid:17) e c , (2.4.2)i.e. ϕ , ψ satisfy conditions (B.1.1), so (cid:16) e A ⋊ G , A , e A ⋊ G e A A , A e A e A ⋊ G , ϕ , ψ (cid:17) is a Moritacontext. Taking into account that the A -module e A A is a finitely generated projec-tive generator and Remark B.1.2 one has a following lemma. Lemma 2.4.1. If (cid:16) A , e A , G (cid:17) is an unital noncommutative finite-fold covering then (cid:16) e A ⋊ G , A , e A ⋊ G e A A , A e A e A ⋊ G , ϕ , ψ (cid:17) is an algebraic Morita equivalence. orollary 2.4.2. Let (cid:16) A , e A , G , π (cid:17) be an unital noncommutative finite-fold covering. Letus define a structure of Hilbert e A ⋊ G − A bimodule on e A ⋊ G e A A given by following prod-ucts h a , b i e A ⋊ G = ϕ ( a ⊗ b ∗ ) , h a , b i A = ψ ( a ∗ ⊗ b ) . (2.4.3) Following conditions hold:(i) A bimodule e A ⋊ G e A A satisfies the associativity condition (a) of the Definition D.6.1,(ii) D e A , e A E e A ⋊ G = e A ⋊ G , D e A , e A E A = A . It follows that e A ⋊ G e A A is a e A ⋊ G − A equivalence bimodule.Proof. (i) From (2.4.2) it follows that products h− , −i e A ⋊ G , h− , −i A satisfy condi-tion (a) of the Definition D.6.1.(ii) From the Lemma 2.4.1 and the definition of algebraic Morita equivalence itturns out ϕ (cid:16) e A ⊗ A e A (cid:17) = D e A , e A E e A ⋊ G = e A ⋊ G , ψ (cid:16) e A ⊗ e A ⋊ G e A (cid:17) = D e A , e A E A = A .Let us consider the situation of the Corollary 2.4.2. Denote by e ∈ G the neutralelement. The unity 1 e A ⋊ G of e A ⋊ G is given by1 e A ⋊ G ( g ) = (cid:26) e A g = e g = e . (2.4.4)From the Lemma 2.4.1 it follows that there are e a , ..., e a n , e b , ..., e b n ∈ e A such that1 e A ⋊ G = ϕ n ∑ j = e a j ⊗ e b ∗ j ! = n ∑ j = De a j , e b j E e A ⋊ G . (2.4.5)From the above equation it turns out that for any g ∈ G ϕ n ∑ j = g e a j ⊗ e b ∗ j ! (cid:0) g ′ (cid:1) = n ∑ j = D g e a j , e b j E e A ⋊ G ! (cid:0) g ′ (cid:1) (cid:26) e A g ′ = g g ′ = g . (2.4.6)50 orollary 2.4.3. (cid:16) A , e A , G , π (cid:17) be an unital noncommutative finite-fold covering. Thegiven by D.5.4 functor A e A e A ⋊ G ⊗ e A ⋊ G ( − ) : Herm e A ⋊ G → Herm A is equivalent to thefunctor of invariant submodule ( − ) G : Herm e A ⋊ G → Herm A ; e H 7→ e H G = n e ξ ∈ e H | g e ξ = e ξ , ∀ g ∈ G o . Proof.
H ∼ = A e A A ⋊ G ⊗ A ⋊ G e H Otherwise any e ξ ∈ A e A A ⋊ G ⊗ A ⋊ G e H equals to 1 e A ⊗ e ξ ′ . From g e A ⊗ e ξ ′ = e A ⊗ g e ξ ′ and g e A = e A for any g ∈ G it turns out that g e ξ ′ = e ξ ′ i.e. e ξ ′ is G -invariant. Lemma 2.4.4.
Let (cid:16) A , e A , G (cid:17) be a noncommutative finite-fold quasi-covering with uniti-zation (cf. Definition 2.1.13). Let us define a structure of Hilbert e A ⋊ G − A bimodule on e A ⋊ G e A A given by products (2.4.3) Following conditions hold:(i) A bimodule e A ⋊ G e A A satisfies associativity conditions (a) of the Definition D.6.1(ii) D e A , e A E A = A , D e A , e A E e A ⋊ G = e A ⋊ G . It follows that e A ⋊ G e A A is a e A ⋊ G − A equivalence bimodule.Proof. (i) From the Definition 2.1.13 it follows that there are unital C ∗ -algebras B , e B and inclusions A ⊂ B , e A ⊂ e B such that A (resp. B ) is an essential ideal of e A (resp. e B ). Moreover there is an unital noncommutative finite-fold covering (cid:16) B , e B , G (cid:17) .From the Corollary 2.4.2 it turns out that a bimodule e B ⋊ G e B B is a e B ⋊ G − B equiv-alence bimodule. Both scalar products h− , −i e A ⋊ G , h− , −i A are restrictions ofproducts h− , −i e B ⋊ G , h− , −i B , so products h− , −i e A ⋊ G , h− , −i A satisfy to condi-tion (a) of the Definition D.6.1.(ii) From (2.4.5) it turns out that there are e a , ..., e a n , e b , ..., e b n ∈ e B such that1 e B ⋊ G = ϕ n ∑ j = e a j ⊗ e b ∗ j ! = n ∑ j = De a j , e b j E e B ⋊ G .51f a ∈ A + is a positive element then there is x ∈ A ⊂ e A such that a = x ∗ x itfollows that a = | G | h x , x i A .Otherwise A is the C -linear span of positive elements it turns out D e A , e A E A = A .For any positive e a ∈ e A + and any g ∈ G denote by y e ag ∈ e A ⋊ G given by y e ag (cid:0) g ′ (cid:1) (cid:26) e a g ′ = g g ′ = g .There is e x ∈ e A such that e x e x ∗ = e a . Clearly e x (cid:0) g e a j (cid:1) , e x e b j ∈ e A and from (2.4.6) itfollows that y e ag = n ∑ j = D x (cid:0) g e a j (cid:1) , x e b j E e A ⋊ G The algebra e A ⋊ G is the C -linear span of elements y e ag , so one has D e A , e A E e A ⋊ G = e A ⋊ G . Theorem 2.4.5. If (cid:16) A , e A , G (cid:17) is a noncommutative finite-fold covering then a Hilbert (cid:16) e A ⋊ G , A (cid:17) bimodule e A ⋊ G e A A is a e A ⋊ G − A equivalence bimodule.Proof.
From the Definition 2.1.17 there is a family ne I λ ⊂ e A o λ ∈ Λ of closed ideals of A such that S λ ∈ Λ e I λ is a dense subset of A and for any λ ∈ Λ there is a naturalnoncommutative finite-fold covering with unitization (cid:16)e I λ T A , e I λ , G (cid:17) . From theLemma 2.4.4 it turns out that e I λ is a e I λ ⋊ G − e I λ T A equivalence bimodule and De I λ , e I λ E e I λ ⋊ G = e I λ ⋊ G , De I λ , e I λ E e I λ T A = e I λ \ A . (2.4.7)for any λ ∈ Λ . The union S λ ∈ Λ e I λ is a dense subset of e A and S λ ∈ Λ e I λ T A is a densesubset of A . So domains of products h− , −i e I λ ⋊ G , h− , −i I λ can be extended upto e A × e A and resulting products satisfy to (a) of the Definition D.6.1. From (2.4.7)it turns out that D e A , e A E e A ⋊ G (resp. D e A , e A E A ) is a dense subset of e A ⋊ G (resp. A ),i.e. the condition (b) of the Definition D.6.1 holds.52 .5 Noncommutative unique path lifting Here we generalize path-lifting property given by the Definition A.2.11. Becausenoncommutative geometry has no points it has no a direct generalization of paths,but there is an implicit generalization. Let e X → X be a covering. The followingdiagram reflects the path lifting problem. e X I = [
0, 1 ] X f pf ′ However above diagram can be replaced with an equivalent diagramHomeo ( e X ) I = [
0, 1 ] Homeo ( X ) f α α | X f ′ where Homeo means the group of homeomorphisms with compact-open topology(See [70]). Above diagram has the following noncommutative generalizationAut ( e A ) I = [
0, 1 ] Aut ( A ) f α α | A f ′ In the above diagram we require that α | A ∈ Aut ( A ) for any α ∈ Aut (cid:16) e A (cid:17) . Thediagram means that f ′ ( t ) | A = f ( t ) for any t ∈ [
0, 1 ] . Noncommutative generaliza-tion of a locally compact space is a C ∗ -algebra, so the generalization of Homeo ( X ) is the group Aut ( A ) of *-automorphisms carries (at least) two different topologiesmaking it into a topological group [72]. The most important is the topology ofpointwise norm-convergence based on the open sets { α ∈ Aut ( A ) | k α ( a ) − a k < } , a ∈ A .53he other topology is the uniform norm-topology based on the open sets ( α ∈ Aut ( A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup a = k a k − k α ( a ) − a k < ε ) , ε > k α k Aut = sup a = k a k − k α ( a ) − a k = sup k a k = k α ( a ) − a k . (2.5.2)Above formula does not really means a norm because Aut ( A ) is not a vectorspace. Henceforth the uniform norm-topology will be considered only. Definition 2.5.1.
Let A ֒ → e A be an inclusion of C ∗ -algebras. Let f : [
0, 1 ] → Aut ( A ) be a continuous function such that f ( ) = Id A . If there is a continuousmap e f : [
0, 1 ] → Aut (cid:16) e A (cid:17) such that e f ( t ) (cid:12)(cid:12)(cid:12) A = f ( t ) for any t ∈ [
0, 1 ] and e f ( ) = Id e A then we say that e f is a π - lift of f . If a lift e f of f is unique then and map e f : [
0, 1 ] → Aut (cid:16) e A (cid:17) is said to be the unique π -lift of f . If e f is the unique π -lift of f then wedenote by lift f def = e f ( ) . (2.5.3) Let π : A ֒ → e A be an inclusion of C ∗ -algebras. Let both f : [
0, 1 ] → Aut ( A ) , f : [
0, 1 ] → Aut ( A ) have unique π -lifts e f and e f respectively. If f ∗ f : [
0, 1 ] → Aut ( A ) , e f ∗ e f : [
0, 1 ] → Aut (cid:16) e A (cid:17) are given by ( f ∗ f ) ( t ) = f ( t ) ≤ t ≤ f (cid:0) (cid:0) t − (cid:1)(cid:1) f ( ) < t ≤ (cid:16) e f ∗ e f (cid:17) ( t ) = e f ( t ) ≤ t ≤ e f (cid:0) (cid:0) t − (cid:1)(cid:1) e f ( ) < t ≤ e f ∗ e f is the unique π -lift of f ∗ f . Hence one has lift f ∗ f = lift f lift f . Itturns out that the set of elements lift f is a subgroup of Aut (cid:16) e A (cid:17) . Definition 2.5.3.
Let π : A ֒ → e A be an inclusion of C ∗ -algebras then the group ofelements lift f (cf. 2.5.2) is said to be the π - lift group .54 emma 2.5.4. Let A ֒ → e A be an inclusion of C ∗ -algebras. Let f : [
0, 1 ] → Aut ( A ) be acontinuous function such that f ( ) = Id A and e f : [
0, 1 ] → Aut (cid:16) e A (cid:17) is the unique π -liftof f . If g ∈ G = n g ∈ Aut (cid:16) e A (cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ A o . Then one has g e f ( ) = e f ( ) g,i.e. G commutes with the π -lift group.Proof. If e f ′ : [
0, 1 ] → Aut (cid:16) e A (cid:17) is given by e f ′ ( t ) = g e f ( t ) g − then e f ′ ( t ) | A = e f ( t ) | A .From the Definition 2.5.1 it turns out e f ′ = e f . It turns out g e f ( ) g − = e f ′ ( ) = e f ( ) ,i.e. one has e f ( ) = g e f ( ) g − and, consequently e f ( ) g = g e f ( ) . Lemma 2.5.5.
Let π : A ֒ → e A be an inclusion of C ∗ -algebras. f : [
0, 1 ] → Aut ( A ) bea continuous function such that f ( ) = Id A . If there are two different π -lifts of f thenthere is a nontrivial continuous map: e f : [
0, 1 ] → Aut (cid:16) e A (cid:17) such that e f | A ( t ) = id A forevery t ∈ [
0, 1 ] .Proof. If e f ′ and e f ′′ are different π -lifts of f then the map e f : [
0, 1 ] → Aut (cid:16) e A (cid:17) given by t e f ′ ( t ) (cid:16) e f ′′ ( t ) (cid:17) − satisfies to conditions of this lemma. Corollary 2.5.6.
Let π : A ֒ → e A be an inclusion of C ∗ -algebras, such that the groupG = n g ∈ Aut (cid:16) e A (cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ A o is finite, and let f : [
0, 1 ] → Aut ( A ) be such that f ( ) = Id A . If e f is a π -lift of f then e f is the unique π -lift of f .Proof. If there are two different lifts of f then there is a nontrivial continuous map: e f : [
0, 1 ] → Aut (cid:16) e A (cid:17) such that e f | A ( t ) = id A . For any t ∈ [
0, 1 ] one has e f ( t ) ∈ G .Since e f is not trivial, the group G is not finite. Theorem 2.5.7.
Let p : e X → X be a topological covering projection. Any path ω : I → Homeo ( X ) such that ω ( ) = Id X can be uniquely lifted to the path e ω : I → Homeo ( e X ) such that e ω ( ) = Id e X , i.e. p ( e ω ( t )( x )) = ω ( t )( p ( x )) , ∀ t ∈ [
0, 1 ] , ∀ x ∈ e X . It means that the natural *-homomorphism C ( X ) ֒ → C (cid:16) e X (cid:17) has the unique pathlifting.Proof. Follows from theorem A.2.12. 55 .6 Coverings of operator spaces
Here we generalize section 2.1.
Definition 2.6.1.
Let Y be an unital operator space, and let C ∗ e ( Y ) be the C ∗ -envelope (cf. Definition D.7.12) of Y . A sub-unital operator space is a pair ( X , Y ) where X is a closed with respect to C ∗ -norm subspace of Y such that X = Y ⊕ C · C ∗ e ( Y ) or X = Y . Remark 2.6.2.
Any C ∗ -algebra A naturally yields a sub-unital operator spacewhich is ( A , A ) is A is unital and ( A , A + ) otherwise. Definition 2.6.3.
If both ( X , Y ) and (cid:16) e X , e Y (cid:17) are sub-unital operator spaces then complete isometry from ( X , Y ) to (cid:16) e X , e Y (cid:17) is a complete unital isometry π Y : Y ֒ → e Y such that π Y ( X ) ⊂ e X . We write (cid:16) π X : X ֒ → e X (cid:17) def = π Y | X ; ( π X , π Y ) : ( X , Y ) ֒ → (cid:16) e X , e Y (cid:17) . (2.6.1) Definition 2.6.4.
If a pair ( X , Y ) is sub-unital operator space then the C ∗ - envelope of ( X , Y ) is the C ∗ -algebra given by C ∗ e ( X , Y ) == ∩ { A ⊂ C ∗ e ( Y ) | A is a C ∗ -subalgebra of C ∗ e ( Y ) AND X ⊂ A } , (2.6.2)i.e. C ∗ e ( X , Y ) is the C ∗ -algebra, generated by X . Definition 2.6.5.
Let both ( X , Y ) and (cid:16) e X , e Y (cid:17) be sub-unital operator spaces and let ( π X , π Y ) : ( X , Y ) ֒ → (cid:16) e X , e Y (cid:17) be a complete isometry from ( X , Y ) to (cid:16) e X , e Y (cid:17) . If thefollowing conditions hold:(a) There is a finite-fold noncommutative covering (cid:16) C ∗ e ( X , Y ) , C ∗ e (cid:16) e X , e Y (cid:17) , G , ρ (cid:17) (cf. Definition 2.1.17) such that π Y = ρ | Y and π X = ρ | X .(b) If e X ′ ⊂ C ∗ e (cid:16) e X , e Y (cid:17) is a C -linear space such that X = e X ′ ∩ C ∗ e ( X , Y ) and G e X ′ = e X ′ then e X ′ ⊆ e X .Then we say that (cid:16) ( X , Y ) , (cid:16) e X , e Y (cid:17) , G , ( π X , π Y ) (cid:17) is a noncommutative finite-fold cov-ering of the sub-unital operator space ( X , Y ) .56 emark 2.6.6. The action G × C ∗ e (cid:16) e X , e Y (cid:17) → C ∗ e (cid:16) e X , e Y (cid:17) is trivial on C ∗ e ( X , Y ) , sofrom (b) of the Definition 2.6.5 it follows that there is the natural action G × e X → e X which is trivial on X . Let ( A , H , D ) be a spectral triple, and let A is the C ∗ -norm completion of A . Let (cid:16) A , e A , G (cid:17) be an unital noncommutative finite-fold covering. Let ρ : A → B ( H ) be a natural representation given by the spectral triple ( A , H , D ) , and let e ρ : e A → B (cid:16) e H (cid:17) be a representation induced by the pair (cid:16) ρ , (cid:16) A , e A , G (cid:17)(cid:17) . The algebra e A is afinitely generated projective A -module, it turns out following direct sum e A M Q ∼ = A n .of A -modules. So there is a projector p ∈ M n ( A ) such that e A ∼ = pA n as A -module. A is dense in A and A is closed with respect homomorphic calculus, it turns outthat there is a projector e p ∈ M n ( A ) such that k e p − p k <
1, so one has e A ∼ = e pA n .From e A ⊂ End A (cid:16) e A (cid:17) and End A (cid:16) e A (cid:17) = e p M n ( A ) e p ⊂ M n ( A ) it follows that thereis the following inclusion of C ∗ -algebras e A ⊂ M n ( A ) . Both e A and M n ( A ) arefinitely generated projective A modules, it turns out that there is an A -module P such that e A M P ∼ = M n ( A ) .Taking into account inclusions e A ⊂ M n ( A ) and M n ( A ) ⊂ M n ( A ) one can definethe intersection of algebras e A def = e A \ M n ( A ) . (2.7.1)From [37] it turns out that M n ( A ) is closed with respect to holomorphic functionalcalculus. Both M n ( A ) and e A are closed with respect to holomorphic functionalcalculus, so e A is closed with respect to holomorphic functional calculus, i.e. e A isa pre- C ∗ -algebra. From M n ( A ) ∼ = e A M (cid:16) P \ M n ( A ) (cid:17) it turns out that e A is a finitely generated projective A module.57 emma 2.7.1. The algebra e A is a dense subalgebra of e A with respect to the C ∗ -normtopology.Proof. There is the isomorphism A n ≈ M n ( A ) of right A -modules. Since e A is aprojective right A -module, there is a projector p M ∈ M n ( A ) it follows that e A ∼ = p M A n The algebra M n ( A ) is dense in M n ( A ) , so there is a projector e p M ∈ M n ( A ) such that such that k e p M − p M k <
1, it turns out e A ∼ = e p M A n (2.7.2)For any e a ∈ e A there is a net { e a α ∈ M n ( A ) } such thatlim e a α = e a in sense of C ∗ -norm topology. The sequence can be regarded as a sequence ne a α ∈ A n o n ∈ N . From (2.7.2) it turns out that if e b α = e p M e a α ∈ e A thenlim e b α = lim e p M e a α = e p M lim e a α = e p M e a = e a .Otherwise from e a α ∈ A n and e p M ∈ M n ( A ) it turns out that e b α = e p M e a α ∈ A n ∼ = M α ( A ) , so one has e b α ∈ e A T M α ( A ) ∼ = e A . Hence for any e a ∈ e A there is asequence ne b α ∈ e A o n ∈ N such that lim e b α = e a . Definition 2.7.2.
In the above situation we say that the unital noncommutativefinite-fold covering (cid:16) A , e A , G (cid:17) is smoothly invariant if G e A = e A . Lemma 2.7.3.
Let us use the above notation. Suppose that the right A-module e A A isgenerated by a finite set { e a , . . . , e a n } , i.e. e A A = n ∑ j = e a j A , such that following conditions hold: a) (cid:10)e a j , e a k (cid:11) e A ∈ A for any j , k =
1, . . . , n.(b) The generated by { e a , . . . , e a n } right A -module is G-invariant, i.e. for all j =
1, . . . , n and g ∈ G one has g e a j = ∑ nk = e a k c k where c k ∈ A for every k =
1, . . . , n.Then following conditions hold:(i) e A ∩ M n ( A ) = ne a ∈ e A | (cid:10)e a j , e a e a k (cid:11) e A ∈ A ; ∀ j , k =
1, . . . , n o (2.7.3) (ii) The unital noncommutative finite-fold covering (cid:16) A , e A , G (cid:17) is smoothly invariant(cf. Definition 2.7.2).Proof. (i) If S ∈ End (cid:16) e A (cid:17) A is given by S = n ∑ j = e a j ih e a j (2.7.4)then S is self-adjoint. Moreover S is represented by a matrix n S jk = (cid:10)e a j , e a k (cid:11) e A o j , k = n ∈ M n ( A ) . From the Corollary 1.1.25 of [41] it turns out that S is strictly positive.Otherwise e A A is a finitely generated right A -module, so from the Exercise 15.Oof [76] it follows that S is invertible, i.e. there is T ∈ End (cid:16) e A (cid:17) A such that ST = TS = End ( e A ) A .If we consider S as element of M m ( A ) then the spectrum of S is a subset of U S U ⊂ C such that • Both U and U are open sets. • U T U = ∅ , • ∈ U and 0 is the unique point of the spectrum of S which lies in U .If π , ψ are homomorphic functions on U S U given by φ | U = ψ | U ≡ φ | U = ψ | U = z z .Then following conditions hold: 59 p = φ ( S ) is a projector, such that e A A ≈ pA n as a right A -module, • p ∈ M n ( A ) , • ψ ( S ) = T ∈ M n ( A ) and TS = ST = End ( e A ) A .If e a ∈ A then from TS = ST = End ( e A ) A it turns out e a = TS e aST = T n ∑ j = e a j ih e a j ! e a n ∑ k = e a k ih e a k ! T = T M e a T where M e a ∈ M n ( A ) is a matrix given by M e a = n M e ajk = (cid:10)e a j , e a e a k (cid:11) e A o j , k = n .From T ∈ M n ( A ) it turns out that M e a ∈ M n ( A ) ⇒ T M e a T ∈ M n ( A ) .Conversely from S ∈ M n ( A ) it follows that T M e a T ∈ M n ( A ) ⇒ ST M e a TS = M e a ∈ M n ( A ) ,so one has M e a ∈ M n ( A ) ⇔ T M e a T ∈ M n ( A ) .Hence e a ∈ M n ( A ) if and only if (cid:10)e a j , e a e a k (cid:11) e A ∈ A for any j , k =
1, . . . n .(ii) Note that given by (2.1.2) product is G -invariant, i.e. De a , e b E e A = D g e a , g e b E e A forany g ∈ G , it follows that (cid:10)e a j , ( g e a ) e a k (cid:11) e A = D g − e a j , e a (cid:16) g − e a k (cid:17)E e A .Otherwise taking into account the condition (b) of these lemma one has g − e a j = n ∑ l ′ = e a l ′ c ′ l ′ , g − e a k = n ∑ l ′′ = e a l ′′ c ′′ l ′′ ,60here c ′ l ′ . c ′′ l ′′ ∈ A for all l ′ , l ′′ =
1, . . . n . It turns out that (cid:10)e a j , ( g e a ) e a k (cid:11) e A = D g − e a j , e a (cid:16) g − e a k (cid:17)E e A = n ∑ l ′ = l ′′ = c ′∗ l ′ h e a l ′ , e a e a l ′′ i e A c l ′′ ∈ A .Hence one has g (cid:16) e A T M n ( A ) (cid:17) = e A T M n ( A ) , or equivalently G (cid:16) e A \ M n ( A ) (cid:17) = e A \ M n ( A ) ,hence from (2.7.1) one has G e A = e A .In the following text we suppose that the unital noncommutative finite-foldcovering (cid:16) A , e A , G (cid:17) is smoothly invariant. From the Proposition E.3.9 it followsthat there is a connection ∇ ′ : e A → e A ⊗ A Ω D .Let us define a connection e ∇ : e A → e A ⊗ A Ω D , e ∇ ( e a ) = | G | ∑ g ∈ G g − (cid:0) ∇ ′ ( g e a ) . (cid:1) (2.7.5)The connection e ∇ is G - equivariant , i.e. e ∇ ( g e a ) = g (cid:16) e ∇ ( e a ) (cid:17) ; for any g ∈ G , e a ∈ e A . (2.7.6) Lemma 2.7.4.
If the unital noncommutative finite-fold covering (cid:16) A , e A , G (cid:17) is smoothlyinvariant then there is the unique G-equivariant connection e ∇ : e A → e A ⊗ A Ω D . Proof.
From the equation (2.7.6) it follows that a G -equivariant connection exists.Let us prove the uniqueness of it. It follows from the Proposition E.3.9 that thespace of connections is an affine space over the vector space Hom A (cid:16) e A , e A ⊗ A Ω D (cid:17) .The space of G -equivariant connections is an affine space over the vector space61om G A (cid:16) e A , e A ⊗ A Ω D (cid:17) of G -equivariant morphisms, i.e. morphisms in the cate-gory M G e A (cf. B.2). However from B .2 it follows that the category M G e A is equivalentto the category M A of A -modules. It turns out that there is a 1-1 correspondencebetween connections ∇ : A → A ⊗ A Ω D = Ω D and G -equivariant connections e ∇ : e A → e A ⊗ A Ω D .It follows that thee is the unique G -equivariant e ∇ connection which correspondsto ∇ : A → A ⊗ A Ω D = Ω D , a [ D , a ] . Let H ∞ def = T ∞ n = Dom D n , and let us define an operator e D : e A ⊗ A H ∞ → e A ⊗ A H ∞ such that if e a ∈ e A and e ∇ ( e a ) = m ∑ j = e a j ⊗ ω j ∈ e A ⊗ A Ω D then e D ( e a ⊗ ξ ) = m ∑ j = e a j ⊗ ω j ( ξ ) + e a ⊗ D ξ , ∀ ξ ∈ H ∞ . (2.7.7)The space e A ⊗ A H ∞ is a dense subspace of the Hilbert space e H = e A ⊗ A H , hence e D can be regarded as an unbounded operator on e H . Definition 2.7.6.
The operator e D given by (2.7.7) is said to be the (cid:16) A , e A , G (cid:17) - lift of D . The spectral triple (cid:16) e A , e H , e D (cid:17) is said to be the (cid:16) A , e A , G (cid:17) - lift of ( A , H , D ) . Let ( A , H , D ) be a spectral triple, let (cid:16) e A , e H , e D (cid:17) is the (cid:16) A , e A , G (cid:17) -lift of ( A , H , D ) .Let V = C n and with left action of G , i.e. there is a linear representation ρ :62 → GL ( C , n ) . Let e E = A ⊗ C n ≈ e A n be a free module over e A , so e E is a pro-jective finitely generated A -module (because e A is a finitely generated projective A -module). Let e ∇ : e E → e E ⊗ e A Ω e D be the trivial flat connection. In 2.7 it is proventhat Ω e D = e A ⊗ A Ω D it follows that the connection e ∇ : e E → e E ⊗ e A Ω e D can beregarded as a map ∇ ′ : e E → e E ⊗ e A e A ⊗ A Ω e D = e E ⊗ A Ω D , i.e. one has a connection ∇ ′ : e E → e E ⊗ A Ω D .From e ∇ ◦ e ∇| E = ∇ ′ ◦ ∇ ′ | E =
0, i.e. ∇ ′ is flat. There is the actionof G on e E = e A ⊗ C n given by g ( e a ⊗ x ) = g e a ⊗ gx ; ∀ g ∈ G , e a ∈ e A , x ∈ C n . (2.8.1)Denote by E = e E G = n e ξ ∈ e E | G e ξ = e ξ o (2.8.2)Clearly E is an A - A -bimodule. For any e ξ ∈ e E there is the unique decomposition e ξ = ξ + ξ ⊥ , ξ = | G | ∑ g ∈ G g e ξ , ξ ⊥ = e ξ − ξ . (2.8.3)From the above decomposition it turns out the direct sum e E = e E G L E ⊥ of A -modules. So E = e E G is a projective finitely generated A -module, it follows thatthere is an idempotent e ∈ End A e E such that E = e e E . The Proposition E.3.9 givesthe canonical connection ∇ : E → E ⊗ A Ω D (2.8.4)which is defined by the connection ∇ ′ : e E → e E ⊗ A Ω D and the idempotent e . Lemma 2.8.1. If ∇ : E → E ⊗ A Ω is the trivial connection and e ∈ End A ( E ) is anidempotent then the given by (E.3.9) connection ∇ e : e E → e E ⊗ Ω ; ξ ( e ⊗ ) ∇ ξ is flat. roof. From ( e ⊗ ) ( Id E ⊗ d ) ◦ ( e ⊗ ) ( Id E ⊗ d ) = e ⊗ d = ∇ e ◦ ∇ e =
0, i.e. ∇ e is flat. Remark 2.8.2.
The notion of the trivial connection is an algebraic version of geo-metrical canonical connection explained in the Section H.2.From the Lemma 2.8.1 it turns out that the given by (2.8.4) connection ∇ is flat. Definition 2.8.3.
We say that ∇ is a flat connection induced by noncommutativecovering (cid:16) A , e A , G (cid:17) and the linear representation ρ : G → GL ( C , n ) , or we say the ∇ comes from the representation ρ : G → GL ( C , n ) .64 .8.2 Mapping between geometric and algebraic constructions The geometric (resp. algebraic) construction of flat connection is explained inthe Section H.2 (resp. 2.8.1). Following table gives a mapping between these con-structions. Geometry Agebra1 Riemannian manifold M . Spectral triple (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) .2 Topological covering e M → M . Noncommutative covering, (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) ,given by the Theorem 1.1.2.3 Natural structure of Reimannian Triple (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S (cid:17) , e / D (cid:17) is themanifold on the covering space e M . (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) -liftof (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) .4 Group homomorphism Action G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) × C n → C n G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) → GL ( n , C ) e M × C n . Free module C ∞ (cid:16) e M (cid:17) ⊗ C n .6 Canonical flat connection on e M × C n Trivial flat connection on C ∞ (cid:16) e M (cid:17) ⊗ C n G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) on e M × C n Action of G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) on C ∞ (cid:16) e M (cid:17) ⊗ C n P = (cid:16) e M × C n (cid:17) / G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . E = (cid:16) C ∞ (cid:16) e M (cid:17) ⊗ C n (cid:17) G ( e M | M ) P Algebraic flat connection on E .65 .9 Unoriented spectral triples Let M be an unoriented Riemannian manifold, and let e M → M be a two-foldcovering by oriented Riemannian manifold e M which admits a spin c structure (cf.Definition E.4.2). There is an action of Z × e M → e M such that M ∼ = e M / Z . Theseconsiderations inspire the following definition. Definition 2.9.1.
Denote by g ∈ Z the unique nontrivial element. An unorientedspectral triple ( A , H , D ) consists of:1. A pre- C ∗ -algebra A .2. An unital noncommutative finite-fold covering (cid:16) A , e A , Z (cid:17) (cf. Definition2.4.2) where A is the C ∗ -norm completion of A .3. A faithful representation ρ : A → B ( H ) .4. A selfadjoint operator D on H , with dense domain Dom D ⊂ H , such that a ( Dom D ) ⊆ Dom D for all a ∈ A .5. An unital oriented spectral triple (cid:16) e A , e H , e D (cid:17) which satisfies to described in[39, 73] axioms.Above objects should satisfy to the following conditions:(a) If e A is the C ∗ -norm completion of e A then the representation e A → e H isinduced by the pair (cid:16) ρ , (cid:16) A , e A , Z (cid:17)(cid:17) (cf. Definition 2.1.9).(b) The action Z × e A → e A of Z ∼ = G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) induces the actions Z × e A → e A , Z × e H → e H , such that g (cid:16)e a e ξ (cid:17) = ( g e a ) (cid:16) g e ξ (cid:17) ; ∀ e a ∈ e A , ∀ e ξ ∈ e H , (cid:16) g e ξ , g e η (cid:17) = (cid:16) e ξ , e η (cid:17) ; ∀ e ξ , e η ∈ e H , ,where ( · , · ) is the scalar product on e H , (2.9.1) g (cid:16) e D e ξ (cid:17) = e D (cid:16) g e ξ (cid:17) ; ∀ e ξ ∈ Dom e D (2.9.2)66c) There are natural isomorphisms A ∼ = e A Z def = ne a ∈ e A | g e a = e a o , H ∼ = e H Z def = n e ξ ∈ e H | g e ξ = e ξ o . (2.9.3)(d) If A (resp. e A ) is a C ∗ -norm completion of A (resp. e A ) then the triple (cid:16) A , e A , Z (cid:17) is an unital noncommutative finite-fold covering and the follow-ing condition hold: A = A \ e A , (2.9.4) D = e D | H = e D | e H Z . (2.9.5) Remark 2.9.2.
The Definition 2.9.1 is motivated by explained in the Section 4.10.4commutative examples. 678 hapter 3
Noncommutative infinitecoverings
This section contains a noncommutative generalization of infinite coverings.
Definition 3.1.1.
Let us consider the following diagram e A e A A π π π where A , e A , e A are C ∗ -algebras and π , π , are noncommutative finite-fold cover-ings. We say that the unordered pair ( π , π ) is compliant if it satisfies to followingconditions:(a) If there is an injective *-homomorphism π : e A → e A such that π = π ◦ π then π is a noncommutative finite-fold covering (cf. Definition 2.1.17).(b) Following condition holds G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) π (cid:16) e A (cid:17) = π (cid:16) e A (cid:17) . (3.1.1)(c) From (3.1.1) it turns out that there is the homomorphism h : G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) → (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) such that π ( h ( g ) a ) = g ◦ π ( a ) (3.1.2)for each a ∈ e A . We claim that h is surjective.(d) If ρ : e A → e A is any injective *-homomorphism such that π = ρ ◦ π thenthere is the unique g ∈ G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) such that ρ = π ◦ g . (3.1.3) Definition 3.1.2.
In the situation of the Definition 3.1.1 we say that the homomor-phism h : G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) → G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) which satisfies to (3.1.2) comes from π . Remark 3.1.3.
From the Definition 3.1.1 it turns out that if ρ ′ : e A → e A and ρ ′ : e A → e A are *-homomorphisms such that π = ρ ′ ◦ π and π = ρ ′′ ◦ π thenboth ρ ′ and ρ ′′ are *-isomorphisms. Definition 3.1.4.
Let Λ be a countable directed set (cf. Definition A.1.3) suchthat there is the unique minimal element λ min ∈ Λ . Let A be a C ∗ -algebra. Letus consider a set noncommutative finite-fold coverings S = { π λ : A ֒ → A λ } λ ∈ Λ indexed by Λ such that A λ min = A , and π λ min = Id A . We do not claim that anynoncommutative finite-fold covering A → e A belongs to S . Suppose that followingconditions hold:(a) For any µ , ν ∈ Λ a pair (cid:0) π µ , π ν (cid:1) is compliant (cf. Definition 3.1.1).(b) For µ , ν ∈ Λ one has µ ≥ ν if and only if there is an *-homomorphism π : A ν ֒ → A µ ;such that π µ = π ◦ π ν . (3.1.4)The set S = { π λ : A ֒ → A λ } λ ∈ Λ , or equivalently S = { ( A , A λ , G ( A λ | A ) , π λ ) } λ ∈ Λ is said to be an algebraical finite covering category . We write S ∈ FinAlg . Remark 3.1.5.
Below for all λ ∈ Λ we implicitly assume that A λ ⊂ C ∗ -lim −→ λ ∈ Λ A λ (cf. Definition 1.2.7) i.e. for any µ , ν ∈ Λ and a µ ∈ A µ , a ν ∈ A ν we suppose thatboth a µ , a ν are elements of the single C ∗ -algebra C ∗ -lim −→ λ ∈ Λ A λ .70 efinition 3.1.6. A subcategory S pt of an algebraical finite covering category S is said to be a pointed algebraical finite covering category if it is equivalent to thepre-order category given by Λ (cf. Definition H.1.1 and Remark H.1.3). We write S pt ∈ FinAlg pt , and use the following notation S pt = (cid:18) { π λ : A ֒ → A λ } λ ∈ Λ , (cid:8) π µν : A µ ֒ → A ν (cid:9) µ , ν ∈ Λ ν > µ (cid:19) ,or simply S pt = (cid:0) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:1) . (3.1.5) Remark 3.1.7.
The Definition 3.1.6 means that for any µ ≥ ν the category S pt contains the unique *-homomorphism π νµ : A ν ֒ → A µ . The family of n π νµ o of*-homomorphisms is said to be a base-point . Let us consider the situation of the Definition 3.1.6. If µ , ν ∈ Λ and µ > ν .Denote by h µν : G (cid:0) A µ (cid:12)(cid:12) A (cid:1) → G ( A ν | A ) the surjective homomorphism whichcomes from π νµ (cf. Definition 3.1.2). Lemma 3.1.9.
Let us use notation of 3.1.8. If µ , λ , ν are such that µ > λ > ν thenh µν = h µλ ◦ h λν .Proof. From the equation (3.1.2) it turns out that for all a ν ∈ A ν and g µ ∈ G (cid:0) A µ (cid:12)(cid:12) A (cid:1) one has π νµ (cid:0) h µν (cid:0) g µ (cid:1) a ν (cid:1) = g µ ◦ π νµ ( a ν ) , (3.1.6) π νλ (cid:16) h λν (cid:0) h µλ (cid:0) g µ (cid:1)(cid:1) a ν (cid:17) = h µλ (cid:0) g µ (cid:1) ◦ π νλ ( a ν ) , (3.1.7) π λµ (cid:0) h µλ (cid:0) g µ (cid:1) π νλ ( a ν ) (cid:1) = g µ ◦ π λµ ( π νλ ( a ν )) = g µ ◦ π νµ ( a ν ) . (3.1.8)If we substitute h µλ (cid:0) g µ (cid:1) π νλ ( a ν ) in the left part of (3.1.8) by the left part of (3.1.7)and taking into account π λµ ◦ π νλ = π νµ one has π λµ (cid:16) π νλ (cid:16) h λν (cid:0) h µλ (cid:0) g µ (cid:1)(cid:1) a ν (cid:17)(cid:17) = π νµ (cid:16) h λν ◦ h µλ (cid:0) g µ (cid:1) a ν (cid:17) = g µ ◦ π νµ ( a ν ) . (3.1.9)Comparison of (3.1.6) and (3.1.9) yields h µν = h µλ ◦ h λν . Definition 3.1.10.
From the Lemma 3.1.9 it turns out that one can define surjecivehomomorphisms h µν : G (cid:0) A µ (cid:12)(cid:12) A (cid:1) → G ( A ν | A ) the surjective homomorphismwhich come from π νµ , hence there is the inverse limit b G = lim ←− λ ∈ Λ G ( A λ | A )
71f groups. For any λ ∈ Λ there is the surjecive homomorphism h λ : b G → G ( A λ | A ) (3.1.10)Wa say that the inverse limit comes from the base point n π νµ o (cf Remark 3.1.7). Lemma 3.1.11.
Let S = { ( A , A λ , G ( A λ | A ) , π λ ) } λ ∈ Λ ∈ FinAlg . be an algebraicalfinite covering category. Consider two pointed subcategories S pt π , S pt ρ having morphisms n π νµ : A ν ֒ → A µ o and n ρ νµ : A ν ֒ → A µ o . Suppose that both (cid:8) d µν : G (cid:0) A µ (cid:12)(cid:12) A (cid:1) → G ( A ν | A ) (cid:9) and (cid:8) e µν : G (cid:0) A µ (cid:12)(cid:12) A (cid:1) → G ( A ν | A ) (cid:9) are surjecive homomorphisms which come from n π νµ o and n ρ νµ o respectively. Sup-pose that both b G = lim ←− λ ∈ Λ G ( A λ | A ) and b H = lim ←− λ ∈ Λ G ( A λ | A ) are inverse lim-its which come from (cid:8) d µν (cid:9) and (cid:8) e µν (cid:9) respectively. If d λ : b G → G ( A ν | A ) and e λ : b H → G ( A ν | A ) are natural surjective homomorphisms then there is the bijective map φ : b G ≈ −→ b H and b g ∈ b G such that e λ ◦ φ = d λ ◦ b g , (3.1.11) ρ νµ = π νµ ◦ d ν ( b g ) ∀ µ > ν (3.1.12) Proof. If µ ≥ ν then there is g ν ∈ G ( A ν | A ) such that ρ νµ = π νµ ◦ g ν . (3.1.13)From π νµ = π λµ ◦ π νλ and ρ νµ = ρ λµ ◦ ρ νλ one has d µν (cid:0) g µ (cid:1) = g ν . The family { g λ } gives an element b g ∈ b G . Any b g ′ ∈ b G corresponds to a family (cid:8) g ′ λ (cid:9) λ ∈ Λ such that d µν (cid:16) g ′ µ (cid:17) = g ′ ν . Otherwise there is the family (cid:8) g ′′ λ (cid:9) λ ∈ Λ such that g ′′ λ = g ′ λ ◦ d λ ( b g ) and e µν (cid:16) g ′′ µ (cid:17) = g ′′ ν . The family (cid:8) g ′′ λ (cid:9) uniquely defines the element b g ′′ ∈ b H . In resultone has the bijective map φ : b G ≈ −→ b H , b g ′ b g ′′ which satisfies to the equation (3.1.11). The equation (3.1.12) follows from (3.1.13).72 emark 3.1.12. The map φ induces the isomorphism of groups ϕ : b G ≈ −→ b H , b g − b g ′ b g φ (cid:0)b g ′ (cid:1) (3.1.14) Remark 3.1.13.
Let S = { ( A , A λ , G ( A λ | A ) , π λ ) } λ ∈ Λ ∈ FinAlg . be an algebraicalfinite covering category. If both n π νµ o and n ρ νµ o are base-points of S then fromthe Lemma 3.1.11 it turns out the category (cid:16) S , n π νµ o(cid:17) is equivalent to (cid:16) S , n ρ νµ o(cid:17) . Remark 3.1.14.
There is a functor from S pt = (cid:8) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:9) to thecategory of finite groups and epimorphisms. The functor is given by ( A ֒ → A λ ) G ( A λ | A ) ; (cid:0) π µν : A ν ֒ → A µ (cid:1) (cid:0) h µν : G (cid:0) A µ (cid:12)(cid:12) A (cid:1) → G ( A ν | A ) (cid:1) . Remark 3.1.15. If ν ≥ µ then since S pt contains the unique injective *-homomorphism π µν : A µ ֒ → A ν , then according to the Remark 3.1.5 we substitute following equa-tions π µν : A µ ֒ → A ν , π µν (cid:0) a µ (cid:1) = ∑ g ∈ G ( A µ | A ν ) ga ν (3.1.15)by A µ ⊂ A ν , a µ = ∑ g ∈ G ( A µ | A ν ) ga ν (3.1.16)Let S pt = (cid:0) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:1) be a pointed algebraical finite coveringcategory (cf. Definition 3.1.6). Let b A = C ∗ - lim −→ λ ∈ Λ A λ be the C ∗ -inductive limit of C ∗ -algebras, and let b G = lim ←− λ ∈ Λ G ( A λ | A ) be the projective limit of groups. Definition 3.1.16.
A faithful representation π : b A ֒ → B ( H ) is said to be equivariant if there is the action b G × H → H such that g ( b a ξ ) = ( g b a ) ( g ξ ) ; ∀ g ∈ b G , b a ∈ b A , ξ ∈ H . (3.1.17) Remark 3.1.17.
If a faithful representation π : b A ֒ → B ( H ) is equivariant then theaction b G × H → H induces the natural action b G × B ( H ) → B ( H ) such that ( ga ) ξ def = g (cid:16) a (cid:16) g − ξ (cid:17)(cid:17) ; ∀ g ∈ b G , a ∈ B ( H ) , ξ ∈ H . (3.1.18)73 emma 3.1.18. Following conditions hold:(i) The universal representation b π : b A ֒ → B (cid:16) b H (cid:17) (cf. Definition D.2.10) is equivari-ant.(ii) The atomic representation π a : b A ֒ → B (cid:16) b H a (cid:17) (cf. Definition D.2.33) is equivariant.Proof. (i) If S is the state space of b A then there is the natural action b G × S → S given by ( gs ) ( b a ) def = s ( ga ) where s ∈ S , g ∈ b G and b a ∈ b A Let s ∈ S , and let L (cid:16) b A , s (cid:17) is the Hilbert space ofthe representation π s : b A → B (cid:16) L (cid:16) b A , s (cid:17)(cid:17) which corresponds to s (cf. D.2.1). If f s : b A → L (cid:16) b A , s (cid:17) is the given by D.2.1 natural b A -linear map then the b A -module f s (cid:16) b A (cid:17) is dense in L (cid:16) b A , s (cid:17) . Since b H = M s ∈ S L (cid:16) b A , s (cid:17) the C -linear span of given by ξ s b a =
0, ..., f s ( b a ) | {z } s th − place , ..., 0 ∈ M s ∈ S L (cid:16) b A , s (cid:17) = b H elements is dense in b H . Hence if for g ∈ b G we define g ξ s b a def =
0, ..., f gs ( g b a ) | {z } gs th − place , ..., 0 ∈ b H then g ξ can be uniquely defined for all ξ ∈ b H i.e. there is the natural action b G × b H → b H . Using the equality b a f s ( b a ) = f s ( b a b a ) one can prove that the action b G × b H → b H satisfies to (3.1.17).(ii) The replacement of the "state" word in (i) with "pure state" one gives (ii).74 efinition 3.1.19. Let S pt = (cid:0) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:1) ∈ FinAlg pt be a pointedalgebraical finite covering category. b A def = C ∗ - lim −→ λ ∈ Λ A λ be the C ∗ -inductive limitof C ∗ -algebras, and let b G def = lim ←− λ ∈ Λ G ( A λ | A ) be the projective limit of groups.Let G λ def = ker (cid:16) b G → G ( A λ | A ) (cid:17) , and let A ∼ λ be given by (1.2.1). Let π : b A → B ( H ) be an equivariant representation. A positive element a ∈ B ( H ) + is saidto be π - special if for any λ ∈ Λ , ε > z ∈ A ∼ λ it satisfies to the followingconditions:(a) If f ε : R → R is a continuous function given by f ε ( x ) = (cid:26) x ≤ ε x − ε x > ε (3.1.19)then for all λ ≥ λ there are a λ , b λ , a ελ ∈ A λ such that ∑ g ∈ G λ π ( z ) ∗ ( ga ) π ( z ) = π ( a λ ) , ∑ g ∈ G λ f ε (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = π ( a ελ ) , ∑ g ∈ G λ (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = π ( b λ ) (3.1.20)where sums of the above series mean the strong convergence in B ( H ) andthe action G λ × B ( H ) → B ( H ) is given by (3.1.18).(b) There is µ ∈ Λ such that µ ≥ λ and ∀ λ ∈ Λ λ ≥ µ ⇒ (cid:13)(cid:13) a λ − b λ (cid:13)(cid:13) < ε (3.1.21)where a λ , b λ ∈ A λ are given by (3.1.20).If π is the atomic representation then a is said to be special . Lemma 3.1.20.
Let S pt = (cid:0) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:1) ∈ FinAlg pt be a pointed al-gebraical finite covering category. b A def = C ∗ - lim −→ λ ∈ Λ A λ be the C ∗ -inductive limit ofC ∗ -algebras, and let b G def = lim ←− λ ∈ Λ G ( A λ | A ) be the projective limit of groups. LetG λ def = ker (cid:16) b G → G ( A λ | A ) (cid:17) , and let A ∼ λ be given by (1.2.1) . Let π : b A → B ( H ) bean equivariant representation. If a ∈ B ( H ) + is a positive element such that said for any λ ∈ Λ , ε > and z ∈ A ∼ λ there is λ ′ ≥ λ such that a satisfies to (3.1.20) for all λ ≥ λ ′ then a satisfies to (3.1.20) for all λ ≥ λ roof. The set Λ is directed, so for any λ ≥ λ there is λ ′′ ∈ Λ such that λ ′′ ≥ λ and λ ′′ ≥ λ ′ . There is the surjective homomorphism G λ ′′ → G λ ′ , such that G λ ′′ / G λ ′ ∼ = G ( A λ ′′ | A λ ′ ) . Let us enumerate G ( A λ ′′ | A λ ′ ) , i.e. G ( A λ ′′ | A λ ′ ) = { g , ..., g n } . If h : G λ ′ → G ( A λ ′′ | A λ ′ ) is the natural sujective homomorphism then for all j =
1, ..., n we select g j ∈ G λ ′ such that h (cid:16) g j (cid:17) = g j . One has ∑ g ∈ G λ ′ π ( z ) ∗ ( ga ) π ( z ) = n ∑ j = g j ∑ g ∈ G λ ′′ π ( z ) ∗ ( ga ) π ( z ) ! = n ∑ j = g j π ( a λ ′′ ) , ∑ g ∈ G λ ′ f ε (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = n ∑ j = g j ∑ g ∈ G λ ′′ f ε (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1)! = n ∑ j = g j π ( a ελ ′′ ) , ∑ g ∈ G λ ′ (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = n ∑ j = g j ∑ g ∈ G λ ′′ (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) ! = n ∑ j = g j π ( b λ ′′ ) where a λ ′′ , b λ ′′ , a ελ ′′ ∈ A λ ′′ . From g j π ( a λ ′′ ) = π (cid:0) g j a λ ′′ (cid:1) , g j π ( b λ ′′ ) = π (cid:0) g j b λ ′′ (cid:1) and g j π (cid:0) a ελ ′′ (cid:1) = π (cid:0) g j a ελ ′′ (cid:1) it follows that ∑ g ∈ G λ ′ π ( z ) ∗ ( ga ) π ( z ) = π ∑ g ′ ∈ G ( A λ ′′ | A λ ′ ) g ′ a λ ′′ , ∑ g ∈ G λ ′ f ε (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = π ∑ g ′ ∈ G ( A λ ′′ | A λ ′ ) g ′ a ελ ′′ , ∑ g ∈ G λ ′ (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = π ∑ g ′ ∈ G ( A λ ′′ | A λ ′ ) g ′ b λ ′′ .However there are a λ ′ , b λ ′ , a ελ ′ ∈ A λ ′ such that π ∑ g ′ ∈ G ( A λ ′′ | A λ ′ ) g ′ a λ ′′ = π ( a λ ′ ) , π ∑ g ′ ∈ G ( A λ ′′ | A λ ′ ) g ′ a ελ ′′ = π ( a ελ ′ ) , π ∑ g ′ ∈ G ( A λ ′′ | A λ ′ ) g ′ b λ ′′ = π ( b λ ′ ) .76 emma 3.1.21. Any π -special element a ∈ B ( H ) + equals to the following strong limita = lim λ ∈ Λ π ( a λ ) (3.1.22) where a λ satisfies to the following equation ∑ g ∈ G λ π (cid:16) A ∼ λ (cid:17) ∗ ( ga ) π (cid:16) A ∼ λ (cid:17) ∼ = ∑ g ∈ G λ ga = π ( a λ ) (3.1.23) (cf. Equation 3.1.20).Proof. The net { π ( a λ ) } λ ∈ Λ is decreasing, so from the Theorem D.1.25 the stronglimit lim α ∈ Λ π ( a λ ) exists. The strong limit coincides with the weak one, so oneshould prove that for all ξ , η ∈ H one has ( ξ , a η ) = lim λ ∈ Λ ( ξ , π ( a λ ) η ) = lim λ ∈ Λ ξ , ∑ g ∈ G λ ga ! η ! . (3.1.24)Element a is positive, so from the the polarization equality (D.4.3) it follows that(3.1.24) is equivalent to ( ξ , a ξ ) = lim λ ∈ Λ ξ , ∑ g ∈ G λ ga ! ξ ! ∀ ξ ∈ H . (3.1.25)For any λ ∈ Λ one has ( ξ , a ξ ) < ξ , ∑ g ∈ G λ ga ! ξ ! . (3.1.26)Otherwise the nonincreasing net n x λ = (cid:16) ξ , (cid:16) ∑ g ∈ ker ( b G → G ( A λ | A ) ) ga (cid:17) ξ (cid:17)o of realnumbers is bounded, so it is convergent. For any ε > λ ε ∈ Λ such that | x λ − lim λ x λ | < ε for all λ ≥ λ ε . It follows that λ ≥ λ ε ⇒⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ , ∑ g ∈ G λ g = e ga ξ − ξ , ∑ g ∈ G λ g = e ga ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε where e is the unity of the group b G . On the other hand from \ λ ∈ Λ ker (cid:16) b G → G ( A λ | A ) (cid:17) = { e }
77t turns out that ∀ λ ∈ Λ λ ≥ λ ε ⇒ ( ξ , a ξ ) + ε > ξ , ∑ g ∈ G λ ga ! ξ ! (3.1.27)The equation (3.1.25) is a direct consequence of the equations (3.1.26) and (3.1.27). Corollary 3.1.22. If b π : b A → B (cid:16) b H (cid:17) is the universal representation then any b π -specialelement a ∈ B (cid:16) b H (cid:17) lies in the enveloping Borel *-algebra B (cid:16) b A (cid:17) of b A (cf. DefinitionD.1.39).Proof.
Since Λ is countable there is an increasing cofinal subset { λ n } n ∈ N ⊂ Λ , soone has lim λ ∈ Λ b π ( a λ ) = lim n → ∞ b π ( a λ n ) .From the above equation and the Definition D.1.39 it turns out that lim λ ∈ Λ b π ( α λ ) ∈ B (cid:16) b A (cid:17) . Taking into account (3.1.22) one has a ∈ B (cid:16) b A (cid:17) . Definition 3.1.23. If a is π - special and b x , b y ∈ b A then we say that the element b = π ( b x ) a π ( b y ) ∈ B ( H ) (3.1.28)is π - weakly special . If π is atomic then we say that b is weakly special . Denote by Ξ π ( S pt ) the b A -bimodule of π -weakly special families. The natural action of b G × H → H induces the action b G × B ( H ) → B ( H ) .Clearly if a ∈ B ( H ) is special then ga satisfies to (a), (b) of the Definition 3.1.19,i.e. ga is also special. Similarly if b is weakly special then gb is also special, i.e.there is the natural action b G × Ξ π (cid:0) S pt (cid:1) → Ξ π (cid:0) S pt (cid:1) (3.1.29)There is the natural C -linear map f Ξ π : Ξ π (cid:0) S pt (cid:1) → B ( H ) (3.1.30)The action (3.1.29) induces the action b G × A π → A π (3.1.31)where A π is the C ∗ -algebra, generated by Ξ π ( S pt ) .78 efinition 3.1.25. Let us consider the situation of 3.1.24. The C ∗ -algebra A π ⊂ B (cid:16) b H (cid:17) generated by the operator space f Ξ π ( Ξ π ( S pt )) is said to be the π - disconnectedalgebraical inverse noncommutative limit of S pt . If G (cid:0) A π (cid:12)(cid:12) A (cid:1) def = b G then there is theaction of G (cid:0) A π (cid:12)(cid:12) A (cid:1) × A π → A π which comes from the action (3.1.29). The triple (cid:0) A , A π , G (cid:0) A π (cid:12)(cid:12) A (cid:1)(cid:1) is said to be the π - disconnected algebraical infinite noncommuta-tive covering of S pt . We say that G (cid:0) A π (cid:12)(cid:12) A (cid:1) is the π - disconnected covering transfor-mation group of S pt . If π is the atomic representation i.e. π = π a : b A → B (cid:16) b H a (cid:17) then we write A instead A π and we say that A is the disconnected algebraical inversenoncommutative limit of S pt . Similarly G (cid:0) A (cid:12)(cid:12) A (cid:1) def = b G . The triple (cid:0) A , A , G (cid:0) A (cid:12)(cid:12) A (cid:1)(cid:1) is said to be the disconnected algebraical infinite noncommutative covering of S pt . Wesay that G (cid:0) A (cid:12)(cid:12) A (cid:1) is the disconnected covering transformation group of S pt . Lemma 3.1.26. If b π : b A → B (cid:16) b H (cid:17) is the universal representation then there is thenatural *-isomorphism A b π ∼ = A def = A b π a . (3.1.32) Proof. If b π a : b A → B (cid:16) b H a (cid:17) is the atomic representation then from the Definition3.1.25 it follows that A def = A b π a ⊂ B (cid:16) b H a (cid:17) . From the Theorem D.2.13 it turns outthat there is the surjective *-homomorphism p : b π (cid:16) b A (cid:17) ′′ → b π a (cid:16) b A (cid:17) ′′ . From theCorollary D.2.35 one has the inclusion B (cid:16) b A (cid:17) ⊂ B (cid:16) b H a (cid:17) where B (cid:16) b A (cid:17) is en-veloping Borel *-algebra of b A (cf. Definition D.1.39). It follows that the restriction p | B ( b A ) : B (cid:16) b A (cid:17) → p (cid:16) B (cid:16) b A (cid:17)(cid:17) is injective hence p | B ( b A ) is an *-isomorphism. On the other hand from the Corol-lary 3.1.22 it turns out that A ⊂ B (cid:16) b A (cid:17) so the restriction p | A b π : A b π → p (cid:0) A b π (cid:1) is *-isomorphism. If a b π ∈ A b π is b π -special then a b π a = p | A b π ( a b π ) ∈ B (cid:16) b H a (cid:17) is b π a -special (cf. Definition 3.1.19). It follows that p (cid:0) A b π (cid:1) ⊂ A b π a so there is the natural inclusion A b π ⊂ A b π a ι H : b H a ֒ → b H is the natural inclusion and p H : b H ֒ → b H a is the natural projectionthen there is the natural inclusion ι : B (cid:16) b H a (cid:17) ֒ → B (cid:16) b H (cid:17) , a ( ξ ι H ( ap H ( ξ ))) ∀ a ∈ B (cid:16) b H a (cid:17) , ξ ∈ b H .If a b π a ∈ B (cid:16) b H a (cid:17) is b π a -special then direct check shows that a b π = ι ( a b π a ) ∈ B (cid:16) b H (cid:17) is b π -special. It follows that there is the natural inclusion A b π a ⊂ A b π . Both inclusions A b π a ⊂ A b π and A b π ⊂ A b π a yield the isomorphism A b π ∼ = A b π a . Lemma 3.1.27.
If we consider the situation of 3.1.24 and the Definition 3.1.25 then thespectrum of b A is the inverse limit of spectra of A λ .Proof. Denote by X λ and b X the spectra of A λ and b A respectively. From the Lemma2.3.7 it turns out that for any µ , ν ∈ Λ such that µ > ν the inclusion π νµ : A ν → A µ induces the surjective continuous map p νµ : X µ → X ν . Otherwise if Y λ and b Y are the state spaces of A λ and b A then one has X λ ⊂ Y λ and b X ⊂ b Y . Moreoverfor any µ , ν ∈ Λ such that µ > ν there is the natural continuous surjective map t νµ : Y µ → Y ν such that p νµ = t νµ (cid:12)(cid:12)(cid:12) X µ From the Corollary 1.2.10 it follows that the state space b Y = lim ←− Y λ of b A is theinverse limit of state spaces of A λ , i.e. for any λ ∈ Λ there is the natural map b t λ : b Y → Y λ such that there is the following diagram b YY µ Y ν b t µ b t ν t νµ is commutative. Similarly there is the inverse limit b X = lim ←− X λ and the followingdiagram b XX µ X ν b p µ b p ν p νµ
80s commutative. Let b x ∈ b X and let b τ : b A → C be the corresponding functional.Suppose that b p : b A → C is a positive functional such that b p ≤ b τ . For any λ ∈ Λ following conditions hold: • The restriction b p | A λ is a positive functional. • There is there is a number t λ ∈ [
0, 1 ] such that b p | A λ = t λ b τ | A λ If µ , ν ∈ Λ are such that µ > ν then b p | A ν = (cid:12)(cid:12) G (cid:0) A µ (cid:12)(cid:12) A ν (cid:1)(cid:12)(cid:12) ∑ g ∈ G ( A µ | A ν ) g b p | A µ , b τ | A ν = (cid:12)(cid:12) G (cid:0) A µ (cid:12)(cid:12) A ν (cid:1)(cid:12)(cid:12) ∑ g ∈ G ( A µ | A ν ) g b τ | A µ (cf. Lemma 2.3.7) it follows that t µ = t ν , hence there is t ∈ [
0, 1 ] such that t λ = t for all λ ∈ Λ . So one has b p = t b τ , i.e. b τ is a pure state. Conversely suppose that b τ is pure and b τ | A λ is not pure. There is a positive functional p λ : A λ → C such that p λ ≤ b τ | A λ and p λ = t τ | A λ for all t ∈ [
0, 1 ] . Using Hahn–Banach theorem one canextent p λ up to a positive functional b p : b A → C such that b p ≤ b τ . But b p = t b τ for all t ∈ [
0, 1 ] , i.e. the state b τ is not pure. Lemma 3.1.28.
Let {H λ } λ ∈ Λ be a family of Hilbert spaces, and let H be the norm com-pletion of the algebraic direct sum ⊕ λ ∈ Λ H λ . The bounded net { a α } α ∈ A ∈ H is stronglyconvergent if and only if for all λ ∈ Λ and ζ ∈ H given by ζ =
0, ... , ζ λ |{z} λ th − place , ..., 0 ∈ M λ ∈ Λ H λ the net { a α ζ } ∈ H is norm convergent.Proof. If { a α } α ∈ A ∈ H is strongly convergent then the net { a α ζ } ∈ H is normconvergent for all ζ ∈ H .Conversely suppose that k a α k < C for all α ∈ A . If ξ = ( ξ λ ) λ ∈ Λ ∈ H and ε > Λ ⊂ Λ and η ∈ ⊕ λ ∈ Λ H λ such that η λ = ( ξ λ λ ∈ Λ λ / ∈ Λ AND η = ( η λ ∈ H λ ) λ ∈ Λ ⇒ k ξ − η k < ε C . (3.1.33)81or any λ ∈ Λ there is α λ ∈ A such that β , γ ≥ α λ ⇒ (cid:13)(cid:13)(cid:0) a β − a γ (cid:1) ξ λ (cid:13)(cid:13) < ε | Λ | C (3.1.34)If α max ≥ a λ for every λ ∈ Λ then one has β , γ ≥ α max ⇒ (cid:13)(cid:13)(cid:0) a β − a γ (cid:1) ξ (cid:13)(cid:13) < ε . Consider the situation of the Lemma 3.1.27. Any ξ ∈ b H a corresponds tothe family { ξ b x ∈ H b x } b x ∈ b X . (3.1.35) Lemma 3.1.30.
Let a ∈ B (cid:16) b H a (cid:17) + is a special element. There is the family { a b x ∈ B ( H b x ) } b x ∈ b X such that for any b ξ ∈ b H a given by the family (3.1.35) element a ξ is represented by thefamily { a b x ξ b x ∈ H b x } b x ∈ b X . Proof.
For all λ ∈ Λ we consider the implicit inclusion A λ ⊂ b A so for any b x ∈ b X there is the representation rep b x : A λ → B ( X ) . From the Lemma 3.1.21 it followsthat there is the decreasing net n a λ ∈ b A + o λ ∈ Λ such that a λ ∈ A λ and there isthe limit a = lim λ ∈ Λ a λ with respect to the strong topology of B (cid:16) b H a (cid:17) . From theLemma D.1.25 for any b x ∈ b X there is the limit a b x = lim λ ∈ Λ rep b x ( a λ ) with respectto the strong topology of B (cid:16) b H b x (cid:17) . If ξ ∈ b H a is given by the family (3.1.35) and ε > F = { b x , ..., b x n } ⊂ b X such that η b x = ( ξ b x b x ∈ F e x / ∈ F AND η = { η b x ∈ H b x } b x ∈ b X ⇒ k ξ − η k < ε k a k . (3.1.36)For any j =
1, ..., n there is λ j ∈ Λ such that µ ≥ ν ≥ λ j ⇒ (cid:13)(cid:13) rep b x (cid:0) a µ − a ν (cid:1) ξ b x (cid:13)(cid:13) < ε n k a k (3.1.37)If λ max ≥ λ AND ... AND λ max ≥ λ n then from the equations (3.1.36) and (3.1.37)it turns out that if ζ is represented by the family { a b x ξ b x ∈ H b x } b x ∈ b X then one has k a ξ − ζ k < ε .82 emark 3.1.31. In the situation of the Lemma 3.1.30 denote by rep b x ( a ) def = lim λ ∈ Λ rep b x ( a λ ) . (3.1.38) Definition 3.1.32.
Any maximal connected C ∗ -subalgebra e A π ⊂ A π is said to bea π - connected component of S . The maximal subgroup G π ⊂ G (cid:0) A π | A (cid:1) amongsubgroups G ⊂ G (cid:0) A π | A (cid:1) such that G e A π = e A π is said to be the e A π - invariantgroup of S pt . Definition 3.1.33.
Let S pt ∈ FinAlg pt , let π : b A → B ( H ) be an equivariant rep-resentation. Suppose that (cid:0) A , A π , G (cid:0) A π (cid:12)(cid:12) A (cid:1)(cid:1) is the π -disconnected infinite non-commutative covering of S , e A π is a π -connected component of S pt , and G π isthe e A π -invariant group of S pt . For any λ ∈ Λ denote by h λ : b G → G ( A λ | A ) the natural surjective homomorphism of groups. We say that S pt is π - good if thefollowing conditions hold:(a) For all λ ∈ Λ the natural *-homomorphism A λ → M (cid:16) e A π (cid:17) is injective.(b) If J ⊂ G (cid:0) A π | A (cid:1) is a set of representatives of G (cid:0) A π (cid:12)(cid:12) A (cid:1) / G π , then thealgebraic direct sum M g ∈ J g e A π is a dense subalgebra of A π .(c) For any λ ∈ Λ the restriction h λ | G π is an epimorphism, i. e. h λ ( G π ) = G ( A λ | A ) .If π is the atomic representation π a then we say e A def = e A π is a connected component of S pt and S pt is good . Definition 3.1.34.
Let S pt ∈ FinAlg pt be an algebraical finite covering category.Let π : b A → B ( H ) a representation such that S pt is π -good. A connected com-ponent e A π ⊂ A π is said to be the π - inverse noncommutative limit of S pt . The e A π -invariant group G π is said to be the π - covering transformation group of S . Thetriple (cid:16) A , e A π , G π (cid:17) is said to be the π - infinite noncommutative covering of S . We willuse the following notation lim ←− π S pt def = e A π , G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17) def = G π .83f π is the atomic representation then the subscript π will be dropped and we usethe words inverse noncommutative limit of S pt , covering transformation group of S and infinite noncommutative covering instead of π -inverse noncommutative limit of S pt , π -covering transformation group of S and π -infinite noncommutative coveringrespectively. In this case we write lim ←− S def = e A and G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) def = G . Lemma 3.1.35.
Any change of base point yields equivalent weak infinite noncommutativecoverings.Proof.
Follows from the Lemma 3.1.11 (cf. Remark 3.1.13)
Remark 3.1.36.
From the condition (b) of the Definition 3.1.34 it turns out that aconnected inverse noncommutative limit is unique up to ∗ -isomorphism. Remark 3.1.37. If b X is the spectrum of the C ∗ -inductive limit C ∗ -lim −→ λ ∈ Λ A λ thenfor all b x and λ ∈ Λ there are the natural representations rep b x : A λ → B ( H b x ) . (3.1.39)From the Lemma 3.1.30 it turns out that there are the following natural represen-tations rep b x : A → B ( H b x ) , rep b x : e A → B ( H b x ) (3.1.40)(cf. Notation (3.1.38)). Remark 3.1.38.
Below for all λ ∈ Λ we implicitly assume that A λ ⊂ B ( H ) . Simi-larly the following natural inclusions b A ⊂ B ( H ) , A π ⊂ B ( H ) , e A π ⊂ B ( H ) ,will be implicitly used. These inclusions enable us replace the Equations 3.1.20with the following equivalent system of equations ∑ g ∈ G λ z ∗ ( ga ) z ∈ A λ , ∑ g ∈ G λ f ε ( z ∗ ( ga ) z ) ∈ A λ , ∑ g ∈ G λ ( z ∗ ( ga ) z ) ∈ A λ . (3.1.41)84 .2 Infinite coverings of operator spaces Here we consider a generalization of the discussed in the Section 3.1 construc-tion.
Consider a pointed algebraical finite covering category (cf. Definition 3.1.6)given by S pt = (cid:18) { π λ : A ֒ → A λ } λ ∈ Λ , (cid:8) π µν : A µ ֒ → A ν (cid:9) µ , ν ∈ Λ ν > µ (cid:19) ,and suppose that S pt is good (cf. Definition 3.1.33). Assume that for every λ ∈ Λ (including λ min cf. 3.1.4) there is a sub-unital operator space ( X λ , Y λ ) (cf. Defini-tion 2.6.1) such that A λ ∼ = C ∗ e ( X λ , Y λ ) suppose that any π λ and π µν corresponds toa noncommutative finite-fold coverings of operator spaces (cid:16) ( X , Y ) , (( X λ , Y λ )) , G ( A λ | A ) , (cid:16) π λ X λ , π λ Y λ (cid:17)(cid:17) and (cid:16)(cid:0) X µ , Y µ (cid:1) , (( X ν , Y ν )) , G (cid:0) A µ (cid:12)(cid:12) A ν (cid:1) , (cid:16) π µν X µ , π µν Y µ (cid:17)(cid:17) respectively. Definition 3.2.2.
Consider the situation 3.2.1. We say that the category S ptop = (cid:0) { ( π X λ , π Y λ ) : ( X , Y ) → ( X λ , Y λ ) } λ ∈ Λ , n(cid:16) π ν X µ , π ν Y µ (cid:17) : (cid:0) X µ , Y µ (cid:1) → ( X ν , Y ν ) o µ , ν ∈ Λ ν > µ ! (3.2.1)is a pointed algebraical finite covering category of operator spaces . We write S ptop ∈ OSp ptop . (3.2.2)
Definition 3.2.3.
Consider the situation of 3.2.1 and 3.2.2. Let e A be the inversenoncommutative limit of S pt . Let b A = C ∗ -lim −→ λ ∈ Λ A λ and π a : b A → B ( H a ) be theatomic representation. An element e x ∈ e A is said to be subordinated to S ptop if thereis a net { x λ ∈ X λ } λ ∈ Λ ⊂ b A such that e x = lim λ ∈ Λ π a ( x λ ) (3.2.3)where the limit of (3.2.3) implies the strong limit in B ( H a ) .85 efinition 3.2.4. Let Ξ op ⊂ e A is the space of subordinated to S ptop elements. The C ∗ -norm completion of the C -linear space Ξ op is said to be the inverse noncommu-tative limit of S ptop . Remark 3.2.5.
Since the inverse noncommutative limit is a C ∗ -norm closed sub-space of C ∗ -algebra e A it has the natural structure of the operator space. Very natural choice of fundamental group is proposed in [48], where the funda-mental group of algebraic manifold is an inverse limit of finite covering groups.However this theory does not yield the fundamental group, it provides its theprofinite completion of the fundamental group. Another way is the constructionof the noncommutative universal covering using construction 3.1. This construc-tion yields fundamental group in the commutative case A = C ( X ) for describedbelow class of spaces. C ∗ -algebras Definition 3.3.1.
Let A be a connected C ∗ -algebra. Let us consider the family { π λ : A ֒ → A λ } λ ∈ Λ of all noncommutative finite-fold coverings of A . Denote by S = { π λ : A ֒ → A λ } λ ∈ Λ and suppose that there is a pointed algebraical finitecovering category category S pt = (cid:16) { π λ : A ֒ → A λ } , n π νµ o(cid:17) which is good. Theconnected infinite noncommutative covering (cid:16) A , e A , G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17) of S is said tobe the universal covering of A . We also say that e A the universal covering of A . Thegroup G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) is said to be the fundamental group of the pair (cid:16) A , n π νµ o(cid:17) . Weuse the following notation π (cid:16) A , n π νµ o(cid:17) def = G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17) . (3.3.1) Remark 3.3.2.
In the Definition 3.3.1 the algebra e A does not depend on a base-point n π νµ o up to isomorphism (cf. Remark 3.1.13). Lemma 3.3.3.
Let b g ∈ b G = lim ←− G (cid:0) A λ | A , (cid:8) π µν (cid:9)(cid:1) and let ρ µν = π µν ◦ h µ ( b g ) then thereis is the natural isomorphism π (cid:16) A , n π νµ o(cid:17) ≈ −→ π (cid:16) A , n ρ νµ o(cid:17) , g b gg b g − . (3.3.2)86 roof. Follows from the Lemma 3.1.11 (cf. Remark 3.1.12).
Here the generalization 3.3.1 is being considered here.
Definition 3.3.4.
Let ( X , Y ) be a sub-unital operator space (cf. Definition 2.6.1).Let us consider the family (( X , Y ) , (( X λ , Y λ )) , G ( A λ | A )) of all noncommutativefinite-fold coverings of ( X , Y ) . Denote by S = { π λ : A ֒ → A λ } λ ∈ Λ and supposethat there is a pointed algebraical finite covering category of operator spaces cate-gory S ptop (cf. Definition 3.2.2) given by S ptop = (cid:0) { ( π X λ , π Y λ ) : ( X , Y ) → ( X λ , Y λ ) } λ ∈ Λ , n(cid:16) π ν X µ , π ν Y µ (cid:17) : (cid:0) X µ , Y µ (cid:1) → ( X ν , Y ν ) o µ , ν ∈ Λ ν > µ ! The inverse noncommutative limit of S ptop universal covering of ( X , Y ) . If C ∗ e ( X , Y ) is the C ∗ - envelope of ( X , Y ) then the fundamental group π (cid:16) C ∗ e ( X , Y ) , n π νµ o(cid:17) is said to be the fundamental group of ( X , Y ) . We write π (cid:16) ( X , Y ) , n π νµ o(cid:17) def = π (cid:16) C ∗ e ( X , Y ) , n π νµ o(cid:17) (3.3.3) Remark 3.3.5. If e X is the universal covering of ( X , Y ) and ^ C ∗ e ( X , Y ) is the uni-versal covering of C ∗ e ( X , Y ) then there is the natural inclusion e X ⊂ ^ C ∗ e ( X , Y ) .Otherwise if π (cid:16) C ∗ e ( X , Y ) , n π νµ o(cid:17) × C ∗ e ( X , Y ) → C ∗ e ( X , Y ) is the natural actionthen π (cid:16) C ∗ e ( X , Y ) , n π νµ o(cid:17) e X = e X , hence from (3.3.3) it follows the existence ofthe natural action π (cid:16) ( X , Y ) , n π νµ o(cid:17) × e X → e X . (3.3.4) Remark 3.3.6.
The both notions of the universal covering and the fundamentalgroup of a C ∗ -algebra are specializations of the universal covering and the funda-mental group of an operator space. Definition 3.4.1.
Let S pt = (cid:0) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:1) ∈ FinAlg pt be a pointed alge-braical finite covering category. Let π : C ∗ -lim −→ λ ∈ Λ A λ → B (cid:16) b H (cid:17) be an equivariant87epresentation such that S pt is π -good (cf. Definition 3.1.33). Let (cid:16) A , e A π , G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17) be an infinite noncommutative covering of S pt . Let K (cid:16) e A π (cid:17) be the Pedersen’sideal of e A π (cf. Definition D.1.33). We say that e A π allows the inner product if forany λ ∈ Λ , and e a , e b ∈ K (cid:16) e A π (cid:17) the series a λ = ∑ g ∈ ker ( G ( e A π | A ) → G ( A λ | A ) ) g (cid:16)e a ∗ e b (cid:17) is convergent with respect to the strict topology (cf. Definition D.1.12)of M (cid:16) e A π (cid:17) and a λ ∈ A λ . Also we say that (cid:16) A , e A π , G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17) allows the inner product . Remark 3.4.2.
Let S pt = (cid:0) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:1) ∈ FinAlg pt be a pointed alge-braical finite covering category. Let (cid:16) A , e A π , G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17) be a weak infinite non-commutative covering of S . Let K (cid:16) e A π (cid:17) be the Pedersen’s ideal of e A π . If there is λ ∈ Λ such that for any e a , e b ∈ K (cid:16) e A π (cid:17) such that c λ = ∑ g ∈ ker ( G ( e A π | A ) → G ( A λ | A )) g (cid:16)e a ∗ e b (cid:17) ∈ A λ then for any λ < λ one has c λ = ∑ g ∈ ker ( G ( e A π | A ) → G ( A λ | A ) ) g (cid:16)e a ∗ e b (cid:17) = ∑ g ∈ G ( A λ | A λ ) gc λ ∈ A λ . Remark 3.4.3. If S pt allows inner product then K (cid:16) e A π (cid:17) is a pre-Hilbert A modulesuch that the inner product is given by De a , e b E A = ∑ g ∈ G ( e A π | A ) g (cid:16)e a ∗ e b (cid:17) ∈ A where the above series is convergent with respect to the strict topology of M (cid:16) e A π (cid:17) .The completion of K (cid:16) e A π (cid:17) with respect to a norm k e a k = q kh e a , e a i A k (3.4.1)88s a Hilbert A -premodule. Denote by X A this completion. The ideal K (cid:16) e A π (cid:17) is aleft e A π -module, so X A is also e A π -module. Sometimes we will write e A π X A instead X A . Moreover since K (cid:16) e A π (cid:17) is A -bimodule and e A π -bimodule e A π X A is also A -bimodule and e A π -bimodule. Since the given by (3.4.1) norm exceeds the C ∗ -normof e A π there is the natural inclusion e A π X A ⊂ e A π (3.4.2) Definition 3.4.4.
Let S pt = (cid:0) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:1) ∈ FinAlg pt and S pt allowsinner product (with respect to π ) then K (cid:16) e A π (cid:17) then we say that given by theRemark 3.4.3 A -Hilbert module e A π X A corresponds to e A π . Definition 3.4.5.
Let S pt = (cid:0) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:1) ∈ FinAlg pt and let (cid:16) A , e A π , G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17) be a weak infinite noncommutative covering of S pt . Suppose e A π allows innerproduct and A -Hilbert module e A π X A corresponds to e A π . Suppose ρ : A → B ( H ) is a representation and e ρ = e A π X A − Ind A e A π ρ : e A π → B (cid:16) e H (cid:17) is given by (D.5.3),i.e. e ρ is the induced representation (cf. Definition D.5.5) The representation e ρ : e A π → B (cid:16) e H (cid:17) is said to be induced by ρ . We also say that e ρ is induced by (cid:16) ρ , (cid:16) A , e A π , G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17)(cid:17) . Remark 3.4.6. If ρ is faithful, then e ρ is faithful. Remark 3.4.7.
There is the action of G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17) on e H which comes from thenatural action of G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17) on the e A π -bimodule K (cid:16) e A π (cid:17) . If the representation e A → B (cid:16) e H (cid:17) is faithful then an action of G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17) on e A π is given by ( g e a ) ξ = g (cid:16)e a (cid:16) g − e ξ (cid:17)(cid:17) ; ∀ g ∈ G , ∀ e a ∈ e A π , ∀ e ξ ∈ e H . (3.4.3) Remark 3.4.8. If e ρ : e A π → B (cid:16) e H (cid:17) is induced by (cid:16) ρ , (cid:16) A , e A π , G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17)(cid:17) then e H is the completion of the pre-Hilbert space K (cid:16) e A π (cid:17) ⊗ A H with given by (cid:16)e a ⊗ ξ , e b ⊗ η (cid:17) e H = (cid:16) ξ , De a , e b E A η (cid:17) H ∀ e a ⊗ ξ , e b ⊗ η ∈ K (cid:16) e A π (cid:17) ⊗ H (3.4.4)89calar product. The action e A π ⊗ e H → e H corresponds to the completion of thefollowing action e A π × (cid:16) K (cid:16) e A π (cid:17) ⊗ H (cid:17) → K (cid:16) e A π (cid:17) ⊗ H ; (cid:16)e a , e b ⊗ ξ (cid:17) e a e b ⊗ ξ . (3.4.5)If e H is the Hilbert norm completion of K (cid:16) e A π (cid:17) ⊗ H then the action (3.4.5) uniquelydefines an action e A π × e H → e H , (3.4.6)so there is the inclusion e A π ⊂ B (cid:16) e H (cid:17) . (3.4.7) Let H λ be a Hilbert completion of A λ ⊗ A H which is constructed in thesection 2.3. There is the natural inclusion of pre-Hilbert spaces K (cid:16) e A π (cid:17) ⊗ A H = K (cid:16) e A π (cid:17) ⊗ A λ ( A λ ⊗ A H ) ֒ → K (cid:16) e A π (cid:17) ⊗ A λ H λ . (3.4.8)such that K (cid:16) e A π (cid:17) ⊗ A H is dense in K (cid:16) e A π (cid:17) ⊗ A λ H λ with respect to pre-Hilbertnorm, i.e. the inclusion 3.4.8 induces the isomorphism of Hilbert completions. Forall λ ∈ Λ there is the action B ( H λ ) × (cid:16) K (cid:16) e A π (cid:17) ⊗ A λ H λ (cid:17) → K (cid:16) e A π (cid:17) ⊗ A λ H λ , ( a λ , ( e a ⊗ ξ )) e a ⊗ a λ ξ ,and taking into account that K (cid:16) e A π (cid:17) ⊗ A λ H λ is dense in e H one has the followingaction B ( H λ ) × e H → e H (3.4.9)and the corresponding inclusion B ( H λ ) ⊂ B (cid:16) e H (cid:17) . (3.4.10)In particular one has the natural inclusion B ( H ) ⊂ B (cid:16) e H (cid:17) . (3.4.11)90oth inclusions A λ ⊂ B ( H λ ) and (3.4.10) yield the inclusion A λ ⊂ B (cid:16) e H (cid:17) . (3.4.12)There is the natural action ( A λ ⊗ A B ( H )) × e H → e H given by (( a ⊗ b ) , ξ ) ab ξ . (3.4.13)where both a ∈ A λ and b ∈ B ( H ) are regarded as elements of B (cid:16) e H (cid:17) . The givenby (3.4.13) action yields the following inclusion A λ ⊗ A B ( H ) ⊂ B (cid:16) e H (cid:17) . (3.4.14)There is the natural action (cid:16) e A π ⊗ A B ( H ) (cid:17) × e H → e H given by (( e a ⊗ b ) , ξ ) ab ξ (3.4.15)where both e a ∈ e A π and b ∈ B ( H ) are regarded as elements of B (cid:16) e H (cid:17) . The givenby (3.4.13) action yields the following inclusion e A π ⊗ A B ( H ) ⊂ B (cid:16) e H (cid:17) . (3.4.16) Definition 3.5.1.
Let ( A , H , D ) be a spectral triple, and let A be the C ∗ -normcompletion of A with the natural representation A → B ( H ) . Let S pt = (cid:0) { π λ : A ֒ → A λ } , (cid:8) π µν (cid:9)(cid:1) ∈ FinAlg pt (3.5.1)be a pointed algebraic finite covering category. Suppose that for any λ ∈ Λ thereis a spectral triple ( A λ , H λ , D λ ) , such that(a) ( A λ , H λ , D λ ) is the ( A , A λ , G ( A λ | A )) -lift of ( A , H , D ) .(b) A λ is the C ∗ -norm completion of A λ .(c) There is an equivariant representation π : C ∗ -lim −→ λ ∈ Λ A λ → B (cid:16) b H (cid:17) such that S pt is π -good.(d) For any µ > ν the spectral triple (cid:0) A µ , H µ , D µ (cid:1) is a (cid:0) A ν , A µ , G (cid:0) A µ (cid:12)(cid:12) A ν (cid:1)(cid:1) -liftof ( A ν , H ν , D ν ) . 91e say that S ( A , H , D ) = { ( A λ , H λ , D λ ) } λ ∈ Λ (3.5.2)is a weakly coherent set of spectral triples . We write S ( A , H , D ) ∈ WCohTriple . Definition 3.5.2.
A weakly coherent set of spectral triples S ( A , H , D ) is said to be a coherent set of spectral triples if π -algebraical infinite noncommutative covering of S (cid:16) A , e A π , G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17) which allows the inner product (cf. Definition 3.4.1). Wewrite S ( A , H , D ) ∈ CohTriple . From the inclusions [ D λ , A λ ] ∈ B ( H λ ) and (3.4.10) one has the natural inclusion [ D λ , A λ ] ⊂ B (cid:16) e H (cid:17) (3.5.3)For any s ∈ N the given by (3.4.9) action B ( H λ ) × e H → e H naturally induces anaction B (cid:16) H s λ (cid:17) × e H s → e H s hence there is an inclusion B (cid:16) H s λ (cid:17) ⊂ B (cid:16) e H s (cid:17) . (3.5.4)If π s λ : A λ ֒ → B (cid:0) H s λ (cid:1) is given by (E.2.1), (E.2.2) then from (3.5.4) it turns out thatthere is the inclusion π s λ ( A λ ) ⊂ B (cid:16) e H s (cid:17) (3.5.5)If Ω D is the module of differential forms associated with the spectral triple ( A , H , D ) (cf. Definition E.3.5) then from Ω D ⊂ B ( H ) and the given by (3.4.16) e A π ⊗ A B ( H ) ⊂ B (cid:16) e H (cid:17) inclusion it follows that there is the following inclusion e A π ⊗ A Ω D ⊂ B (cid:16) e H (cid:17) . (3.5.6) Definition 3.5.3.
Let us consider the situation of the Definition 3.5.2. Let Ω D bethe module of differential forms associated with the spectral triple ( A , H , D ) (cf.Definition E.3.5). An element e a ∈ e A π is said to be S ( A , H , D ) - smooth if the followingconditions hold: 92a) The element e a lies in the Pedersen’s ideal of e A π , i.e. e a ∈ K (cid:16) e A π (cid:17) .(b) For any λ ∈ Λ the series a λ = ∑ g ∈ ker ( G ( e A π | A ) → G ( A λ | A ) ) g e a is convergent with respect to strict topology of M (cid:16) e A π (cid:17) and a λ ∈ A λ wherethe inclusion A λ ⊂ A λ ⊂ M (cid:16) e A π (cid:17) is implied.(c) If for any λ ∈ Λ and s ∈ N the representation π s λ : A λ ֒ → B (cid:0) H s λ (cid:1) is givenby (E.2.1), (E.2.2) and inclusion π s λ ( A λ ) ⊂ B (cid:16) e H s (cid:17) (cf. (3.5.5)) is impliedthen the net (cid:8) π s λ ( a λ ) (cid:9) λ ∈ Λ is convergent with respect to the strong topologyof B (cid:16) e H s (cid:17) .(d) If for any λ ∈ Λ the given by (3.5.3) inclusion [ D λ , A λ ] ⊂ B (cid:16) e H (cid:17) is impliedthen the net { [ D λ , a λ ] } λ ∈ Λ is convergent with respect to the strong topologyof B (cid:16) e H s (cid:17) . Moreover one haslim λ ∈ Λ [ D λ , a λ ] ∈ K (cid:16) e A π (cid:17) ⊗ A Ω D ⊂ B (cid:16) e H (cid:17) where left part of the above equation means the limit with respect to thestrong topology of B (cid:16) e H (cid:17) , and the right part implies the inclusion K (cid:16) e A π (cid:17) ⊗ A Ω D ⊂ e A π ⊗ A Ω D ⊂ B (cid:16) e H (cid:17) (cf. (3.5.6)).Denote by e a s def = lim λ ∈ Λ π s λ ( a λ ) ∈ B (cid:16) e H s (cid:17) in sense the strong convergence of B (cid:16) e H s (cid:17) , and denote by e W ∞ the space of smooth elements. There is a subalgebra e A smooth ⊂ e A generated by smooth elements. For any s > k·k s on e A smooth given by k e a k s def = k e a s k = lim λ ∈ Λ k π s λ ( a λ ) k . (3.5.7) Definition 3.5.5.
The completion of e A smooth in the topology induced by the semi-norms k·k s is said to be a smooth algebra of the coherent set (3.5.2) of spectral triples.This algebra is denoted by e A . We say that the set of spectral triples is good if e A isdense in e A . 93 .5.6. For any e a ∈ e W ∞ we denote by e a D = lim λ ∈ Λ [ D λ , a λ ] = k ∑ j = e a jD ⊗ ω j ∈ K (cid:16) e A (cid:17) ⊗ A Ω D (3.5.8)where a λ def = ∑ g ∈ ker ( G ( e A π | A ) → G ( A λ | A ) ) e a ∈ A λ . If H ∞ = T ∞ n = Dom D n then for any e a ⊗ ξ ∈ e W ∞ ⊗ A H ∞ we denote by e D ( e a ⊗ ξ ) def = k ∑ j = e a jD ⊗ ω j ( ξ ) + e a ⊗ D ξ ∈ K (cid:16) e A (cid:17) ⊗ A H , (3.5.9)i.e. e D is a C -linear map from W ∞ ⊗ A H ∞ to K (cid:16) e A (cid:17) ⊗ A H . The space e W ∞ ⊗ A H ∞ is dense in e H , hence the operator e D can be regarded as an unbounded operatoron e H . Definition 3.5.7.
Let S pt ∈ FinAlg pt is given by (3.5.1). Let (cid:16) A , e A , G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17) be an infinite noncommutative covering of S . Let (3.5.2) be a good coherentset of spectral triples. Let e D be given by (3.5.9). We say that (cid:16) e A , e H , e D (cid:17) is a (cid:16) A , e A , G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17) - lift of ( A , H , D ) . Also we say that (cid:16) e A , e H , e D (cid:17) is the limit of thecoherent set S ( A , H , D ) = { ( A λ , H λ , D λ ) } λ ∈ Λ . Let us consider the situation of the Definition 3.5.2. If e π : e A → B (cid:16) e H (cid:17) isa faithful representation then for every λ ∈ Λ the injective *-homomorphisms A λ ֒ → M (cid:16) e A (cid:17) and M (cid:16) e A (cid:17) induce the faithful representation π λ : A λ ֒ → B (cid:16) e H (cid:17) .If e D is an unbounded operatoron e H then similarly to (E.2.1) and (E.2.2) we candefine representations e π s λ : A λ ֒ → B (cid:16) e H s (cid:17) = B (cid:16) e H (cid:17) .We imply that e π s λ ( a ) is bounded for any a ∈ A λ . Using the Proposition E.3.3 onecan define *-representation e π λ : Ω ∗ A λ → B (cid:16) e H (cid:17) , e π λ ( a da ... da n ) = a h e D , a i ... h e D , a n i ∀ a j ∈ A λ . (3.5.10)of the reduced universal algebra Ω ∗ A λ (cf. (E.3.1)).94 efinition 3.5.8. An unbounded operator e D on e H is said to be (cid:16) A , e A , G (cid:16) e A (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17) - axiomatic lift of the weakly coherent set of spectral triples S ( A , H , D ) = { ( A λ , H λ , D λ ) } λ ∈ Λ (cf. Definition 3.5.1) if following conditions hold:(a) If π s λ : A λ ֒ → B (cid:0) H s λ (cid:1) is given by (E.2.1) and (E.2.2) then for any a ∈ A λ onehas k e π s λ ( a ) k = k π s λ ( a ) k .(b) If π λ : Ω ∗ A λ → B ( H λ ) , π λ ( a da ... da n ) = a [ D λ , a ] ... [ D λ , a n ] ∀ a j ∈ A λ . (3.5.11)and e π λ is given by (3.5.10) then ker π λ = ker e π λ , where both ker π λ andker e π λ are graded ideals given by the equation (E.3.2). Remark 3.5.9. If J = ker π λ and e J = ker e π λ are graded two-sided ideals given by(E.3.2) then from J = e J it follows that J + dJ = e J + d e J ,so one has Ω ∗ A λ / ( J + dJ ) = Ω ∗ A λ / (cid:16)e J + d e J (cid:17) (3.5.12)and taking into account (E.3.4) we conclude that Ω D λ = Ω e D .Thus the representation A λ → B ( H λ ) and the operator D λ yield a graded involu-tive algebra Ω D λ which coincides with the algebra Ω e D given by the representation A λ → B (cid:16) e H (cid:17) and the operator e D . Definition 3.5.10.
Let us consider the situation of the Definition 3.5.8. An element e a ∈ e A is said to be S ( A , H , D ) - axiomatically smooth if the following conditions hold:(a) The element e a lies in the Pedersen’s ideal of e A , i.e. e a ∈ K (cid:16) e A (cid:17) .(b) For any λ ∈ Λ the series a λ = ∑ g ∈ ker ( G ( e A | A ) → G ( A λ | A ) ) g e a is convergent with respect to strict topology of M (cid:16) e A (cid:17) and a λ ∈ A λ for any λ ∈ Λ . 95c) If e π s : e A → B (cid:16) e H s (cid:17) is constructed similarly to equations (E.2.1) and (E.2.2)from the representation e π : e A → B (cid:16) e H (cid:17) and the operator e D , then for all s ∈ N following condition holds k e π s ( e a ) k < ∞ . (3.5.13) Definition 3.5.11.
Let e A ax smooth ⊂ e A is a generated by axiomatically smooth el-ements involutive algebra. The completion of e A ax smooth in the topology inducedby the seminorms k·k s = k e π s ( · ) k is said to be a axiomatically smooth algebra of thecoherent set (3.5.2) of spectral triples. This algebra is denoted by e A ax . We say thatthe set of spectral triples is axiomatically good if e A ax is dense in e A . Remark 3.5.12.
It is not known whether the axiomatic approach is equivalent tothe constructive one. Also it is not known whether the axiomatic approach yieldsthe unique result. 96 hapter 4
Coverings of commutative C ∗ -algebras Lemma 4.1.1.
Let X be a locally compact Hausdorff space. Let f ′ , f ′′ ∈ C ( X ) be suchthat supp f ′ ⊂ supp f ′′ , and let U = { x ∈ X | f ′′ ( x ) = } . Suppose f : X → C begiven by f ( x ) = ( f ′ ( x ) f ′′ ( x ) x ∈ U x / ∈ U (4.1.1) One has f ∈ C ( X ) if and only if following conditions hold:(a) For any x ∈ supp f ′′ \ U and any net { x α ∈ U } α ∈ A such that lim α x α = x following condition holds lim α ∈ A f ′ ( x α ) f ′′ ( x α ) = (b) For any ε > there is a compact V ⊂ X such thatx ∈ U ∩ ( X \ V ) ⇒ (cid:12)(cid:12)(cid:12)(cid:12) f ′ ( x ) f ′′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) < ε . Proof. If x ∈ U then f ′′ ( x ) = f is continuous at x . If x ∈ X \ supp f ′′ thenthere is an open neighborhood of x and f ′′ ( X \ supp f ′′ ) = f ( X \ supp f ′′ ) = { } ,hence f is continuous at x . If x ∈ supp f ′′ \ U then from the equation (4.1.1) andthe condition (a) it turns out that f is continuous at x . Condition (b) means that f ∈ C ( X ) (cf. Definition A.1.21). 97 efinition 4.1.2. If f ′ , f ′′ ∈ C ( X ) then f ′ is a multiple of f ′′ if there is f ∈ C ( X ) such that f ′ = f f ′′ and f ′′ ( x ) = ⇒ f ( x ) =
0. Denote by div (cid:0) f ′ , f ′′ (cid:1) def = f ∈ C ( X ) . (4.1.2) Remark 4.1.3. If f ′ , f ′′ ∈ C ( X ) then f ′ is a multiple of f ′′ if and only if f ′ and f ′′ satisfy to the Lemma 4.1.1. Let X be a locally compact Hausdorff space, and let f ε be given by (3.1.19).For every f ∈ C ( X ) + and and ε > f ε ( f ) , f satisfies to the Lemma4.1.1, i.e. f ε ( f ) ∈ C ( X ) is a multiple of f . If F ε ( f ) def = div ( f ε ( f ) , f ) (4.1.3)then one has k f − F ε ( f ) f k ∼ = k f − f ε ( f ) k ≤ ε . (4.1.4)The set supp f ε is compact, so from supp f ε = supp F ε ( f ) it turns out that F ε ( f ) ∈ C c ( X ) (4.1.5)(cf. Definition A.1.20). Lemma 4.1.5. If X is a locally compact, Hausdorff space then for any x ∈ X and anyopen neighborhood U ⊂ X of x there is a continuous function a : X → [
0, 1 ] such thatfollowing conditions hold: • a ( X ) = [
0, 1 ] . • a ( x ) = . • supp a ⊂ U . • There is an open neighborhood
V ⊂ U of x which satisfies to the following conditiona ( V ) = { } . (4.1.6) Proof.
From the Exercise A.1.12 it turns out that that X is completely regular (cf.Definition A.1.10), i.e. there is a continuous function a : X → [
0, 1 ] such that a ( x ) = a ( X \ U ) = { } . The set V = (cid:8) x ∈ X | a ( x ) > (cid:9) is open. If f : R → R is a continuous function given by f ( t ) = t ≤ t − < t ≤ t > then f ( a ) : X → [
0, 1 ] satisfies to conditions of this lemma.98 orollary 4.1.6. Let X be a locally compact, Hausdorff space. For any x ∈ X , and anyopen neighborhood U of x there is an open neighborhood V of x such that the closure of V is a subset of U .Proof. If a satisfies to the Lemma 4.1.5 then set V = { x ∈ X | a ( x ) > } is openand the closure of V is a subset of U . Let X be a locally compact, Hausdorff space, x ∈ X and f is given by theLemma 4.1.5 then we denote by f x def = f . (4.1.7)Let p : e X → X be a covering. Let e x ∈ e X , and let e U be an open neighborhoodof e x such that the restriction p | e U : e U ≈ −→ U def = p (cid:16) e U (cid:17) is a homeomorphism. Since e X is locally compact and Hausdorff there is e f ∈ C c (cid:16) e X (cid:17) and open subset e V suchthat { e x } ⊂ e V ⊂ e U , e f (cid:16) e V (cid:17) = e f (cid:16) e X (cid:17) = [
0, 1 ] and supp e f ⊂ e U . We write e f e x def = e f . (4.1.8) Lemma 4.2.1.
Suppose X is a locally compact Hausdorff space, and {V α } α ∈ A is a count-able family of of open subsets of X such that X = ∪V α . If for all α ∈ A one has: • the set V α is connected, • the closure of V α is compactthen for any x ∈ X there is a finite or countable sequence U $ ... $ U n $ ... of connectedopen subsets of X such that • x ∈ U . • For any n ∈ N the closure of U n is compact. • ∪ U n = X .Proof. Let us define a finite or countable sequence U $ ... $ U n $ ... by induction.1. Let us select a α ∈ A such that x ∈ V α , and let U = V α .99. If U n is already defined then we looking for V α such that V α
6⊂ U n ; V α ∩ U n = ∅ . (4.2.1)It there is no V α which satisfies to (4.2.1) then the sequence is competed.Otherwise one sets U n + = U n ∪ V α .Clearly that for every n ∈ N the set U n is an open connected and have the compactclosure. If X 6 = ∪ U n then X = ∪ U n G ∪ α ∈ A V α ; where V α \ ∪ U n = ∅ ; ∀ α ∈ A ,i.e. X is not connected. So one has X = ∪ U n . Lemma 4.2.2. If X is a locally compact, connected, locally connected, Lindelöf (cf. Defini-tion A.1.28), Hausdorff space then there is a family {V α } α ∈ A of open subsets of X whichsatisfies to conditions of the Lemma 4.2.1.Proof. The space X is a locally compact, so for any point there is an open neigh-borhood V x such that the closure of V x is compact. Moreover one can assumethat V x is connected since X is locally connected. The space is X is Lindelöf, sothere is a finite or countable family {V α } α ∈ A such that {V α } α ∈ A ⊂ {V x } x ∈X and X = ∪ α ∈ A V α . If consider the situation of the Lemma 4.2.1 and suppose that X is para-compact, then from the Theorem A.1.25 it turns out that there is a partition ofunity ∑ λ ∈ Λ f λ dominated by {V λ } λ ∈ Λ . From the proof of the Lemma 4.2.1 it turns out that forany n ∈ N there is a finite subset Λ n ⊂ Λ such that U n = ∪ λ ∈ Λ n V λ . If f n = ∑ λ ∈ Λ n f λ (4.2.2)then there is a point-wise limit 1 C b ( X ) = lim n → ∞ f n . Lemma 4.2.4.
If a countable indexed family of subsets {U α } α ∈ A of topological space X islocally finite (cf. Definition A.1.22) and a subspace Y of X is compact then there is finitelymany α such that U α ∩ Y 6 = ∅ . roof. Denote by A ′ = { α ∈ A | U α ∩ Y 6 = ∅ } and for any α ∈ A ′ we select x α ∈ U α ∩ Y . If A ′ is not finite then since Y is compact then there is an infinite directed subset A ′′ ⊂ A ′ such that the net { x α } α ∈ A ′′ is convergent. If y = lim α ∈ A ′′ x α then any neighborhood of y intersectswith infinitely many sets U α . It is a contradiction with the Definition A.1.22. Corollary 4.2.5.
Let X be a topological space, and let C b ( X ) = ∑ α ∈ A f α be a partition of unity (cf. A.1.23). For any compact subset Y ⊂ X there is a finite subset A Y ⊂ A such that ∑ α ∈ A Y f α ( x ) = ∀ x ∈ Y . (4.2.3) Proof.
From the Definition A.1.23 it follows that the family { supp f α } α ∈ A is locallyfinite, from the Lemma 4.2.4 it turns out that the set A Y def = { α ∈ A | supp f α ∩ Y 6 = Y } is finite.
Definition 4.2.6.
In the situation of the Corollary 4.2.5 we say that the finite sum ∑ α ∈ A Y f α = m ∑ i = f α j is a covering sum for Y . The notion of "regular covering" has no any good noncommutative generaliza-tion. So instead the "regular covering" we will use "transitive covering" which isexplained below.
Definition 4.3.1.
A covering p : e X → X is said to be transitive if X is connectedand for every x ∈ X the group of covering transformations (cf. Definition A.2.3) G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) transitively acts on p − ( x ) , i.e. ∀ e x ∈ e X G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) e x = p − ( p ( e x )) . (4.3.1)101 emark 4.3.2. From the Corollary A.2.17 it turns out that any regular covering istransitive.
Lemma 4.3.3.
Let X be a connected, locally compact, Hausdorff space, and let p : e X → X be a covering of connected spaces. If g ∈ Homeo (cid:16) e X (cid:17) is such that p ◦ g = p then onehas ∃ e x ∈ e X g e x = e x ⇔ ∀ e x ∈ e X g e x = e x . Proof.
The implication ⇐ is evident. Suppose that ∃ e x ∈ e X g e x = e x . There areopen neighborhoods e U ′ and e U ′′ of e x and g e x such that e U ′ ∩ e U ′′ = ∅ . Otherwisethere is open neighborhood e V of e x such that g e V ⊂ e U ′′ . It follows that g e y = e y forall e y ∈ e U ′ ∩ e V . Hence the set n e y ∈ e X (cid:12)(cid:12)(cid:12) g e y = e y o is open, or equivalently the set f W = n e y ∈ e X (cid:12)(cid:12)(cid:12) g e y = e y o is closed. Suppose that e x ∈ e X is such that e x = g e x . There is a connected openneighborhood e U of e x such that p | e U is injective. On the other hand the restriction p | e U can be regarded as a homeomorphism from e U onto p (cid:16) e U (cid:17) . Otherwise there isan open neighborhood e V of e x such that g e V ⊂ e U . Taking into account p ◦ g = p onehas ∀ e y ∈ e U ∩ e V g e y = (cid:0) p | e U (cid:1) − ◦ p ◦ g ( e y ) = (cid:0) p | e U (cid:1) − ◦ p ( e y ) = e y , so f W is open.From ∃ e x ∈ e X g e x = e x it turns out that f W 6 = ∅ and taking into account that X isconnected we conclude that f W = e X . It equivalently means that ∀ e x ∈ e X g e x = e x . Corollary 4.3.4.
Let X be a connected, locally compact, Hausdorff space, and let p : e X →X be a covering of connected spaces. If there is e x ∈ e X such that the mapG (cid:16) e X | X (cid:17) ≈ −→ p − p ( e x ) ; g g e x . is biective then p is a transitive covering. orollary 4.3.5. If p : e X → X is a transitive covering then there is the natural homeo-morphism
X ∼ = e X / G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) . Remark 4.3.6.
The Corollary 4.3.5 can be regarded as a generalization of the The-orem A.2.16.
Lemma 4.3.7.
Consider a commutative triangle of connected topological spaces and con-tinuous maps e X e X X pp p Suppose X is a locally compact, locally connected, Hausdorff space and p , p are cover-ings. For all g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) the following condition holds ∃ e x ∈ e X p ◦ g ( e x ) = p ( e x ) ⇔ ∀ e x ∈ e X p ◦ g ( e x ) = p ( e x ) . Proof.
The implication ⇐ is evident. Suppose that ∃ e x ∈ e X p ( g e x ) = p ( e x ) .There are open neighborhoods e U ′ and e U ′′ of p ( e x ) and p ( e gx ) such that e U ′ ∩ e U ′′ = ∅ . Otherwise there is open neighborhood e V of e x such that p (cid:16) g e V (cid:17) ⊂ e U ′′ . Itfollows that p ( g e y ) = p ( e y ) for all e y ∈ e U ′ ∩ e V . Hence the set n e y ∈ e X (cid:12)(cid:12)(cid:12) p ( g e y ) = p ( e y ) o is open, or equivalently the set f W = n e y ∈ e X (cid:12)(cid:12)(cid:12) p ( g e y ) = p ( e y ) o is closed. Suppose that e x ∈ e X is such that e x = g e x . There is an open neigh-borhood e U ′ of e x such that p | e U ′ is injective. There is an open neighborhood e U ′ of p ( e x ) such that p | e U ′ is injective. Since X is locally connected there is aconnected open neighborhood U of p ( e x ) such that U ⊂ p (cid:16) e U ′ (cid:17) ∩ p (cid:16) e U ′ (cid:17) . If e U = e U ′ ∩ p − ( U ) and e U = e U ′ ∩ p − ( U ) then both p | e U and p | e U are injectiveand may be regarded as homeomorphisms from both e U and e U onto U . For any e y ∈ e U one has p ( g e y ) = (cid:16) p | e U (cid:17) − ◦ p ( g e y ) .103n the other hand from g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) it follows that p ( g e y ) = p ( e y ) , hencethe following condition holds p ( g e y ) = (cid:16) p | e U (cid:17) − ◦ p ( e y ) = p ( e y ) .The above equation is true for all e y ∈ e U so the set f W is open. From ∃ e x ∈ e X p ◦ g ( e x ) = p ( e x ) it turns out that f W = ∅ and taking into account that X isconnected we conclude that f W = e X . It equivalently means that ∀ e x ∈ e X p ◦ g ( e x ) = p ( e x ) . Corollary 4.3.8.
Consider the situation of the Lemma 4.3.7. If the map p is surjective andthe covering p is transitive then map p is a transitive covering.Proof. From the Corollary A.2.6 it follows that p is a covering. There is the naturalhomomorphic inclusion G (cid:16) e X (cid:12)(cid:12)(cid:12) e X (cid:17) ⊂ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) . Let e x ∈ e X be a point and x = p ( e x ) . If e x ∈ e X is such that p ( e x ) = e x then p − ( x ) = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) e x because the covering e p is transitive. It follows that p − ( e x ) = G e x e x where G e x = n g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17)(cid:12)(cid:12)(cid:12) p ( g e x ) = p ( e x ) o .However from the Lemma 4.3.7 it follows that G e x = G (cid:16) e X (cid:12)(cid:12)(cid:12) e X (cid:17) , hence p − ( e x ) = G (cid:16) e X (cid:12)(cid:12)(cid:12) e X (cid:17) e x . Definition 4.3.9.
Let X be a connected, locally connected, locally compact, Haus-dorff space. Let {U α } α ∈ A be a finite or countable family of connected open sub-sets of X , such that X = ∪ U α and for all α the closure of U α is compact. Anordered finite subset ( U α , ..., U α n ) of the family {U α } is said to be an {U α } - path if U α j ∩ U α j + = ∅ for all j =
1, ..., n − Lemma 4.3.10.
Consider the situation of the Definition 4.3.9. If the space X is Lindelöfthen for any U α ′ , U α ′′ ∈ X there is a {U α } -path ( U α , ..., U α n ) such that U α = U α ′ and U α n = U α ′′ .Proof. If x ′ ∈ U α ′ and x ′′ ∈ U α ′′ then from the Lemma 4.2.1 it follows that there isthe sequence of connected compact sets (cid:8) U j (cid:9) j ∈ N such that x ′ ∈ U and X = ∪ U j .104et n ∈ N be such that x ′′ ∈ U n and x ′′ / ∈ U n − . From the proof of the Lemma4.2.1 it turns out that there is a path (cid:16) U α ′ , ..., U α ′ k (cid:17) such that x ′ lies in the closureof U α and x ′′ lies in the closure of U α k The path {U α } -path (cid:16) U α ′ , U α ′ , ..., U α ′ k , U α ′′ (cid:17) satisfies to this lemma. Definition 4.3.11.
If both p ′ = (cid:16) U α ′ , ..., U α ′ k (cid:17) and p ′′ = (cid:16) U α ′′ , ..., U α ′′ l (cid:17) are pathssuch that U α ′ k = U α ′′ then the path p = (cid:16) U α ′ , ..., U α ′ k , U α ′′ , ..., U α ′′ l (cid:17) is said to be the composition of p ′ and p ′′ . We write p ′ ◦ p ′′ def = p . (4.3.2) Let X be a connected, locally connected, second-countable, locally compactHausdorff space. Let p : e X → X is a transitive covering such that G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) iscountable. There is a countable set {U α } α ∈ A set of open connected subsets of X such that following conditions hold: • X = ∪ α ∈ A U α . • The closure of U α is a compact set for all α ∈ A . • U α is evenly covered by p . • The family {U α } is locally finite (cf. Definition A.1.22).From the Theorem A.1.25 it turns out that there is a partition of unity ∑ α ∈ A a α ( x ) = {U α } α ∈ A . Consider the situation 4.3.12. For any α ∈ A we selectan open connected set e U α ∈ e X . The family n g e U α o ( g , α ) ∈ G ( e X | X ) × A is locally finiteand e X = [ ( g , α ) ∈ G ( e X | X ) × A g e U α Definition 4.3.13.
In the above situation we say that the family {U α } α ∈ A is compli-ant with p , and the family n g e U α o ( g , α ) ∈ G ( e X | X ) × A is the p - lift of {U α } α ∈ A . Remark 4.3.14.
Sometimes we denote by f A def = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × A and n e U e α o e α ∈ f A def = n g e U α o ( g , α ) ∈ G ( e X | X ) × A . (4.3.3)105 .3.15. Let X be a connected, locally connected, second-countable, Hausdorffspace, and let p : e X → X be a transitive covering, such that e X is connected.Consider a described in 4.3.12 family {U α } α ∈ A of open subsets of X . Denote by n e U e α o e α ∈ f A the p -lift of {U α } α ∈ A (cf. Definition 4.3.13 and Equation (4.3.3)). Definition 4.3.16.
Consider the situation 4.3.15. If e p = (cid:16) e U e α , ..., e U e α n (cid:17) is a n e U e α o -path then clearly p = (cid:16) p (cid:16) e U e α (cid:17) , ..., p ( U e α n ) (cid:17) is a {U α } -path. We say that p is the p - descend of e p and we write p def = desc p e p . (4.3.4) Lemma 4.3.17. If p = ( U α , ..., U α n ) is a {U α } -path then for any e U e α ′ ∈ n e U e α o such thatp (cid:16) e U e α ′ (cid:17) = U α there is the unique n e U e α o -path e p such that p = desc p e p and the firstelement of e p is e U e α ′ .Proof. From the condition of this lemma the first element of e p is e U e α ′ , other elementswill be constructed by the induction. Suppose that we already have j elements e U e α , ..., e U e α j of the path e p . If e U ′ ∈ n e U e α o is such that p (cid:16) e U ′ (cid:17) = U α j + then from U α j ∩ U α j + = ∅ it turns out that there is the unique g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such that g e U ′ ∩ e U e α j = ∅ . We set g e U ′ as j + th element of e p , i.e. U e α j + def = g e U ′ . In result thepath e p def = (cid:16) e U e α , ..., U e α n (cid:17) satisfies to the requirements of this lemma. The above construction is unique so e p is uniquely defined by the conditions of this lemma. Definition 4.3.18.
The given by the Lemma 4.3.17 path e p is said to be the p - U e α ′ - lift of p . We write lift U e α ′ p p def = e p . (4.3.5) In the situation 4.3.15 select e x ∈ e X and e U ∈ n e U e α o is such that e x ∈ e U .Let U = p (cid:16) e U (cid:17) and x = p ( e x ) . From the Lemma 4.3.10 it follows that for all g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) there is a n e U e α o path e p g = (cid:16) e U e α , ..., e U e α n (cid:17) such that e U e α = e U ; e U e α n = g e U (4.3.6)106 path which satisfies to (4.3.6) is not unique. One has p g = desc p e p g ⇔ p (cid:0) p g (cid:1) = p ( p ) = (cid:16) p (cid:16) e U e α (cid:17) , ..., p ( U e α n ) (cid:17) ⇒⇒ p (cid:16) e U e α (cid:17) = p (cid:16) e U e α n (cid:17) = U . (4.3.7)If g ′ , g ′′ ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) then from (4.3.7) it follows that there is the composition desc p e p g ′ ◦ desc p e p g ′′ (cf. Definition 4.3.11). Definition 4.3.20.
In the situation 4.3.19 we say that the path e p g corresponds tog ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) . This path is not unique. Lemma 4.3.21.
If we consider the situation 4.3.19 and suppose that e p g ′ = (cid:16) e U e α ′ , ..., e U e α ′ k (cid:17) , e p g ′′ = (cid:16) e U e α ′′ , ..., e U e α ′′ l (cid:17) . If e p = lift e U p (cid:0) desc p e p g ′ ◦ desc p e p g ′′ (cid:1) , e p = (cid:16) e U e α , ..., e U e α k + l − (cid:17) then e U e α k + l − = g ′ g ′′ e U .Proof. From e U e α ′ k = g ′ e U it follows that e p = (cid:16) e U e α ′ , ..., e U e α ′ k , g ′ e U e α ′′ , ..., g ′ e U e α ′′ l (cid:17) and taking into account e U e α ′′ l = g ′ e U one has e U e α k + l − = e U e α ′′ k = g ′ g ′′ e U . In the situation 4.3.19 consider a commutative triangle of connected topo-logical spaces and transitive coverings. e X ′ e X ′′ X pp ′ p ′′ Suppose that X is a connected, locally connected, locally compact, Hausdorffspace. Consider a described in 4.3.12 family {U α } α ∈ A of open subsets of X . Letboth n e U e α ′ o e α ′ ∈ f A ′ , n e U e α ′′ o f α ′′ ∈ f A ′′ be the p ′ -lift and p ′′ -lift of {U α } α ∈ A respectively107cf. Definition 4.3.13 and Equation (4.3.3)). Let e U ′ ∈ n e U e α ′ o , and e x ∈ e U ′ . Denoteby e U ′′ = p (cid:16) e U ′ (cid:17) ⊂ e X ′′ , U = p ′ (cid:16) e U ′ (cid:17) ⊂ X , e x ′′ = p ( e x ′ ) ⊂ e U ′′ , x = p ′ ( e x ′ ) ⊂ U .Both p ′ and p ′′ are transitive coverings, hence from the Corollary 4.3.4 it followsthat there are bijecive maps G (cid:16) e X ′ | X (cid:17) ≈ −→ p ′− p ′ (cid:0)e x ′ (cid:1) , g g e x ′ ; G (cid:16) e X ′′ | X (cid:17) ≈ −→ p ′′− p ′′ (cid:0)e x ′′ (cid:1) , g g e x ′′ .On the other hand the map p induces the surjective map p ′− p ′ ( e x ′ ) → p ′′− p ′′ ( e x ′′ ) so one has the surjecive map h : G (cid:16) e X ′ | X (cid:17) → G (cid:16) e X ′′ | X (cid:17) (4.3.8) Lemma 4.3.23.
In the situation 4.3.22 following condition holds:(i) The given by (4.3.8) surjective map h : G (cid:16) e X ′ | X (cid:17) → G (cid:16) e X ′′ | X (cid:17) is a homo-morphism of groups.(ii) If we consider the natural inclusion G (cid:16) e X ′ | X ′′ (cid:17) ⊂ G (cid:16) e X ′ | X (cid:17) then G (cid:16) e X ′ | X ′′ (cid:17) is a normal subgroup of G (cid:16) e X ′ | X (cid:17) (cf. Definition B.3.3).Proof. (i) Let g ′ , g ′ ∈ G (cid:16) e X ′ | X (cid:17) and let both e p ′ g ′ = (cid:16) e U ′ e α ′ , ..., e U ′ e α ′ k (cid:17) , e p ′ g ′ = (cid:16) e U ′ e α ′ , ..., e U ′ e α ′ l (cid:17) are n e U ′ e α ′ o -paths which correspond to g ′ and g ′ respectively (cf. Definition 4.3.20).Clearly both desc p e p ′ g ′ and desc p e p ′ g ′ are n e U e α ′′ o -paths which correspond to h ( g ) and h ( g ) respectively. From the Lemma 4.3.21 it follows that both e p ′ def = lift e U ′ p ′ (cid:0) desc p ′ e p g ◦ desc p ′ e p g (cid:1) , e p ′′ def = lift e U ′′ p ′′ (cid:0) desc p ′ e p g ◦ desc p ′ e p g (cid:1) are n e U ′ e α ′ o and n e U ′′ e α ′′ o -paths which correspond to g g and h ( g ) h ( g ) respec-tively. It means that that e p ′ = (cid:16) e U ′ e α ′ , ..., e U ′ e α ′ k + l − (cid:17) then e U ′ e α ′ k + l − = g g e U ′ . On the otherhand e p ′′ = desc p e p ′ = (cid:16) p (cid:16) e U ′ e α ′ (cid:17) , ..., p (cid:16) e U ′ e α ′ k + l − (cid:17)(cid:17) , so p (cid:16) e U ′ e α ′ k + l − (cid:17) = h ( g g ) e U ′′ , i.e.108 p ′′ corresponds to h ( g g ) . It follows that h ( g g ) = h ( g ) h ( g ) , i.e. h is a homo-morphism.(ii) Follows from G (cid:16) e X ′′ | X ′ (cid:17) = ker h . Corollary 4.3.24.
Let X , X ′ and X ′′ be connected, locally connected, second-countable,locally compact, Hausdorff spaces. If p ′ : X ′ → X and p ′′ : X ′′ → X ′ are finite-foldtransitive coverings such that p ′′ ◦ p ′ is transitive then(i) There is an exact sequence of group homomorphisms { e } → G (cid:0) X ′′ (cid:12)(cid:12) X ′ (cid:1) → G (cid:0) X ′′ (cid:12)(cid:12) X (cid:1) → G (cid:0) X ′ | X (cid:1) → { e } . (ii) For any g ∈ G ( X ′′ | X ) one has gC ( X ′ ) = C ( X ′ ) . Proof. (i) Directly follows from the Lemma 4.3.23.(ii) Suppose a ′ ∈ C ( X ′′ ) . One has a ′ ∈ C ( X ′ ) if and only if g ′′ a ′ = a ′ forall g ′′ ∈ G ( X ′′ | X ′ ) . Since G ( X ′′ | X ′ ) is a normal subgroup of G ( X ′′ | X ) for each g ∈ G ( X ′′ | X ) and g ′′ ∈ G ( X ′′ | X ′ ) there is g ′′ ∈ G ( X ′′ | X ′ ) suchthat g ′′ g = gg ′′ , it turns out g ′′ ga ′ = gg ′′ a ′ = ga ′ , i.e. g ′′ ( ga ′ ) = ga ′ so one has ga ′ ∈ C ( X ′ ) . Definition 4.3.25.
Let X be a connected, locally connected, locally compact, second-countable, Hausdorff space. Let us consider the category FinCov - X such that • Objects of
FinCov - X are transitive finite-fold coverings e X → X of X wherethe space e X is connected. • A morphism form p : e X → X to p : e X → X is a surjective continuousmap p : e X → e X such that p ◦ p = p .We say that FinCov - X is the finite covering category of X . Remark 4.3.26.
From the Corollary 4.3.8 it follows that p is a transitive covering. Remark 4.3.27.
We also use the alternative notation in which objects of
FinCov - X are covering spaces, i.e. a covering e X → X is replaced with the space e X . Lemma 4.3.28.
If p : e X → X , p : e X → X are objects of FinCov - X and p ′ : e X → e X , p ′′ : e X → e X are morphisms of FinCov - X then there is g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such thatp ′′ = g ◦ p ′ .109 roof. The covering p is transitive, so from p ′ ◦ p = p ′′ ◦ p it turns out that ∀ e x ∈ e X there is the unique g e x ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) p ′ ( e x ) = g e x ◦ p ′′ ( e x ) .There is the map h : e X → G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) ; e x g e x and since both p ′ and p ′′ are local isomorphisms the map h is continuous. How-ever the space G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) of homeomorphisms is discrete and the space e X isconnected, hence the map h is constant, i.e. ∃ g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) ∀ e x ∈ e X p ′′ ( e x ) = g ◦ p ′ ( e x ) . Definition 4.3.29. If X is a set (resp. topological space) and x ∈ X is a base-pointthen we say that the pair ( X , x ) is a pointed set (resp. pointed space ). If both ( X , x ) and ( Y , y ) are pointed spaces and ϕ : X → Y is such that ϕ ( x ) = y then wesay that ϕ is a pointed map . We write ϕ : ( X , x ) → ( Y , y ) (4.3.9) Definition 4.3.30.
Let ( X , x ) be a pointed space such that X is a connected, locallyconnected, locally compact, second-countable, Hausdorff space. Let us considerthe category FinCov - ( X , x ) such that • Objects of
FinCov - ( X , x ) are transitive finite-fold coverings (cid:16) e X , e x (cid:17) → ( X , x ) of X preserving base-points. • A morphism form p : (cid:16) e X , e x (cid:17) → ( X , x ) to p : (cid:16) e X , e x (cid:17) → ( X , x ) is apreserving base-points surjective continuous map p : (cid:16) e X , e x (cid:17) → (cid:0) X , x (cid:1) such that p ◦ p = p .We say that FinCov - ( X , x ) is the topological pointed finite covering category of ( X , x ) . Remark 4.3.31.
The category
FinCov - ( X , x ) is a subcategory of FinCov - X . Remark 4.3.32.
If both p : (cid:16) e X , e x (cid:17) → ( X , x ) and p : (cid:16) e X , e x (cid:17) → ( X , x ) areobjects of FinCov - ( X , x ) then there is no more than one morphism from p to p .110 emma 4.3.33. Let X be a connected, locally connected, locally compact, Hausdorff, Lin-delöf space. Suppose that e X → X be a transitive finite-fold covering. Then for any x ∈ X there is a finite or countable sequence U $ ... $ U n $ ... of connected open subsets of X such that(i) x ∈ U .(ii) For any n ∈ N the closure of U n is compact.(iii) ∪ U n = X .(iv) The space p − ( U n ) is connected for any n ∈ N .Proof. (i)-(iii) From the Lemma 4.2.1 it follows that there is a finite or countablesequence U $ ... $ U n $ ... of connected open subsets of X such that conditions(i)-(iii) holds.(iv) Let e x ∈ p − ( U n ) be any point, and let e U e x n be a connected component of p − ( U n ) such that e x ∈ e U e x n . If G n = n g ∈ G (cid:16) e X | X (cid:17)(cid:12)(cid:12)(cid:12) g e x ∈ U e x n o then there is isa nondecreasing sequence of subgroups G ⊆ ... ⊆ G n ⊆ ...Since G (cid:16) e X | X (cid:17) is finite there is m ∈ N such that G k = G m for every k ≥ m . If J ∈ G (cid:16) e X | X (cid:17) is a set of representatives of G (cid:16) e X | X (cid:17) / G m and e U e x = ∪ n ∈ N e U e x n then one has e X = [ n ∈ N p − ( U n ) = G g ∈ J g e U e x Since the space e X is connected the set J is a singleton. It follows that p − ( U k ) = e U e x k for every k ≥ m i.e. p − ( U k ) is connected. So the finite or countable subse-quence U m $ ... $ U k $ ... of U $ ... $ U n $ ... satisfies to the condition (iv). Lemma 4.3.34.
Let X be a connected, locally connected, locally compact, Hausdorff space,and let U ⊂ X be a connected open subspace. If p : e X → X is a transitive covering, suchthat such that the space e U = p − ( U ) is connected then the restriction p | e U : e U → U is atransitive covering. Moreover the maph : G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) → G (cid:16) e U (cid:12)(cid:12)(cid:12) U (cid:17) , g g | e U (4.3.10) is the isomorphism of groups. roof. The covering p is transitive, hence if e x ∈ e U then there is a bijection G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) ≈ −→ p − ◦ p ( e x ) it follows that the map (4.3.10) is injective. If g ′ ∈ G (cid:16) e U (cid:12)(cid:12)(cid:12) U (cid:17) then for any e x ∈ e U there is g e x ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such that g ′ e x = g e x ′ e x . There is a continuous map e U → G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) ; e x g e x ′ .Since e U is connected and G (cid:16) e X | X (cid:17) is a discrete group of homeomorphisms themap is constant, so that there is the unique g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such that g ′ = g | e U , i.e.the map h is surjecive. This section is not exclusively devoted to commutative C ∗ -algebras. It containsa more general theory, where C ∗ -algebras are represented by locally trivial bun-dles. Homogeneous C ∗ algebras and with continuous trace C ∗ -algebras can berepresented as locally trivial bundles. Any commutative C ∗ -algebra is also a vec-tor bundle with one-dimensional fibers. So this theory is used in this chapter andthe chapters 5, 6, 7. Remark 4.4.1.
The section A.3 describes the theory of vector bundles with finite-dimensional fibers. However many results of the theory can be applied to vectorbundles where any fiber is a more general Banach space. Below the word vectorbundle means that a fiber may be arbitrary Banach space.
Let X be a locally compact Hausdorff space and ξ = ( E , π , X ) is a vectorbundle (cf. Definition A.3.4 and Remark 4.4.1) such that any fiber is a Banach C -space. The space Γ ( E , π , X ) of sections of ( E , π , X ) is a C -space. For any a ∈ Γ ( E , π , X ) the given by norm a : X → R , x
7→ k a x k (4.4.1)112ap is continuous. Denote by Γ c ( E , π , X ) def = { a ∈ Γ ( E , π , X ) | norm a ∈ C c ( X ) } , Γ ( E , π , X ) def = { a ∈ Γ ( E , π , X ) | norm a ∈ C ( X ) } , Γ b ( E , π , X ) def = { a ∈ Γ ( E , π , X ) | norm a ∈ C b ( X ) } . (4.4.2)There is the norm k a k def = k norm a k on the above spaces such that both Γ ( E , π , X ) and Γ b ( E , π , X ) are Banach spaces and Γ c ( E , π , X ) is dense in Γ ( E , π , X ) . If f : Y → X is a continuous map and ( E × X Y , ρ , Y ) is the inverse image of ( E , π , X ) by f (cf. Definition A.3.7) then the natural projection E × X Y → E yields thenatural C -linear maps Γ ( E , π , X ) ֒ → Γ ( E × X Y , ρ , Y ) , Γ b ( E , π , X ) ֒ → Γ b ( E × X Y , ρ , Y ) (4.4.3)such that the second map is an isometry. Consider a category Top - X such that • Objects of are continuous maps
Y → X . • Morphism Y ′ → X to Y ′′ → X is a continuous map f : Y ′ → Y ′′ such thatthe following diagram Y ′ Y ′′ X f is commutative.For any vector bundle ( E , π , X ) equations (4.4.2) and (4.4.3) yield two contravari-ant functors Γ and Γ b form Top - X to the category of vector spaces given suchthat Γ ( Y ) def = Γ ( E × X Y , ρ , Y ) , Γ b ( Y ) def = Γ b ( E × X Y , ρ , Y ) , Γ ( f ) def = (cid:0) Γ (cid:0) E × X Y ′′ , ρ , Y ′′ (cid:1) → Γ (cid:0) E × X Y ′ , ρ , Y ′ (cid:1)(cid:1) , Γ b ( f ) def = (cid:0) Γ b (cid:0) E × X Y ′′ , ρ , Y ′′ (cid:1) → Γ b (cid:0) E × X Y ′ , ρ , Y ′ (cid:1)(cid:1) (4.4.4)and Γ b is a functor to the category of Banach spaces and isometries.113 .5 Continuous structures lift and descent Let X be a locally compact, paracompact, Hausdorff space; and for each x in X ,let A x be a (complex) Banach space. Let us consider a continuity structure F for X and the { A x } x ∈X (cf. Definition D.8.27). Denote by C ( X , { A x } , F ) def = ( { a x } x ∈X ∈ ∏ x ∈X A x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { a x } is a continuous section ) (4.5.1)(cf. Definition D.8.28). Remark 4.5.1.
From the Lemma D.8.31 it follows that C ( X , { A x } , F ) is C ( X ) -module. Lemma 4.5.2. (i) If F is a continuity structure for X and the { A x } (cf. DefinitionD.8.27), then C ( X , { A x } , F ) is a continuity structure for X and the { A x } .(ii) C ( X , { A x } , F ) = C ( X , { A x } , C ( X , { A x } , F )) . (4.5.2) Proof. (i) One needs check conditions (a)-(c) of the Definition D.8.27.(a) Follows from the Lemma D.8.30.(b) Follows from the inclusion
F ⊂ C ( X , { A x } , F ) .(c) If A x is a C ∗ -algebra for all x ∈ X then from (c) of the Definition D.8.27 itturns out that F is closed under multiplication and involution. Select x ∈ X and ε >
0. For all a ∈ C ( X , { A x } , F ) there is an open neighborhood U of x and a family { a ′ x } ∈ F such that k a x − a ′ x k < ε for all x ∈ U . It follows that k a ∗ x − a ′∗ x k < ε for all x ∈ U , and taking into account a ′∗ x ∈ F we concludethat a ∗ ∈ C ( X , { A x } , F ) . If a , b ∈ C ( X , { A x } , F ) then for any δ > δ + δ ( k a k + k b k ) < δ there are open neighborhoods U ′ , U ′′ of x andfamilies { a ′ x } ∈ F such that k a x − a ′ x k < δ for all x ∈ U ′ and k b x − b ′ x k < δ for all x ∈ U ′′ . It turns out that k a x b x − a ′ x b ′ x k < ε for all x ∈ U ′ ∩ U ′′ . Itfollows that the family { a x b x } lies in C ( X , { A x } , F ) .(ii) From F ⊂ C ( X , { A x } , F ) it follows that C ( X , { A x } , F ) ⊂ C ( X , { A x } , C ( X , { A x } , F )) .Let a ∈ C ( X , { A x } , C ( X , { A x } , F )) , let x ∈ X and ε >
0. There is an openneighborhood U ′ of x and a ′ ∈ C ( X , { A x } , F ) such that k a x − a ′ x k < ε for all114 ∈ U ′ . Otherwise there is an open neighborhood U ′′ of x and a ′′ ∈ F suchthat k a ′ x − a ′′ x k < ε for every x ∈ U ′′ . It follows that k a x − a ′′ x k < ε for every x ∈ U ′ ∩ U ′′ , hence one has a ∈ C ( X , { A x } , F ) . In result we conclude that C ( X , { A x } , C ( X , { A x } , F )) ⊂ C ( X , { A x } , F ) .Any vector bundle ( E , π , X ) (cf. Definition A.3.4 and Remark 4.4.1) yields afamily of fibers { E x } x ∈X . The given by (4.4.2) spaces are a continuity structuresfor X and the { E x } x ∈X . Definition 4.5.3.
Let ( E , π , X ) be vector bundle with the family of fibers { E x } x ∈X .Any linear subspace F ⊂ Γ ( E , π , X ) (cf. 4.4.2) is said to be ( E , π , X ) - continuousstructure if Γ ( E , π , X ) ∼ = C ( X , { E x } , F ) . Lemma 4.5.4.
In the situation of the Definition 4.5.3 the space Γ ( E , π , X ) is a ( E , π , X ) -continuous structure.Proof. Clearly one has Γ ( E , π , X ) ⊂ C ( X , { E x } , Γ ( E , π , X )) . Let x ∈ X be anypoint, and select an open neighborhood U of x such that ( E , π , X ) | U is trivial (cf.A.3.2), i.e. E | U ∼ = U × E x . The map s : X → E such that π ◦ s = Id X is continuousat x if there is an open neighborhood V of x such that V ⊂ U and s | V correspondsto the continuous map f s : V → E x . For any ε > t ∈ C ( X , { E x } , Γ ( E , π , X )) there is s ∈ Γ ( E , π , X ) and an open neighborhood U of x such that k s x − t x k < ε /2 for all x ∈ U . Otherwise there is an an open neighborhood U of x such that U ⊂ U and k f s ( x ) − f s ( x ) k < ε /2 for any x ∈ U . If t | U corresponds to a map f t : U → E x such then for any x ∈ V = U ∩ U one has k f t ( x ) − f t ( x ) k < ε .It follows that f t is continuous at x , so t corresponds to a continuous section of ( E , π , X ) at x . However x ∈ X is an arbitrary, so t is continuous at any pointof X , equivalently t is continuous, i.e. t is a section of ( E , π , X ) . It follows that t ∈ Γ ( E , π , X ) and C ( X , { E x } , Γ ( E , π , X )) ⊂ Γ ( E , π , X ) .For any a ∈ C ( X , { A x } , F ) denote bynorm a : X → R , x
7→ k a x k . (4.5.3)Let us define k·k : C ( X , { A x } , F ) → [ ∞ ) given by k a k def = k norm a k . (4.5.4)115 efinition 4.5.5. The space C b ( X , { A x } , F ) given by C b ( X , { A x } , F ) def = { a ∈ C ( X , { A x } , F ) | k a k < ∞ } . (4.5.5)is said to be the space of bounded continuous sections . Lemma 4.5.6.
One hasa ∈ C b ( X , { A x } , F ) ⇔ norm a ∈ C b ( X ) . Proof.
From the Lemma D.8.30 it follows that the map x
7→ k a x k is continuous,from the Equation (4.5.5) it follows that x
7→ k a x k is bounded. Definition 4.5.7. If a ∈ C ( X , { A x } , F ) is presented by the family { a x ∈ A x } thenthe closure of { x ∈ X | k a x k > } is said to be the support of a . The support of a coincides with the support of the norm a , hence we write supp a def = supp norm a . (4.5.6) Definition 4.5.8.
The C b (cid:16) e X (cid:17) -module C c ( X , { A x } , F ) = { a ∈ C b ( X , { A x } , F ) | supp a is compact } (4.5.7)is said to be the compactly supported submodule . The norm completion of C c ( X , { A x } , F ) is said to be converging to zero submodule and we denote it by C ( X , { A x } , F ) . Wealso use the following notation C c ( A ) def = C c ( X , { A x } , A ) , (4.5.8) C ( A ) def = C ( X , { A x } , A ) , (4.5.9) C b ( A ) def = C b ( X , { A x } , A ) , (4.5.10) C ( A ) def = C ( X , { A x } , A ) , (4.5.11) C ( X , { A x } , F ) | U def = { a ∈ C ( X , { A x } , F ) | supp a ⊂ U } . (4.5.12) Remark 4.5.9.
Both C c ( X , { A x } , A ) and C ( X , { A x } , A ) are C ( X ) -modules (cf.Remark 4.5.1). Remark 4.5.10. If C ( X , { A x } , F ) is a C ∗ -algebra with continuous trace then thenotation (4.5.12) complies with (D.2.7). 116 emma 4.5.11. Following conditions holdC c ( A ) = { a ∈ C b ( A ) | norm a ∈ C c ( X ) } , (4.5.13) C ( A ) = { a ∈ C b ( A ) | norm a ∈ C ( X ) } (4.5.14) (cf. Equations (4.5.3) , (4.5.9) , (4.5.10) ).Proof. The equation (4.5.13) is evident. If a ∈ C ( A ) then there is a C ∗ -normconvergent net { a α ∈ C c ( A ) } α ∈ A such that a = lim a α . For any α , β ∈ A one has (cid:13)(cid:13)(cid:13) norm a α − norm a β (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13) a α − a β (cid:13)(cid:13) . (4.5.15)From (4.5.15) it turns out that the net { norm a α } is uniformly convergent. Fromnorm a α ∈ C c ( X ) and the Definition 4.5.8 it turns out lim α norm a α ∈ C ( X ) . Con-versely let a ∈ C ( A ) be such that norm a ∈ C ( X ) . If F ε is given by (4.1.3) and a ε ∈ C c ( A ) is such that a ε = F ε ( norm a ) a then from (4.1.4) it turns out that k F ε ( norm a ) norm a − norm a k ≤ ε ,or, equivalently k F ε ( norm a ) ( x ) a x − a x k ≤ ε ∀ x ∈ X ,and taking into account (4.5.4) one has k F ε ( norm a ) a − a k = k a ε − a k ≤ ε . (4.5.16)From (4.1.5) it follows that supp F ε ( norm a ) = supp a ε is compact. (4.5.17)From (4.5.16) and (4.5.17) it turns out that there is the norm limit a = lim ε → a ε such that a ε ∈ C c ( A ) . Taking into account the Definition 4.5.8 one has a ∈ C ( A ) . Corollary 4.5.12.
One has X is compact ⇒ C c ( A ) = C ( A ) = C b ( A ) = C ( A ) . (4.5.18) Lemma 4.5.13. (i) If F is a continuity structure for X and the { A x } (cf. DefinitionD.8.27), then all of C c ( X , { A x } , F ) , C ( X , { A x } , F ) and C b ( X , { A x } , F ) area continuity structures for X and the { A x } . ii) C ( X , { A x } , F ) = C ( X , { A x } , C c ( X , { A x } , F )) = C ( X , { A x } , C ( X , { A x } , F )) = C ( X , { A x } , C b ( X , { A x } , F )) (4.5.19) Proof. (i) One needs check conditions (a)-(c) of the Definition D.8.27.(a) Follows from the Lemma D.8.30.(b) For any x denote by f x ∈ C c ( X ) such that f x ( X ) = [
0, 1 ] and there is anopen neighborhood U of x which satisfies to f x ( U ) = { } (cf. 4.1.7). Forany a ∈ F one has f x a ∈ C c ( X , { A x } , F ) , and from f x ( x ) = A = (cid:8) a x ∈ A x | ∃ a ′ ∈ F a x = a ′ x (cid:9) ⊂⊂ A c = (cid:8) a x ∈ A x | ∃ a ′ ∈ C c ( X , { A x } , F ) a x = a ′ x (cid:9) .Since A is dense in A x , so A c is dense in A x . If A = (cid:8) a x ∈ A x | ∃ a ′ ∈ C ( X , { A x } , F ) a x = a ′ x (cid:9) , A b = (cid:8) a x ∈ A x | ∃ a ′ ∈ C b ( X , { A x } , F ) a x = a ′ x (cid:9) then from A ⊂ A c ⊂ A ⊂ A b ⊂ A x it turns out that A and A b are dense in A x .(c) The proof is similar to the proof of (c) in the proof of the Lemma 4.5.2.(ii) From the inclusions C c ( X , { A x } , F ) ⊂ C ( X , { A x } , F ) ⊂ C b ( X , { A x } , F ) ⊂ C ( X , { A x } , F ) it follows that C ( X , { A x } , C c ( X , { A x } , F )) ⊂ C ( X , { A x } , C ( X , { A x } , F )) ⊂⊂ C ( X , { A x } , C b ( X , { A x } , F )) ⊂ C ( X , { A x } , F ) . (4.5.20)Let a ∈ C ( X , { A x } , C ( X , { A x } , F )) , let x ∈ X and ε >
0. There is an openneighborhood U ′ of x and a ′ ∈ C ( X , { A x } , F ) such that k a x − a ′ x k < ε forall x ∈ U ′ . Otherwise if f x is given by the Corollary 4.1.6 and a ′′ def = f x a ′ ∈ C c ( X , { A x } , F ) . There is an open neighborhood U ′′ of x such that f x ( U ′′ ) = { } it turns out that k a x − a ′′ x k < ε for all x ∈ U ′ ∩ U ′′ . It follows that a ∈ C ( X , { A x } , C c ( X , { A x } , F )) , so one has C ( X , { A x } , F ) ⊂ C ( X , { A x } , C c ( X , { A x } , F )) ,and taking into account (4.5.20) one obtains (4.5.19).118 orollary 4.5.14. Following conditions hold:C c (cid:0) X , { A x } x ∈X , C c (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) == C c (cid:0) X , { A x } x ∈X , C (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) == C c (cid:0) X , { A x } x ∈X , C b (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) == C c (cid:0) X , { A x } x ∈X , C (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) = C c (cid:0) X , { A x } x ∈X , F (cid:1) , C (cid:0) X , { A x } x ∈X , C c (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) == C (cid:0) X , { A x } x ∈X , C (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) == C (cid:0) X , { A x } x ∈X , C b (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) == C (cid:0) X , { A x } x ∈X , C (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) = C (cid:0) X , { A x } x ∈X , F (cid:1) , C b (cid:0) X , { A x } x ∈X , C c (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) == C b (cid:0) X , { A x } x ∈X , C (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) == C b (cid:0) X , { A x } x ∈X , C b (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) == C b (cid:0) X , { A x } x ∈X , C (cid:0) X , { A x } x ∈X , F (cid:1)(cid:1) = C b (cid:0) X , { A x } x ∈X , F (cid:1) . (4.5.21) Lemma 4.5.15.
Let F be a continuity structure for X and the { A x } x ∈X (cf. DefinitionD.8.27), and let a ∈ C ( X , { A x } , F ) is represented by a family { a x ∈ A x } . Suppose norm a is multiple of f ∈ C ( X ) (cf. Definition 4.1.2). Let U = { x ∈ X | f ( x ) = } . If { b x ∈ A x } x ∈X is the family given byb x = ( a x f ( x ) x ∈ U x / ∈ U (4.5.22) then there is b ∈ C ( X , { A x } , F ) represented by the family { b x } x ∈X . Moreover k b x k = ( k a x k f ( x ) x ∈ U x / ∈ U (4.5.23) Proof.
Firstly we proof that the family { b x } is continuous with respect to F at any x ∈ X (cf. Definition D.8.28). Consider two alternative cases:(i) x ∈ U . Let f x ∈ C c ( X ) be such that f x ( X ) = [
0, 1 ] and supp f x ⊂ U andthere is an open neighborhood V of x which satisfies to f x ( V ) = { } (cf.Lemma 4.1.5). From the Lemma 4.1.1 it turns out f x is a multiple of f , i.e.there is f ′ ∈ C ( X ) such that f x = f ′ f . If a f ′ = f ′ a ∈ C c ( X , { A x } , F ) then a f ′ x = b x for any x ∈ V . 119ii) x / ∈ U . From the Lemma 4.1.1 it turns that out for any ε > W of x such that k b x k < ε for any x ∈ W . So if a def = ∈ C ( X , { A x } , F ) then k b x − a x k < ε for all x ∈ W .So the family { b x } yields the element b ∈ C ( X , { A x } , F ) . Otherwise from norm b = div ( norm a , f ) if follows that norm b ∈ C ( X ) and taking into account (4.5.14) onehas b ∈ C ( X , { A x } , F ) . The equation (4.5.23) follows from (4.5.22). Definition 4.5.16.
In the situation of the Lemma 4.5.15 we say that a is a multiple of f . we write div ( a , f ) def = b ∈ C ( X , { A x } , F ) . (4.5.24) If F is a continuity structure F for X and the { A x } then from the LemmaD.8.31 it follows that C ( X ) F ⊂ C ( X , { A x } , F ) . Lemma 4.5.18.
Let F be a continuity structure F for X and the { A x } , such that F ⊂ C ( X , { A x } , F ) . If X is paracompact (cf. Definition A.1.24) then the space C ( X ) F isdense in C ( X , { A x } , F ) with respect to the norm topology.Proof. Suppose a ∈ C ( X , { A x } , F ) is represented by the family { a x ∈ A x } andlet ε >
0. For any x ′ ∈ X we select an open neighborhood U x ′ ⊂ X and a x ′ ∈ F represented by the family { a ′ x ∈ A x } such that (cid:13)(cid:13)(cid:13) a x − a x ′ x (cid:13)(cid:13)(cid:13) < ε /4 for all x ∈ U x ′ .From X = ∪ x ′ ∈X U x ′ and the Theorem A.1.25 there exists a partition of unity ∑ x ′ ∈X f x ′ ( x ) = ∀ x ∈ X dominated by {U x ′ } . The set U def = { x ∈ X | k a x k ≥ ε /4 } is compact so there is afinite covering sum ∑ nj = f x j for U (cf. Definition 4.2.6), such that ∑ nj = f x j ( x ) = x ∈ U . From (cid:13)(cid:13)(cid:13) a x − a x ′ x (cid:13)(cid:13)(cid:13) < ε /4 for all x ∈ U x ′ it turns out that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = f x j ( x ) a x j x − a x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε /4 < ε ∀ x ∈ U .and taking into account x ∈ X \ U ⇒ ≤ n ∑ j = f x j ( x ) ≤ k a x k < ε /4 AND (cid:13)(cid:13)(cid:13) a x − a x ′ x (cid:13)(cid:13)(cid:13) < ε /4 ⇒⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = f x j ( x ) a x j x − a x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = f x j ( x ) a x j x − a x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε ∀ x ∈ X ⇔ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = f x j a x j − a (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε .From ∑ nj = f x j a x j ∈ C ( X ) F it turns out that the space C ( X ) F is dense in C ( X , { A x } , F ) . Corollary 4.5.19.
Let F be a continuity structure for X and the { A x } , such that F ⊂ C ( X , { A x } , F ) . If X is paracompact C ( X ) F ⊂ F and F is norm closed thenthe the natural inclusion F ֒ → C ( X , { A x } , F ) is the C ( X ) -isomorphism F ∼ = C ( X , { A x } , F ) . Remark 4.5.20.
From the Corollary 4.5.19 it follows that C ( X , { A x } , F ) is a min-imal norm closed subspace of C b ( X , { A x } , F ) which contains all products f a where a ∈ F and f ∈ C ( X ) . Let F be a continuity structure for X and the { A x } Select x ∈ X . From (b)of the Definition D.8.27 it turns out that the C -space (cid:8) a x ∈ A x | ∃ a ′ ∈ C b ( X , { A x } , F ) a x = a ′ x (cid:9) is dense in A x . So there is the map ϕ x : C b ( X , { A x } , F ) → A x . (4.5.25)Denote by I x = { f ∈ C b ( X ) | f ( x ) = } , and let I x C ( X , { A x } , F ) be a submodulesuch that ϕ x ( I x C b ( X , { A x } , F )) = C x = C b ( X , { A x } , F ) / I x C b ( X , { A x } , F ) (4.5.26)then there is a C -linear map φ x : C x → A x .Let us define a norm on C x given by k c k C x = inf a ∈ C ( X , { A x } , F ) c = a + I x C ( X , { A x } , F ) k a k (4.5.27)Clearly k c k C x ≥ k φ x ( c ) k . If a ′ ∈ : C b ( X , { A x } , F ) is a representative of c givenby the family { a ′ x ∈ A x } then for all ε > U of x k a ′ x k − (cid:13)(cid:13) a ′ x ′ (cid:13)(cid:13) < ε for all x ′ ∈ U . From the Exercise A.1.12 it turns outthat X is completely regular (cf. Definition A.1.10). If f : X → [
0, 1 ] is such that supp f ⊂ U and f ( x ) = f a ′ is a representative of c and k f a ′ k < k φ x ( c ) k + ε .It follows that k c k C x < k φ x ( c ) k + ε and taking into account k c k C x ≥ k φ x ( c ) k onehas k c k C x = k φ x ( c ) k , i.e. φ x : C x → A x is an isometry, hence A x can be definedas the norm completion of C x . If β X is the Stone- ˇCech compactification of X (cf.Definition A.1.33) one can define A β x for any β x ∈ β X . For any β x ∈ β X there isan ideal I β x ⊂ C ( β X ) = C b ( X ) and similarly to above construction there is thenormed space C β x = C b ( X , { A x } , F ) / I β x C b ( X , { A x } , F ) . Definition 4.5.22.
Let F be a continuity structure for X and the { A x } x ∈X , suchthat X is completely regular (cf. Definition A.1.10). A family (cid:8) β A β x (cid:9) x ∈ β X suchthat β A β x is a completion of C β x is said to be the Stone- ˇCech extension or the β - extension of { A x } x ∈X .If x ∈ X then clearly A x ∼ = β A x . For any a ∈ C b ( X , { A x } , F ) on can define afamily β ( a ) = (cid:8) a β x ∈ β A β x (cid:9) such that a β x = φ β x (cid:0) a + I β x C b ( X , { A x } , F ) (cid:1) . (4.5.28)From norm a ∈ C b ( X ) one has norm β ( a ) ∈ C ( β X ) , so the space β F of families4.5.28 is a continuity structure for β X and the (cid:8) β A β x (cid:9) . Since β F is norm closedand β F is compact one has β F = C c (cid:16) β X , (cid:8) β A β x (cid:9) x ∈ β X , β F (cid:17) = C (cid:16) β X , (cid:8) β A β x (cid:9) x ∈ β X , β F (cid:17) == C b (cid:16) β X , (cid:8) β A β x (cid:9) x ∈ β X , β F (cid:17) = C (cid:16) β X , (cid:8) β A β x (cid:9) x ∈ β X , β F (cid:17) . (4.5.29)From the equation (4.5.28) it turns out that for all β x ∈ β X there is the C -linearmap φ β x : C b ( X , { A x } , F ) → β A β x (4.5.30)such that φ β x ( C b ( X , { A x } , F )) is dense in A β x . Definition 4.5.23.
The continuity structure β F for β X and the (cid:8) β A β x (cid:9) given by(4.5.28) is said to be the Stone- ˇCech extension or the β - extension of F . Remark 4.5.24.
From the the construction the family β ( a ) = (cid:8) a β x ∈ β A β x (cid:9) uniquelydepends on a ∈ C b ( X , { A x } , F ) . It turns out that there is the natural C b ( X ) -isomorphism β F ∼ = C b ( X , { A x } , F ) (4.5.31)122 emma 4.5.25. Let β F be the given by families (cid:8) a β x (cid:9) (cf. (4.5.28) ) continuity structurefor β X and the (cid:8) β A β x (cid:9) . If A ⊂ β F the norm bounded and norm closed C ( β X ) -submodule which is a continuity structure for β X and the (cid:8) β A β x (cid:9) then there is thenatural isomorphism A ∼ = C b ( X , { A x } , F ) . (4.5.32) Proof.
The inclusion A ⊂ β F induces the inclusion A ⊂ C b (cid:0) β X , (cid:8) β A β x (cid:9) , β F (cid:1) .From the Corollaries 4.5.12 and 4.5.19 it follows that A = C b (cid:0) β X , (cid:8) β A β x (cid:9) , β F (cid:1) ,and taking into account C b (cid:0) β X , (cid:8) β A β x (cid:9) , β F (cid:1) ∼ = C b ( X , { A x } , F ) one has 4.5.32. Definition 4.5.26.
For any family of Banach spaces { A x } x ∈X the union \ { A x } def = ∪ x ∈X A x is said to be the total set of { A x } x ∈X . The natural map \ { A x } → X , a x x is said to be the total projection . Let us consider a continuity structure F for X and the { A x } x ∈X (cf. Defi-nition D.8.27). Let U ⊂ X be an open subset and let s ∈ C ( X , { A x } , F ) . For any ε > O ( U , s , ε ) def = n a x ∈ \ { A x } (cid:12)(cid:12)(cid:12) x ∈ U k a x − s x k < ε o (4.5.33) Lemma 4.5.28.
A collection of given by the equation (4.5.33) subsets satisfies to (a) and(b) of the Definition A.1.1.Proof.
Condition (a) is evident, let us proof (b). Let a x ∈ \ { A x } and consider O ( U ′ , s ′ , ε ′ ) , O ( U ′′ , s ′′ , ε ′′ ) ⊂ \ { A x } such that a x ∈ O ( U ′ , s ′ , ε ′ ) ∩ O ( U ′′ , s ′′ , ε ′′ ) . Let s ∈ C ( X , { A x } , F ) is C ( X ) be such that a x = s x . One has k s ′ x − s x k < ε ′ and k s ′′ x − s x k < ε ′′ . If ε < min ( ε ′ − k s ′ x − s x k , ε ′′ − k s ′′ x − s x k ) then there are opensubsets U ′ , U ′′ ⊂ X such that ∀ x ∈ U ′ (cid:13)(cid:13) s ′ x − s x (cid:13)(cid:13) < ε ′ − ε , ∀ x ∈ U ′′ (cid:13)(cid:13) s ′′ x − s x (cid:13)(cid:13) < ε ′′ − ε .From the above equations it follows that O ( U , s , ε ) ⊂ O ( U ′ , s ′ , ε ′ ) ∩ O ( U ′′ , s ′′ , ε ′′ ) .123 efinition 4.5.29. A topological T ( X , { A x } , F ) space such that • The space T ( X , { A x } , F ) which coincides with the total set \ { A x } def = ∪ x ∈X A x as a set. • The topology of T ( X , { A x } , F ) generated by given by (4.5.33) sets (cf. Def-inition A.1.1).is said to be the total space for ( X , { A x } , F ) . Remark 4.5.30.
The total projection T ( X , { A x } , F ) → X (cf. Definition 4.5.26) isa continuous map. Lemma 4.5.31.
Following conditions hold.(i) If p : T ( X , { A x } , F ) → X is the total projection then any continuous mapj : X → T ( X , { A x } , F ) such that p ◦ j = Id X then the family { j ( x ) ∈ A x } iscontinuous.(ii) Conversely any continuous section { a x } corresponds to a continuous mapj : X → T ( X , { A x } , F ) , x a x . (4.5.34) Proof. (i) If ε > x ∈ X then from (b) of the Definition D.8.27 it follows thatthere is a ∈ F such that k a x − j ( x ) k < ε /2. If U ′ is an open neighborhood of x then from j ( x ) ∈ O ( U ′ , a , ε ) , and since j is continuous it follows that U def = j − ( O ( U ′ , a , ε )) is open. So for all x ∈ U one has k a x − j ( x ) k < ε , i.e. the family { j ( x ) } is continuous (cf. Definition D.8.28).(ii) Suppose ε > x ∈ X . If that j ( x ) ∈ O ( U ′ , b , ε ) then k a x − b x k < ε .From the Lemma D.8.30 it follows that there is an open neighborhood U of x such that k a x − b x k < ε for all x ∈ U . So one has U ⊂ j − ( O ( U , b , ε )) , i.e. j is continuous at x . Similarly one can prove that j is continuous at any point of X . Definition 4.5.32.
For any subset
V ⊂ X denote by C ( V , { A x } , F ) the C spaceof continuous maps j : V → T ( X , { A x } , F ) such that if p is the total projec-tion then p ◦ j = Id V . The space C ( V , { A x } , F ) is said to be the V - restriction of C ( X , { A x } , F ) . 124 .5.33. Similarly to the Equation 4.5.3 any a ∈ C ( V , { A x } , F ) yields a continuousmap norm a : V → R , x
7→ k a x k . (4.5.35)Similarly to (4.5.4) define k·k : C ( X , { A x } , F ) → [ ∞ ) given by k a k def = k norm a k . (4.5.36)Denote by C b ( V , { A x } , F ) def = { a ∈ C ( V , { A x } , F ) | norm a ∈ C b ( V ) } , C ( V , { A x } , F ) def = { a ∈ C ( V , { A x } , F ) | norm a ∈ C ( V ) } . (4.5.37) Remark 4.5.34.
Both given by the Equations (4.5.37) spaces are closed with respectto the given by (4.5.36) norm .
Remark 4.5.35. If V is compact then one has C ( V , { A x } , F ) = C b ( V , { A x } , F ) = C ( V , { A x } , F ) . Lemma 4.5.36.
Let X be a locally compact, normal space; and for each x in X , letA x be a (complex) Banach space. Let us consider a continuity structure F for X andthe { A x } x ∈X (cf. Definition D.8.27). If V ⊂ X is compact then the restriction mapC ( X , { A x } , F ) → C ( V , { A x } , F ) is surjective.Proof. One needs prove that for any a ∈ C ( V , { A x } , F ) there is b ∈ C ( V , { A x } , F ) such that a = b | V . (4.5.38)Our proof has two parts:(i) For all a ∈ C ( V , { A x } , F ) and ε > b ∈ C ( X , { A x } , F ) such that k a − b | V k < ε , k b k < k a k + ε . (4.5.39).(ii) Looking for b which satisfies to (4.5.38).(i) For any x ∈ V we select s x ∈ C ( X , { A x } , F ) such that k s xx − a x | < ε . There isan open neighborhood U ′ x of x such that (cid:13)(cid:13)(cid:13) s xy (cid:13)(cid:13)(cid:13) < ε + k a k for all y ∈ U ′ x . Clearly a x ∈ O ( U ′ x , s x , 2 ε ) . There is an open neighborhood U x of x such that a xy ∈ O ( U ′ x , s , 2 ε ) y ∈ U x . One has V ⊂ ∪ x ∈V U x , and since V is compact there is finite { x , ..., x n } ∈ V such that V ⊂ ∪ nj = U x j . If 1 C ( V )= ∑ nj = φ ′ j is a subordinated by V ⊂ ∪ nj = U x j partition of unity then from the Theorem A.1.15 it follows that forany j =
1, ..., n there is φ ′′ j ∈ C ( X ) such that φ ′′ j ( X ) = [
0, 1 ] and φ ′′ j (cid:12)(cid:12)(cid:12) V = φ j .Similarly from the Theorem A.1.15 it turns out that there is ψ ∈ C ( X ) such that ψ ( X ) = [
0, 1 ] and ψ ( x ) = ( x ∈ V x / ∈ ∪ nj = U x j For all j =
1, ..., n denote by φ j def = ψφ ′ j . There is s ∈ C ( V , { A x } , F ) given by b def = ∑ nj = φ j s x j . From our construction it follows that b satisfies to the Equations(4.5.39).(ii) Let ε >
0. Using induction we construct the sequence { a n } n ∈ N ⊂ C ( V , { A x } , F ) , { b n } n ∈ N ⊂ C ( X , { A x } , F ) such that k a n k < ε n − , n > ⇒ k b n k < ε n − . (4.5.40)From (4.5.39) it follows that there is b ∈ C ( X , { A x } , F ) such that k a − b | V k < ε , k b k < k a k + ε .If a = a − b | V then one has k a k < ε . Applying (4.5.39) once again (replacing ε with ε /2) one can obtain b ∈ C ( U , { A x } , F ) such that k a − b | V k < ε k b k < k a k + ε < ε a = a = b | V then a , a , b , b satisfy to the Equations (4.5.40). If a n and b n areknown then using (4.5.39) one can find b n + such that k a n − b n + | V k < ε n , k b n + k < k a n k + ε n < ε n .126rom k b n + k < ε n it follows that the series b def = ∑ ∞ n = b n is norm convergent.Otherwise from (4.5.40) it follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − m ∑ n = b n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε m − ,hence one has a = b | V .Below in this Section we assume that all topological spaces are paracompact,connected and locally connected. If p : e X → X is a covering then the family { A x } naturally induces thefamily n e A e x o e x ∈ e X such that for any e x ∈ e X there is the isomorphism c e x : e A e x ∼ = A ( p ( e x )) . (4.5.41) Definition 4.5.38.
Let F be a continuity structure for X and the { A x } x ∈X , let p : e X → X be a covering. For any a ∈ C ( X , { A x } , F ) we define the family ne a e x = c − e x (cid:16) a p ( e x ) (cid:17) ∈ e A e x o e x ∈ e X (4.5.42)is said to be the p - lift of a and we write { e a e x } def = lift p ( a ) . (4.5.43) Lemma 4.5.39.
Let F be a continuity structure for X and the { A x } x ∈X , let p : e X → X be a covering. The space e F = n lift p ( a ) (cid:12)(cid:12)(cid:12) a ∈ C (cid:0) X , { A x } x ∈X , F (cid:1)o (4.5.44) is continuity structure for e X and the n e A e x o e x ∈ e X .Proof. Check (a) - (c) of the Definition D.8.27.(a) For every e x ∈ e X there is an open neighborhood e U such that the restriction p | e U is injective. For any a ∈ C (cid:0) X , { A x } x ∈X , F (cid:1) the function norm a is continuousat x = p ( e x ) . Otherwise for all e x ∈ e U one has norm lift p ( a ) ( e x ) = norm a ( p ( e x )) ,hence norm lift p ( a ) is continuous at e x .(b) The subspace { a x ∈ A x | a ∈ C ( X , { A x } , F ) } A x so the space n lift p ( a ) e x ∈ e A e x (cid:12)(cid:12)(cid:12) a ∈ C ( X , { A x } , F ) o is dense in e A e x .(c) If A x is a C ∗ -algebra for all x ∈ X then from the condition (c) of the DefinitionD.8.27 it follows that F is closed under pointwise multiplication and involution,i.e. for any a , b ∈ F represented by families { a x } and { b x } respectively the families { a ∗ x } and { a x b x } lie in F . If (cid:8)e a ∗ e x (cid:9) = lift p ( a ∗ ) and ne a e x e b e x o = lift p ( ab ) then from(4.5.42) it turns out that lift p ( a ∗ ) , lift p ( ab ) ∈ e F . Definition 4.5.40.
Let e F be given by (4.5.44). The space of continuous sections(with respect to) e F vector fields is said to be p - lift of C ( X , { A x } , F ) . We write lift p [ C ( X , { A x } , F )] def = C (cid:16) e X , n e A e x o , e F (cid:17) , (4.5.45) lift p [ F ] def = C (cid:16) e X , n e A e x o , e F (cid:17) (4.5.46)Following Lemma is a direct consequence of the Lemma 4.5.39 and the Defini-tion 4.5.40. Lemma 4.5.41. . If F is a continuity structure for X and the { A x } then one has:(i) For any covering p : e X → X there is the natural injective C b ( X ) -linear map. lift p : C ( X , { A x } , F ) ֒ → lift p [ C ( X , { A x } , F )] . (4.5.47) (ii) If p ′ : X ′ → X and p ′′ : X ′′ → X ′ are coverings then one has lift p ′ ◦ p ′′ [ C ( X , { A x } , F )] = lift p ′ h lift p ′′ [ C ( X , { A x } , F )] i , (4.5.48) ∀ a ∈ C ( X , { A x } , F ) lift p ′ ◦ p ′′ ( a ) = lift p ′ ◦ lift p ′′ ( a ) . (4.5.49)The following Lemma is evident. Lemma 4.5.42.
For any a ∈ C b ( X , { A x } , F ) one has k a k = (cid:13)(cid:13)(cid:13) lift p ( a ) (cid:13)(cid:13)(cid:13) , (4.5.50) hence the map (4.5.47) induces the norm preserving map lift p : C b ( X , { A x } , F ) ֒ → C b (cid:16) e X , n e A e x o , e F (cid:17) ,128 .e. one has the isometry lift p : C b ( X , { A x } , F ) ֒ → C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) . (4.5.51) Lemma 4.5.43.
If p : e X → X is a finite-fold covering then one hasC c ( X , { A x } , F ) ⊂ C c (cid:16) e X , n e A e x o , e F (cid:17) == C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , C ( X , { A x } , F ) ⊂ C (cid:16) e X , n e A e x o , e F (cid:17) == C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) . (4.5.52) Proof. If a ∈ C c ( X , { A x } , F ) then supp a is compact. Since p is a finite-fold cover-ing supp lift p ( a ) = p − ( supp a ) ⊂ e X is compact, hence lift p ( a ) ∈ C c (cid:16) e X , n e A e x o , F (cid:17) .For any a ∈ C ( X , { A x } , F ) there is a net { a α ∈ C c ( X , { A x } , F ) } α ∈ A such thatthere is a norm limit lim α a α = a . Otherwise for any α , β ∈ A one has (cid:13)(cid:13) a α − a β (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) lift p ( a α ) − lift p (cid:0) a β (cid:1)(cid:13)(cid:13)(cid:13) it follows that the net n lift p ( a α ) ∈ C c (cid:16) e X , n e A e x o , e F (cid:17)o isconvergent and lim α lift p ( a α ) = lift p ( a ) . Remark 4.5.44.
Similarly to the Remark 4.5.20 C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) is theminimal norm closed subspace of C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) which contains allproducts f lift p ( a ) where a ∈ C ( X , { A x } , F ) and f ∈ C (cid:16) e X (cid:17) . Lemma 4.5.45.
Let F be a continuity structure for X and the { A x } x ∈X , let p : e X → X be a transitive covering.(i) There is the natural actionG (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × lift p [ C ( X , { A x } , F )] → lift p [ C ( X , { A x } , F )] which yields actionsG (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) → C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) → C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) → C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) .129 ii) If p ′ : e X ′ → e X is a transitive covering thenG (cid:16) e X ′ (cid:12)(cid:12)(cid:12) X (cid:17) lift p ′ (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) == lift p ′ (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) X (cid:17) lift p ′ (cid:16) C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:17) == lift p ′ (cid:16) C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:17) . (4.5.53) Moreover if p ′ is a finite-fold covering then one hasG (cid:16) e X ′ (cid:12)(cid:12)(cid:12) X (cid:17) lift p ′ (cid:16) C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:17) == lift p ′ (cid:16) C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:17) , G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) X (cid:17) lift p ′ (cid:16) C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:17) == lift p ′ (cid:16) C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:17) . (4.5.54) (iii) If A x is a C ∗ -algebra for any x ∈ X then any g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) yields automor-phisms of involutive algebras lift p [ C ( X , { A x } , F )] , C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) ,C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) and C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) .Proof. (i) For every g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) from (4.5.41) it turns out that e A e x ∼ = A p ( e x ) ∼ = e A g e x If e a ∈ lift p [ C ( X , { A x } , F )] corresponds to a family { e a e x } e x ∈ e X then we define g e a such that it is given by the family (cid:8)e a g e x (cid:9) e x ∈ e X . The given by (4.5.44) continuousstructure e F is G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) - invariant, it turns out that (cid:8)e a g e x (cid:9) e x ∈ e X is continuous (withrespect to e F ) (cf. Definition D.8.28). So for any g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) one has anisomorphism lift p [ C ( X , { A x } , F )] ≈ −→ lift p [ C ( X , { A x } , F )] , e a g e a .Any g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is in fact a homeomorphism it follows that supp e a is homeo-morphic to supp g e a . In particular if supp e a is compact then supp g e a is also compact,130o one has G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) = C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) .Taking into account that C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) is the norm completion of C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) we conclude G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) = C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) .From k e a k = k g e a k one concludes G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) = C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) .(ii) If e a ∈ lift p [ C ( X , { A x } , F )] then e a corresponds to a family ne a e x ∈ e A e x o e x ∈ e X where e A e x ∼ = A p ( e x ) for each e x ∈ e X . The element e a ′ = lift p ( e a ) ∈ lift p ◦ p ′ [ C ( X , { A x } , F )] corresponds to the family ne a ′ e x ′ = c − e x ′ (cid:16)e a p ′ ( e x ′ ) (cid:17) ∈ e A ′ e x o e x ′ ∈ e X ′ where c e x ′ is given by (4.5.42). For any e g ∈ G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) e X (cid:17) one has p ′ ( e x ) = p ′ ( e g e x ) it turns out that e g e a ′ = e a ′ . Conversely if e a ′ = e g e a ′ for all e g ∈ G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) e X (cid:17) then clearly there is e a ∈ lift p [ C ( X , { A x } , F )] suchthat e a ′ = lift p ′ ( e a ) , i.e. lift p ′ (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) == n e a ′ ∈ lift p ◦ p ′ [ C ( X , { A x } , F )] (cid:12)(cid:12)(cid:12) e a ′ = e g e a ′ ∀ e g ∈ G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) e X (cid:17)o (4.5.55)The subgroup G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) e X (cid:17) ⊂ G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) X (cid:17) is normal (cf. Lemma 4.3.23), so forany g ∈ G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) X (cid:17) and e g ∈ G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) e X (cid:17) there is e g ′ ∈ G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) e X (cid:17) such that e gg = g e g ′ . If e a ′ ∈ lift p ′ (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) then e g ( g e a ′ ) = g ( e g ′ e a ′ ) = g e a ′ ,hence from (4.5.55) it turns out that g e a ′ ∈ lift p ′ (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) . If e a ∈ C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) then supp e a ∈ e X is compact. Moreover if p ′ is a finite-fold covering then supp lift p ′ ( e a ) ∈ e X ′ is compact. Since any g ′ ∈ G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) X (cid:17) is ahomeomorphism supp g ′ lift p ′ ( e a ) = g ′ supp lift p ′ ( e a ) is also compact so one has g ′ lift p ′ ( e a ) ∈ C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) (4.5.56)131f e b ∈ C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) then there is a net ne b α ∈ C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)o such that e b = lim α e b α . Taking into account (4.5.56) one has g ′ lift p ′ (cid:16)e b (cid:17) = lim α g ′ lift p ′ (cid:16)e b α (cid:17) ∈ C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) . (4.5.57)The equations (4.5.56) and (4.5.57) yield (4.5.54).(iii) If e a , e b ∈ lift p [ C ( X , { A x } , F )] correspond to ne a e x ∈ e A e x o e x ∈ e X and ne b e x ∈ e A e x o e x ∈ e X respectively, then the product e a e b corresponds to the families ne a e x e b e x o . For any g ∈ G (cid:16) e X ′ (cid:12)(cid:12)(cid:12) X (cid:17) elements g e a , g e b , ( g e a ) (cid:16) g e b (cid:17) correspond to (cid:8)e a g e x (cid:9) , ne b g e x o , ne a g e x e b g e x o .Otherwise g (cid:16)e a e b (cid:17) corresponds to ne a g e x e b g e x o , so one has ( g e a ) (cid:16) g e b (cid:17) = g (cid:16)e a e b (cid:17) . Theelements e a ∗ , g e a ∗ correspond to the families ne a ∗ e x ∈ e A e x o , ne a ∗ g e x ∈ e A e x o and takinginto account that g e a corresponds to (cid:8)e a g e x (cid:9) we conclude that ( g e a ) ∗ = g e a ∗ . Thus g is a *-automorphism of the involutive algebra lift p [ C ( X , { A x } , F )] , and tak-ing into account (i) of this Lemma we conclude that g is an automorphism ofthe involutive algebras C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) and C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) . If B ⊂ C ( A ) = C ( X , { A x } , A ) is a norm closed C -space then for any x ∈ X there is a normed C -subspace˚ B x = (cid:8) b x ∈ A x | ∃ b ′ ∈ B b x = b ′ x (cid:9) (4.5.58)Denote by B x the norm closure of ˚ B x and suppose B x = { } for any x ∈ X . The C -space B is a continuity structure for X and the { B x } x ∈X . Definition 4.5.47.
In the situation 4.5.46 the family { B x } x ∈X is the B - restriction of { A x } x ∈X . If B is a C ( X ) -module then from the Corollary 4.5.19 it turns out thatthere is the natural isomorphism B ∼ = C ( X , { B x } , B ) . (4.5.59) Definition 4.5.49.
In the situation 4.5.48 we define C c ( B ) = B ∩ C c ( A ) = { b ∈ B | norm b ∈ C c ( X ) } (4.5.60) C ( B ) = B ∩ C ( A ) = { b ∈ B | norm b ∈ C ( X ) } (4.5.61)132here are following inclusions C c ( B ) ⊂ C ( B ) = B (4.5.62)such that C c ( B ) is dense in C ( B ) . Lemma 4.5.50.
Let us consider a continuity structure A for X and the { A x } x ∈X suchthat A ∼ = C ( X , { A x } , A ) . Let both B ′ , B ′′ ⊂ A are C -subspaces, such that B ′ is a normclosed C ( X ) -module. If both { B ′ x } and { B ′′ x } are B ′ and B ′′ -restrictions of { A x } (cf.Definition 4.5.47) then one has ∀ x ∈ X B ′′ x ⊂ B ′ x ⇒ B ′′ ⊂ B ′ Proof.
From the Corollary 4.5.19 it follows that B ′ = C (cid:0) X , (cid:8) B ′ x (cid:9) , B ′ (cid:1) .Select b ′′ ∈ B ′′ and x ∈ X . For any ε > b ′ = { b ′ x } ∈ C ( X , { B ′ x } , B ′ ) such that (cid:13)(cid:13) b ′ x − b ′′ x (cid:13)(cid:13) < ε . The map x
7→ k b ′ x − b ′′ x k is continuous (cf. LemmaD.8.30), it turns out that there is an open neighborhood U of x such that k b ′ x − b ′′ x k < ε for every x ∈ U . If follows that b ′′ is continuous with respect to B ′ (cf. Defini-tion D.8.28). Taking into account that norm b ′′ ∈ C ( X ) it turns out that b ′′ ∈ C ( X , { B ′ x } , B ′ ) = B ′ , hence B ′′ ⊂ B ′ . Corollary 4.5.51.
Let us consider a continuity structure A for X and the { A x } x ∈X suchthat A ∼ = C ( X , { A x } , A ) . Let both B ′ , B ′′ ⊂ A be closed C ( X ) -modules. If both { B ′ x } and { B ′′ x } are B ′ and B ′′ -restrictions of { A x } (cf. Definition 4.5.47) then one has ∀ x ∈ X B ′′ x = B ′ x ⇒ B ′′ = B ′ . Let F be a continuity structure for X and the { A x } x ∈X , let p : e X → X bea covering. Let B ⊂ C ( X , { A x } , F ) is a norm closed C (cid:16) e X (cid:17) -module such thatthere is the B -restriction { B x } x ∈X of { A x } x ∈X (cf. Definition 4.5.47). If p : e X →X is a finite-fold covering then there are p -lifts n e A e x o e x ∈ e X and n e B e x o e x ∈ e X of both { A x } x ∈X and { B x } x ∈X respectively (cf. 4.5.38). For any x ∈ X there is the naturalinclusion B x ⊂ A x , which yields the inclusion e B e x = c − e x ( B x ) ⊂ e A e x ∀ e x ∈ e X c e x is given by (4.5.41). (4.5.63)From (4.5.63) and B ⊂ C ( X , { A x } , F ) for any b ∈ B one has lift p ( b ) ∈ lift p [ C ( X , { A x } , F )] .133aking into account the Definition 4.5.40 and equations (4.5.13), (4.5.14), (4.5.50)one has lift p [ C ( X , { B x } , B )] ⊂ lift p [ C ( X , { A x } , F )] , C c (cid:16) lift p [ C ( X , { B x } , B )] (cid:17) ⊂ C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , C (cid:16) lift p [ C ( X , { B x } , B )] (cid:17) ⊂ C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , C b (cid:16) lift p [ C ( X , { B x } , B )] (cid:17) ⊂ C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) . (4.5.64)The space lift p ( B ) is a continuity structure for e X and the n e B e x o (cf. DefinitionD.8.27). Definition 4.5.53.
In the above situation the space e B of continuous (with respectto lift p ( B ) ) vector fields n b e x ∈ e B e x o e x ∈ e X (cf. Definition D.8.28) is said to be the p - lift of B . We write lift p [ B ] def = e B . (4.5.65) Remark 4.5.54.
From the Definition 4.5.53 it turns out that there are injective C b ( X ) -linear maps lift p : B ֒ → lift p [ B ] , lift p : C b ( B ) ֒ → C b (cid:16) lift p [ B ] (cid:17) . (4.5.66)Moreover if p is a finite-fold covering there are following inclusions lift p : C c ( B ) ֒ → C c (cid:16) lift p [ B ] (cid:17) ; lift p : C ( B ) ֒ → C (cid:16) lift p [ B ] (cid:17) (4.5.67)(cf. Definition 4.5.49). Remark 4.5.55.
Similarly to (4.5.48) one has lift p ′ ◦ p ′′ [ B ] = lift p ′ h lift p ′′ [ B ] i . (4.5.68) Definition 4.5.56.
Let F continuity structure for X and the { A x } x ∈X , let p : e X →X be a covering. Let e U be an open set such that the restriction p | e U is injective. Let134 = p (cid:16) e U (cid:17) . If a ∈ C ( X , { A x } , F ) is such that supp a ⊂ U and e a = lift p ( a ) thenthe family e a e x = ( c − e x (cid:16) a p ( e x ) (cid:17) e x ∈ e U e x / ∈ e U c e x is given by (4.5.41) (4.5.69)is said to be the p - e U - lift or simply the e U - lift of a . Otherwise we say that a is the p - descent of e a . We write e a def = lift p e U ( a ) or simply e a def = lift e U ( a ) , a def = desc p ( e a ) or simply a def = desc ( e a ) . (4.5.70) Remark 4.5.57. if a = desc p ( e a ) then a is represented by the family { a x } x ∈X suchthat a x = ( c e x ( e a e x ) x ∈ U AND e x ∈ e U AND p ( e x ) = x x / ∈ U (4.5.71)where c e x is given by (4.5.41). Remark 4.5.58.
From the Equation (4.5.42) it turns out a = desc p (cid:16) lift p e U ( a ) (cid:17) , e a = lift p e U (cid:0) desc p ( e a ) (cid:1) . (4.5.72) Lemma 4.5.59.
Let ( E , π , X ) be vector bundle with the family of fibers { E x } x ∈X and let Γ ( E , π , X ) be a ( E , π , X ) -continuous structure (cf. Lemma 4.5.4). If p : e X → X isa covering and (cid:16) E × X e X , ρ , e X (cid:17) is the inverse image of ( E , π , X ) by p (cf. DefinitionA.3.7) then there is the natural C -isomorphism lift p [ C ( X , { E x } , Γ ( E , π , X ))] ∼ = Γ (cid:16) E × X e X , ρ , e X (cid:17) . Proof.
From the equation (4.4.3) it follows that there is the natural inclusion lift p ( C ( X , { E x } , Γ ( E , π , X ))) ⊂ Γ (cid:16) E × X e X , ρ , e X (cid:17) and taking into account the Lemma 4.5.4 one has lift p [ C ( X , { E x } , Γ ( E , π , X ))] ⊂⊂ C (cid:18) X , n(cid:16) E × X e X (cid:17) e x o e x ∈ e X , Γ (cid:16) E × X e X , ρ , e X (cid:17)(cid:19) = Γ (cid:16) E × X e X , ρ , e X (cid:17) .135et e s ∈ Γ (cid:16) E × X e X , ρ , e X (cid:17) be represented the a family ne s e x ∈ (cid:16) E × X e X (cid:17) e x o . Let e x ∈ e X be any point and ε >
0. Let e U is an open neighborhood of e x such that therestriction p | e U is injective. Let f e x ∈ C c (cid:16) e X (cid:17) is such that supp f e x ⊂ e U and there is aneighborhood e V of e x which satisfies to the condition f e x (cid:16) e V (cid:17) = { } (cf. Equation(4.1.8)). Denote by e t def = f e x e s ∈ Γ (cid:16) E × X e X , ρ , e X (cid:17) . The family { t x } x ∈X given by t p ( e x ) = c − e x (cid:0)e t e x (cid:1) p ( e x ) ∈ p (cid:16) e U (cid:17) AND e x ∈ e U p ( e x ) / ∈ p (cid:16) e U (cid:17) where c e x is given by (4.5.41)yields an element t ∈ Γ ( E , π , X ) . On the other hand one has lift p ( t ) e x = e s e x forall e x ∈ e V . It follows that e s continuous (with respect to lift p ( Γ ( E , π , X )) ) at x (cf. Definition D.8.28). Otherwise e x is arbitrary it turns out e s continuous (withrespect to lift p ( Γ ( E , π , X )) ), hence from the Definition 4.5.45 it follows that e s ∈ lift p [ Γ ( E , π , X )] . C ∗ -algebras Lemma 4.6.1.
Let X be a locally compact, Hausdorff space called the base space; and foreach x in X , let A x be a C ∗ -algebra. If F is a continuity structure for X and the { A x } x ∈X (cf. Definition D.8.27), then C b ( X , { A x } , A ) is a C ∗ -algebra.Proof. Follows from the condition (c) of the Definition D.8.27 and the LemmasD.8.29, D.8.31.
Since converging to zero submodule C ( X , { A x } , A ) (cf. Definition 4.5.8) is C ∗ -norm closed subalgebra of C b ( X , { A x } , A ) it is a C ∗ -algebra. Lemma 4.6.3.
In the situation of 4.6.2 there is the natural inclusionC b ( X , { A x } , F ) ֒ → M ( C ( X , { A x } , F )) of C ∗ -algebras.Proof. If a ∈ C ( X , { A x } , F ) and b ∈ C b ( X , { A x } , F ) then ab ∈ C b ( X , { A x } , F ) .Form the Lemma 4.5.11 it follows that norm a ∈ C ( X ) and from the Lemma4.5.6 it follows that norm a , norm ab ∈ C b ( X ) . Taking into account norm ab ≤ a norm b we conclude that norm ab ∈ C ( X ) , hence from the Lemma 4.5.11one has ab ∈ C ( X , { A x } , F ) . Similarly we prove that ba ∈ C ( X , { A x } , F ) , so b ∈ M ( C ( X , { A x } , F )) . If A = C ( X , { A x } , A ) is a C ∗ -algebra with continuous trace and U is anysubset then denote by C ( V , { A x } , A ) the V - restriction of C ( X , { A x } , A ) (cf. Def-inition 4.5.32). The space C ( V , { A x } , A ) has the natural structure of *-algebrawith C ∗ -norm. It is norm closed (cf. Remark 4.5.34), so C ( V , { A x } , A ) is a C ∗ -algebra. Lemma 4.6.5.
Consider the above situation. If X is a normal space and V ⊂ X is compactthen the following conditions hold.(i) If A | V is given by the Equation D.2.8 then there is the natural *-isomorphismA | V ∼ = C ( V , { A x } , A ) .(ii) The spectrum of A | V coincides with V .Proof. (i) From D.2.8 it follows that there is the surjective *-homomorphism p ′ : A → A | V . From the Lemma 4.5.36 it turns out that is the surjective *-homomorphism p ′ : A → C ( V , { A x } , A ) . From ker p ′ = ker p ′′ = A | X \U it follows that A | V ∼ = C ( V , { A x } , A ) .(ii) The spectrum of C ( V , { A x } , A ) equals to V so from(i) of this Lemma it followsthat the spectrum of A | V coincides with V .Below in this Section we assume that all topological spaces are paracompact,connected and locally connected. Lemma 4.6.6. If F is a continuity structure for X and the { A x } x ∈X , such that A x is aC ∗ -algebra for all x ∈ X and p : e X → X is a covering then(i) The map lift p : C ( X , { A x } , F ) ֒ → lift p [ X , { A x } , F ] is injective homomorphismof involutive algebras.(ii) The restriction lift p (cid:12)(cid:12)(cid:12) C b ( X , { A x } , F ) : C b ( X , { A x } , F ) ֒ → C b (cid:16) lift p [ X , { A x } , F ] (cid:17) is a homomorphism of C ∗ -algebras.(iii) If p is a finite-fold covering then there is an injective *-homomorphism of C ∗ -algebras lift p (cid:12)(cid:12)(cid:12) C ( X , { A x } , F ) : C ( X , { A x } , F ) ֒ → C (cid:16) lift p [ X , { A x } , F ] (cid:17) .137 roof. (i) If a ∈ C (cid:0) X , { A x } x ∈X , F (cid:1) , b ∈ C (cid:0) X , { A x } x ∈X , F (cid:1) then a , b and ab are represented by families { a x ∈ A x } , { b x ∈ A x } , { a x b x ∈ A x } .Otherwise lift p ( a ) , lift p ( a ) , lift p ( ab ) ∈ lift p [ X , { A x } , F ] are represented by fami-lies ne a e x = a p ( e x ) ∈ A p ( e x ) o e x ∈ e X , ne b e x = b p ( e x ) ∈ A p ( e x ) o e x ∈ e X , ne a e x e b e x = a p ( e x ) b p ( e x ) = ( ab ) p ( e x ) ∈ A p ( e x ) o e x ∈ e X ,hence one has lift p ( ab ) = lift p ( a ) lift p ( b ) . (4.6.1)Element a ∗ is represented by the family { a ∗ x ∈ A x } and lift p ( a ) is represented bythe family ne a ∗ e x = a ∗ p ( e x ) ∈ A p ( e x ) o e x ∈ e X it turns out that lift p ( a ∗ ) = lift p ( a ) ∗ . (4.6.2)(ii) Follows from (i) and (4.5.51).(iii) Follows from (i) and (4.5.52). Let X be a connected, locally connected, locally compact, Hausdorff spacecalled the base space; and for each x in X , let A x be a C ∗ -algebra. Let us considera continuity structure F for X and the { A x } x ∈X (cf. Definition D.8.27). Denote by A def = C (cid:0) X , { A x } x ∈X , F (cid:1) . (4.6.3)Let us consider the category FinCov - X given by the Definition 4.3.25. From 4.6.2it turns out that C ( X , { A x } , F ) is a C ∗ -algebra. If p is a finite-fold covering thenfrom the Lemma 4.6.6 it follows that lift p induces the injective *-homomorphism C ( X , { A x } , F ) ֒ → C (cid:16) lift p [ X , { A x } , F ] (cid:17) of C ∗ -algebras. From the Lemma 4.5.41 it follows contravariant functor A from FinCov - X to the category of C ∗ -algebras such that138 If p : e X → X is an object of the category
FinCov - X , i.e. p is a transitivefinite-fold covering then A ( p ) def = (cid:16) C ( X , { A x } , F ) ֒ → C (cid:16) lift p [ X , { A x } , F ] (cid:17)(cid:17) . (4.6.4)We also write A (cid:16) e X (cid:17) instead of A ( p ) and use the notation A (cid:16) e X (cid:17) def = C (cid:16) lift p [ X , { A x } , F ] (cid:17) (4.6.5)which is alternative to (4.6.4) (cf. Remark 4.3.27). • If p : e X → e X is morphism form p : e X → X to p : e X → X , i.e. p = p ◦ p , then A (cid:16) p (cid:17) def = lift p (cid:12)(cid:12)(cid:12) A ( e X ) : A (cid:16) e X (cid:17) ֒ → A (cid:16) e X (cid:17) . (4.6.6) Definition 4.6.8.
The described in 4.6.7 contravariant functor A is from the cate-gory the category FinCov - X to the category of C ∗ - algebras and *-homomorphismsis said to by the finite covering functor associated with A = (cid:0) X , { A x } x ∈X , F (cid:1) . Definition 4.6.9.
Let X be a connected, locally connected, locally compact, Haus-dorff space and for each x in X , let A x be a C ∗ -algebra. Let us consider a con-tinuity structure F for X and the { A x } x ∈X (cf. Definition D.8.27). Suppose that A def = (cid:0) X , { A x } x ∈X , F (cid:1) . If p : e X → X is a transitive covering then we use thefollowing notation for C ∗ -algebras and their injective *-homomorphisms: A c ( X ) def = C c ( X , { A x } , F ) , A ( X ) def = C ( X , { A x } , F ) , A b ( X ) def = C b ( X , { A x } , F ) , A c (cid:16) e X (cid:17) def = C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , A (cid:16) e X (cid:17) def = C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , A b (cid:16) e X (cid:17) def = C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , A b ( p ) def = lift p (cid:12)(cid:12)(cid:12) A ( X ) : A ( X ) ֒ → A b (cid:16) e X (cid:17) , A b ( p ) def = lift p (cid:12)(cid:12)(cid:12) A b ( X ) : A b ( X ) ֒ → A b (cid:16) e X (cid:17) (4.6.7)(cf. (ii) of the Lemma 4.6.6). 139 emark 4.6.10. If A x is a C ∗ -algebra for any x ∈ X then for any a , b ∈ C ( X , { A x } , F ) and e a , e b ∈ C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) one has supp e a ⊂ e U OR supp e b ⊂ e U ⇒ desc p (cid:16)e a e b (cid:17) = desc p ( e a ) desc p (cid:16)e b (cid:17) , supp a ⊂ U OR supp b ⊂ U ⇒ lift p e U ( ab ) = lift p e U ( a ) lift p e U ( b ) , supp e a ⊂ e U ⇒ desc p ( e a ∗ ) = desc p ( e a ) ∗ , supp a ⊂ U ⇒ lift p e U ( a ∗ ) = lift p e U ( a ) ∗ . (4.6.8) Consider a special case of the Definition 4.6.8 Let X be a locally compactHausdorff space. If we consider a family { C x } x ∈X of Banach spaces each of whichis isomorphic to C then C ( X ) is a continuity structure for X and the { C x } .Moreover if p : e X → X is a finite fold covering then one has C b (cid:16) e X (cid:17) ∼ = C b (cid:16) lift p [ C ( X , { C x } , C ( X ))] (cid:17) , C (cid:16) e X (cid:17) ∼ = C (cid:16) lift p [ C ( X , { C x } , C ( X ))] (cid:17) . (4.6.9) Definition 4.6.12.
Consider the finite covering functor associated with A = ( X , { C x } , C ( X )) (cf. Definition 4.6.8) from the category FinCov - X to the category of C ∗ - algebrasand *-homomorphisms is said to be the finite covering algebraic functor . We denotethis functor by C . The restriction of C on FinCov - ( X , x ) is also said to be the finite covering algebraic functor and also denoted by C . Remark 4.6.13.
If we use described in the Remark 4.3.27 alternative notation thenfor any object p : e X → X of FinCov - X the value of C coincides with C (cid:16) e X (cid:17) , i.e.this notation is tautological. If p : e X → e X is a morphism form p : e X → X to p : e X → X , i.e. p is a continuous map p : e X → e X such that p ◦ p = p (cf.Definition 4.3.25) then C (cid:16) p (cid:17) def = C (cid:16) p (cid:17) : C ( X ) ֒ → C ( X ) . (4.6.10)is the natural injective *-homomorphism. Lemma 4.6.14.
Let X be a connected, locally connected, locally compact, Hausdorff space,and let F be continuity structure for X and the { A x } x ∈X (cf. Definition D.8.27) whereA x is a C ∗ -algebra for every x ∈ X . Let p : e X → X be a covering. Let e U be an open set uch that the restriction p | e U is injective and U = p (cid:16) e U (cid:17) . If the closure of U is compact,both C ( X , { A x } , F ) | U and C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:12)(cid:12)(cid:12) e U are given by (4.5.12) thenthere is the natural *-isomorphismC ( X , { A x } , F ) | U ∼ = C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:12)(cid:12)(cid:12) e U . Proof.
There are two C -linear maps lift p e U : C ( X , { A x } , F ) | U → C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:12)(cid:12)(cid:12) e U , desc p : C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:12)(cid:12)(cid:12) e U → C ( X , { A x } , F ) | U and from (4.5.70) it turns out that the maps are mutually inverse. Moreover from(4.6.8) it follows that both maps are ∗ -homomorphisms. Lemma 4.6.15.
Let X be a connected, locally connected, locally compact, Hausdorff space,and let F be continuity structure for X and the { A x } x ∈X (cf. Definition D.8.27) whereA x is a C ∗ -algebra for every x ∈ X . Suppose that A def = C (cid:0) X , { A x } x ∈X , F (cid:1) .(i) If p : e X → X is a transitive covering then there is the natural injective *-homomorphism M ( A ( p )) : M ( A ( X )) ֒ → M (cid:16) A (cid:16) e X (cid:17)(cid:17) . (4.6.11) (ii) There is the natural action G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × M (cid:16) A (cid:16) e X (cid:17)(cid:17) → M (cid:16) A (cid:16) e X (cid:17)(cid:17) .(iii) If p : e X → X is a transitive finite-fold covering then M ( A ( p )) induces thefollowing *-isomorphismM ( A ( X )) ∼ = M (cid:16) A (cid:16) e X (cid:17)(cid:17) G ( e X | X ) . (4.6.12) Proof. (i) From the Theorem D.8.33 it turns out than any a ∈ M ( A ( p )) cor-responds to the strictly continuous section { a x ∈ M ( A x ) } x ∈X . The section de-fines the section ne a e x = a p ( e x ) ∈ M (cid:16) e A e x (cid:17) = M (cid:16) A p ( e x ) (cid:17)o e x ∈ e X . Let e x ∈ e X be anypoint consider an open neighborhood e U of e x such that the restriction p | e U : e U ≈ −→ U = p (cid:16) e U (cid:17) is a injective. If e f e x is given by (4.1.7) such that supp e f e x ⊂ e U and there is an open neighborhood e V of e x such that e f e x (cid:16) e V (cid:17) =
1. If e c ∈ (cid:16) e X (cid:17) = C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) , then supp e f e x e c ⊂ e U . Let c = desc p ( e a ) (cf. Definition 4.5.56). From the Theorem D.8.33 it follows that for any ε thereis an open V ′ neighborhood of x = p ( e x ) and the element b ∈ F such that k c x ( a x − b x ) k + k ( a x − b x ) c x k < ε for every x in V ′ . One can suppose V ′ ⊂ V . Let f x = desc p (cid:16) e f e x (cid:17) , e a = lift p e U ( f x a ) , e b = lift p e U ( f x b ) . Direct check shows that k e c e x ( e a e x − e a e x ) k + k ( e a e x − e a e x ) e c e x k < ε for all e x ∈ p − V ′ ∩ e U . From the the Theorem D.8.33 it turns out that e a = { e a e x } ∈ M (cid:16) C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:17) ∼ = M (cid:16) A (cid:16) e X (cid:17)(cid:17) .The map a e a is the required by this lemma injective *-homomorphism from M ( A ( X )) to M (cid:16) A (cid:16) e X (cid:17)(cid:17) .(ii) If e a ∈ M (cid:16) A (cid:16) e X (cid:17)(cid:17) represented by the section ne a e x ∈ M (cid:16) A p ( e x ) (cid:17)o e x ∈ e X and g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) then we define g e a ∈ M (cid:16) A (cid:16) e X (cid:17)(cid:17) as the element represented bythe section ne a g e x ∈ M (cid:16) A p ( e x ) (cid:17)o e x ∈ e X .(iii) Every e a ∈ M (cid:16) A (cid:16) e X (cid:17)(cid:17) G ( e X | X ) represented by a section ne a e x ∈ M (cid:16) A p ( e x ) (cid:17)o e x ∈ e X such that e a e x = e a g e x for each g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) . If turns out that p ( e x ′ ) = p ( e x ′′ ) ⇒ e a e x ′ = e a e x ′′ , hence there is the section n a x ∈ M (cid:16) A p ( x ) (cid:17)o x ∈X such that e a e x = a p ( e x ) forevery e x ∈ X . It turns out M ( A ( X )) ( a ) = e a ,i.e. one has the surjective *-homomorphism M ( A ( X )) → M (cid:16) A (cid:16) e X (cid:17)(cid:17) G ( e X | X ) .However the *-homomorphism M ( A ( X )) → M (cid:16) A (cid:16) e X (cid:17)(cid:17) is injective and M (cid:16) A (cid:16) e X (cid:17)(cid:17) G ( e X | X ) ⊂ M (cid:16) A (cid:16) e X (cid:17)(cid:17) hence the map M ( A ( X )) → M (cid:16) A (cid:16) e X (cid:17)(cid:17) G ( e X | X ) is injective, so one has the*-isomorphism M ( A ( X )) ∼ = M (cid:16) A (cid:16) e X (cid:17)(cid:17) G ( e X | X ) .142 emma 4.6.16. Let X be a connected, locally connected, locally compact, Hausdorff space,and let F be continuity structure for X and the { A x } x ∈X (cf. Definition D.8.27) whereA x is a C ∗ -algebra for every x ∈ X . Let p : e X → X be a transitive covering. Suppose that
U ⊂ X is a connected open subset evenly covered by e U ⊂ e X . Let a ∈ C c ( X , { A x } , F ) be such that supp a ⊂ U . If e a = lift e U ( a ) then following conditions hold:(i) The series ∑ g ∈ G ( e X | X ) g e a . is convergent in the strict topology of M (cid:16) C (cid:16) lift p [ C ( X , { A x } , F )] e X (cid:17)(cid:17) (cf. Def-inition D.1.12),(ii) ∑ g ∈ G ( e X | X ) g e a = a = lift p ◦ desc p ( e a ) ∈∈ C b (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) ⊂⊂ M (cid:16) C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:17) . (4.6.13) Proof. (i) If e b ∈ C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) and ε > e V ⊂ e X such that (cid:13)(cid:13)(cid:13)e b ( e x ) (cid:13)(cid:13)(cid:13) < ε k e a k for any e x ∈ e X \ e V . Let us prove that the set G = n g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17)(cid:12)(cid:12)(cid:12) e V ∩ g e U 6 = ∅ o If G is not finite then there is an infinite closed set e X ′ = (cid:8)e x g (cid:9) g ∈ G such that e x g ∈ e V ∩ e U . If g ′ = g ′′ then both g ′ e U and g ′′ e U are open neighborhoods of both e x g ′ and e x g ′′ such that g ′ e U ∩ g ′′ e U = ∅ . It means that the set e X ′ is discrete. From e X ′ ⊂ e V it follows that e X ′ is compact, however any infinite discrete set is notcompact. From this contradiction we conclude that the set G is finite. For any G ′ ⊂ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such that G ⊂ G ′ a following condition holds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)e b ∑ g ∈ G ′ g e a − ∑ g ∈ G g e a !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G ′ g e a − ∑ g ∈ G g e a ! e b (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε .143he above equation means that the series is convergent in the strict topology of M (cid:16) C (cid:16) lift p [ C ( X , { A x } , F )] e X (cid:17)(cid:17) .(ii) If e b ∈ C (cid:16) lift p [ C ( X , { A x } , F )] e X (cid:17) and e x ∈ e X then (cid:16)e a e b (cid:17) e x = a p ( e x ) e b e x . If a p ( e x ) = (cid:16)e a e b (cid:17) e x =
0. Otherwise there is e x ′ ∈ e U such that p ( e x ′ ) = p ( e x ) .From (4.5.42) it turns out e a e x ′ = a p ( e x ′ ) . The covering p is transitive, so there is theunique g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such that e x = g e x ′ . So one has ( g e a ) e x = a p ( e x ) . If g ′ = g then g e U T e U = ∅ . It turns out that ( g ′ e a ) e x = g ′ = g . Hence one has ∑ g ′ ∈ G ( e X | X ) g ′ e a e x = ( g e a ) e x = c − e x (cid:16) a p ( e x ) (cid:17) where c e x is given by (4.5.41),it follows that ∑ g ′ ∈ G ( e X | X ) g ′ e a e b e x = e b ∑ g ′ ∈ G ( e X | X ) g ′ e a e x = e b e x a p ( e x ) .The above equation means that ∑ g ∈ G ( e X | X ) g e a = a = lift p ◦ desc p ( e a ) ∈ M (cid:16) C (cid:16) lift p [ C ( X , { A x } , F )] (cid:17)(cid:17) . Suppose that K = K ( H ) is a simple C ∗ -algebra (cf. Definition D.1.31) ofcompact operators where H = C or H = C n or H = ℓ ( N ) . If A is C ∗ -algebrasuch that the spectrum X of A is Hausdorff then A is a CCR algebra (cf. RemarkD.8.24). There is a family { rep x ( A ) } x ∈X such that rep x ( A ) is simple C ∗ -algebrafor all x ∈ X . From the Lemma D.8.26 it follows that A is a continuity structurefor X and the { A x } (cf. Definition D.8.27). Otherwise from the Lemma 4.5.4 itturns out that A ∼ = C (cid:16) X , n A x def = rep x ( A ) o x ∈X , A (cid:17) (4.6.14)where A x ∼ = K ( H ) for all x ∈ X . If p : e X → X is a covering then A (cid:16) e X (cid:17) def = C (cid:16) e X , n e A e x o , e F (cid:17) are C ∗ -algebra with continuous trace and e X is the spectrum of144 (cid:16) e X (cid:17) . If e a ∈ A (cid:16) e X (cid:17) ) is represented by the family { e a e x } e x ∈ e X then one has e a e x = rep e x ( e a ) (4.6.15)where rep e x : A (cid:16) e X (cid:17) → B ( H e x ) is the irreducible representation which corre-sponds to e x (cf. Equation D.2.3). If π a : A (cid:16) e X (cid:17) ֒ → B ( H a ) is the atomic represen-tation (cf. Definition D.2.33) then H a = L e x ∈ e X H e x and π a = L e x ∈ e X rep e x . Otherwisethe representation π a can be uniquely extended up to π a : A b (cid:16) e X (cid:17) → B ( H a ) .From (4.6.7) it turns out there is the injective *-homomorphism lift p : A ( X ) ֒ → A b (cid:16) e X (cid:17) so A ( X ) can be regarded as a subalgebra of A b (cid:16) e X (cid:17) . Lemma 4.6.18.
Consider the situation 4.6.17. Suppose that
U ⊂ X is a connected opensubset evenly covered by e U ⊂ e X . Let e a ∈ A (cid:16) e X (cid:17) be an element such that supp e a ⊂ e U .The following conditions hold:(i) The series ∑ g ∈ G ( e X | X ) π a ( g e a ) . (4.6.16) is convergent in the strong topology of B ( H a ) (cf. Definition D.1.20 ),(ii) ∑ g ∈ G ( e X | X ) π a ( g e a ) = π a (cid:0) desc p ( e a ) (cid:1) (4.6.17) where desc p ( e a ) ∈ A is the p-descend e a of (cf. Definition 4.5.56) and A is regarded as asubalgebra of A b (cid:16) e X (cid:17) (cf. Lemma 4.6.16).Proof. (i) Since H a is the Hilbert completion of the algebraic direct sum L e x ∈ e X H e x one can apply the Lemma 3.1.28. If ζ =
0, ... , ζ e x |{z} e x th − place , ..., 0 For all g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) one has ( g e a ) ζ = ( e x / ∈ g e U ( g e a ) ζ = desc p ( e a ) ζ e x ∈ g e U ,145ence for each finite subset G ⊂ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) the following condition holds ∑ g ∈ G g e a ! ζ = ( e x / ∈ ∪ g ∈ G g e U ( g e a ) ζ = desc p ( e a ) ζ e x ∈ ∪ g ∈ G g e U (4.6.18)From the Equitation (4.6.18) it follows that the series (cid:16) ∑ g ∈ G ( e X | X ) g e a (cid:17) ζ is normconvergent. From the Lemma it turns out that the series 4.6.16 is convergent inthe strong topology of B ( H a ) .(ii) Follows from the Equation (4.6.18). Lemma 4.6.19.
Let A def = C (cid:0) X , { A x } x ∈X , F (cid:1) be a C ∗ -algebra described in 4.6.17, andsuppose that X is a connected, locally connected, locally compact, Hausdorff space. Letboth p : e X → X , p : e X → X are transitive coverings. If π : A (cid:16) e X (cid:17) → A (cid:16) e X (cid:17) is an injective *-homomorphism such that A ( p ) = π ◦ A ( p ) (cf. Notation (4.6.4) )then following conditions hold:(a) There is the transitive covering p : e X → e X such that π = A (cid:0) p (cid:1) (cf. Equation (4.6.7) ).(b) If G j = n g ∈ Aut (cid:16) A (cid:16) e X j (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ∀ a ∈ A o , j =
1, 2 (4.6.19) then for there are the natural isomorphismsG j ∼ = G (cid:16) e X j (cid:12)(cid:12)(cid:12) X (cid:17) , j =
1, 2. (4.6.20)
Moreover one has G π (cid:16) A (cid:16) e X (cid:17)(cid:17) = π (cid:16) A (cid:16) e X (cid:17)(cid:17) . (c) The natural homomorphism h : G → G such that π ( h ( g ) a ) = g ◦ π ( a ) , ∀ a ∈ A (cid:16) e X (cid:17) is surjective.(d) If ρ : A (cid:16) e X (cid:17) → A (cid:16) e X (cid:17) is any injective *-homomorphism such that A ( p ) = ρ ◦ A ( p ) then there is the unique g ∈ G such that ρ = π ◦ g .146 roof. (a) Let e x ∈ e X . Let e U ′ is a connected open subset neighborhood of e x which is mapped homeomorphically onto p (cid:16) e U ′ (cid:17) . There is a connected opensubset e U ′ ⊂ e X which is mapped homeomorphically onto p (cid:16) e U ′ (cid:17) and p ( e x ) ∈ p (cid:16) e U ′ (cid:17) . Let us select an open connected neighborhood U of p (cid:0) e × (cid:1) such that U ⊂ p (cid:16) e U ′ (cid:17) ∩ p (cid:16) e U ′ (cid:17) and let e U = e U ′ ∩ p − ( U ) and e U = e U ′ ∩ p − ( U ) . If theideals A | U ⊂ A , A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) e U ⊂ A (cid:16) e X (cid:17) and A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) e U ⊂ A (cid:16) e X (cid:17) are given bythe Equation (D.2.7) ideals then the generated by A ( p ) ( A | U ) and A ( p ) ( A | U ) hereditary subalgebras of A (cid:16) e X (cid:17) and A (cid:16) e X (cid:17) are equal to M g ∈ G ( e X | X ) g A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) e U ⊂ A (cid:16) e X (cid:17) , M g ∈ G ( e X | X ) g A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) e U ⊂ A (cid:16) e X (cid:17) , (4.6.21)i.e. there are direct sum decompositions of both (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , A ( p ) (cid:17) and (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , A ( p ) (cid:17) (cf. Definition 2.2.16). The system of equa-tions (4.6.21) is a special instance of the equations 2.2.13. If p e : L g ∈ G ( e X | X ) g A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) e U → A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) e U is the natural projection then fromthe Lemma 2.2.17 it follows that there is the unique g ∈ G such that p e ◦ π (cid:18) g A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) e U (cid:19) = { } . Both homeomorphisms e U ∼ = U and g e U ∼ = U yield ahomeomorphism ϕ : e U → g e U ∼ = U . Assume that p ( e x ) def = ϕ ( e x ) . This defini-tion does not depend on choice of the sets U , e U , e U because if we select U ′ ⊂ U , e U ′ ⊂ e U , e U ′ ⊂ e U such that e x ∈ e U ′ then we obtain the same p ( e x ) . In resultone has a continuous map p : e X → e X which is surjective because π is injec-tive. From the Lemma 2.2.17 it follows that there is the surjecive homomorphism φ : G → G such that π (cid:18) g A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) e U (cid:19) ⊂ M g ∈ ker φ g A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) e U It turns out that (cid:0) p (cid:1) − (cid:16) g e U (cid:17) = G g ∈ ker φ g e U ,147.e. p is a covering. From the Corollary 4.3.8 it turns that the map p is a transitivecovering.(b) If g j ∈ G j then from (a) of this lemma it follows that there is a covering q j : e X j → e X j such that q j = p j ◦ q j and A (cid:0) q j (cid:1) = g . From the Definition A.2.3 it turnsout that q j ∈ G (cid:16) e X j (cid:12)(cid:12)(cid:12) X (cid:17) and one has the isomorphism G j ∼ = G (cid:16) e X j (cid:12)(cid:12)(cid:12) X (cid:17) , g j q j .The equation G π (cid:16) A (cid:16) e X (cid:17)(cid:17) = π (cid:16) A (cid:16) e X (cid:17)(cid:17) .follows from (4.5.54) and the isomorphism G j ∼ = G (cid:16) e X j (cid:12)(cid:12)(cid:12) X (cid:17) .(c) Follows from (4.6.20) and the natural surjective homomorphism G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) → G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) .(d) If ρ : A (cid:16) e X (cid:17) → A (cid:16) e X (cid:17) is any *-homomorphism then from (a) of this lemmait turns out that there is a covering q : e X → X such that p ◦ q = p and ρ = C ( q ) .From the Lemma 4.3.28 it follows that there is g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such that q = g ◦ p .It follow that C ( q ) = C (cid:0) p (cid:1) ◦ g (4.6.22)where g is regarded as element of Aut (cid:16) A (cid:16) e X (cid:17)(cid:17) . Taking into account ρ = C ( q ) and π = C (cid:0) p (cid:1) the equation (4.6.22) is equivalent to ρ = π ◦ g . If we consider the situation of the Definition H.4.6 then inverse image sheaf f − G on X be the sheaf associated to the presheaf U 7→ lim V⊇ f ( U ) G ( V ) , where U is anyopen set in X , and the limit is taken over all open sets V of V containing f ( U ) . Ifthe map f is surjective then one has lim V⊇ f ( X ) G ( V ) ∼ = G ( X ) it follows that thereis the injective homomorphism p − G : G ( X ) ֒ → f − G ( Y ) (4.7.1)of Abelian groups. 148et F be a continuity structure for X and the { A x } x ∈X , and let C ( X , { A x } , F ) the C b ( X ) -module of continuous sections (with respect to) F vector fields. For any a ∈ C ( X , { A x } , F ) there is a family { a x ∈ A x } x ∈X . For any open subset U ⊂ X there is the restriction a | U which is a family { a x ∈ A x } x ∈U . The image of the map a a | U is an Abelian group we denote it by G ( X , { A x } , F ) ( U ) . Clearly the map U 7→ G ( X , { A x } , F ) ( U ) (4.7.2)yields a presheaf. Definition 4.7.1.
The sheaf associated with the given by (4.7.2) presheaf is the sheafof the continuity structure F . We denote it by C ( X , { A x } , F ) . Lemma 4.7.2.
Let X be a locally compact, locally connected, Hausdorff space. Let F be acontinuity structure for X and the { A x } x ∈X . Let p : e X → X be a covering andC (cid:16) e X , n e A e x o , e F (cid:17) = lift p [ C ( X , { A x } , F )] is the p-lift of C ( X , { A x } , F ) (cf. Definition 4.5.40). If both C ( X , { A x } , F ) and C (cid:16) e X , n e A e x o , e F (cid:17) are sheaves of the continuity structures F and e F respectively then C (cid:16) e X , n e A e x o , e F (cid:17) is the inverse image of C ( X , { A x } , F ) (cf. Definition H.4.6), i.e. C (cid:16) e X , n e A e x o , e F (cid:17) = p − ( C ( X , { A x } , F )) .Proof. From the Definition H.4.6 it follows that p − ( C ( X , { A x } , F )) is associ-ated to the presheaf G given by e U 7→ lim
U ⊇ p ( e U ) C ( X , { A x } , F ) ( U ) , where e U is any open set in e X , and the limit is taken over all open sets U of X contain-ing p (cid:16) e U (cid:17) . In particular if e U is homeomorphically mapped onto p (cid:16) e U (cid:17) then G (cid:16) e U (cid:17) ∼ = C ( X , { A x } , F ) (cid:16) p (cid:16) e U (cid:17)(cid:17) . Let e s ∈ p − ( C ( X , { A x } , F )) (cid:16) e U (cid:17) . From(2) of H.4.5 it follows that for any e x ∈ e U there is an open neighborhood e V of e x such that e V ⊂ e U and the element e t ∈ G (cid:16) e V (cid:17) such that for all e y ∈ e V thegerm e t e y of e t at e y is equal to e s ( e y ) . For any e x we select an neighborhood e V e x of e x such that e V e x satisfies to (2) of H.4.5 and e V e x is homeomorphically mapped onto p (cid:16) e V e x (cid:17) . From G (cid:16) e V x (cid:17) ∼ = C (cid:0) X , (cid:8) A y (cid:9) , F (cid:1) (cid:16) p (cid:16) e V x (cid:17)(cid:17) it follows that the restriction e s | e V x corresponds to element s e x ∈ C (cid:0) X , (cid:8) A y (cid:9) , F (cid:1) (cid:16) p (cid:16) e V x (cid:17)(cid:17) . From the Definition4.7.1 it follows that s e x corresponds to a continuous (with respect to F ) family (cid:8) a y ∈ A y (cid:9) y ∈ p ( e V x ) . So if c e y : e A e y → A p ( e y ) is given by (4.5.41) for any e y ∈ e V x e F ) family ne a e y = c − (cid:16) a p ( e y ) (cid:17)o e y ∈ e V e x . Let e x ′ , e x ′′ ∈ e U . If both ne a ′ e y ′ o e y ′ ∈ e V e x ′ and ne a ′′ e y ′′ o e y ′′ ∈ e V e x ′′ are families corresponding to both e s | e V x ′ and e s | e V x ′′ then from e s | e V x ′ (cid:12)(cid:12)(cid:12) e V x ′ ∩ e V x ′′ = e s | e V x ′′ (cid:12)(cid:12)(cid:12) e V x ′ ∩ e V x ′′ it follows that e a ′ e y = e a ′′ e y for all e y ∈ e V x ′ ∩ e V x ′′ . So there is the single family (cid:8)e a e y (cid:9) e y ∈ e U which is the combination ofcontinuous (with respect to e F ) families (cid:8)e a e y (cid:9) e y ∈ e V e x The family (cid:8)e a e y (cid:9) e y ∈ e U continuous(with respect to e F ) so it corresponds the element of C (cid:16) e X , n e A e x o , e F (cid:17) (cid:16) e U (cid:17) . If we consider the situation of proof of the Lemma 4.7.2 then the map lift p : C ( X , { A x } , F ) ֒ → lift p [ C ( X , { A x } , F )] is in fact an injective homomorphism lift p : C ( X , { A x } , F ) ( X ) ֒ → p − ( C ( X , { A x } , F )) (cid:16) e X (cid:17) of Abelian groups. One the other hand from (4.7.1) it follows that there is aninjective homomorphism p − C ( X , { A x } , F ) : C ( X , { A x } , F ) ( X ) ֒ → p − ( C ( X , { A x } , F )) (cid:16) e X (cid:17) .From the given by proof of the Lemma 4.7.2 construction one has p − C ( X , { A x } , F ) = lift p . (4.7.3) Example 4.7.4.
Let X be a locally compact Hausdorff space. If we consider afamily { C x } x ∈X of Banach spaces each of which is isomorphic to C then C ( X ) isa continuity structure for X and the { C x } . Moreover one has C ( X ) = C ( X , { C x } , C ( X )) If p : e X → X is a covering then C b (cid:16) e X (cid:17) ∼ = C b (cid:16) lift p [ C ( X , { C x } , C ( X ))] (cid:17) and C (cid:16) e X (cid:17) ∼ = C (cid:16) lift p [ C ( X , { C x } , C ( X ))] (cid:17) . There is the sheaf C (cid:16) e X , { C e x } , C ( X ) (cid:17) of continuous functions. 150 xample 4.7.5. If ( S , π , X ) is a Hermitian vector bundle then there is a family {S x } x ∈X . From the Lemma 4.5.4 it turns out that Γ ( X , S ) is a continuity structurefor X and the {S x } . Moreover one has Γ ( X , S ) = C ( X , {S x } , Γ ( X , S )) If p : e X → X is a covering and e S is the inverse image of S by p then from theLemma 4.5.59 it follows that Γ (cid:16) e X , e S (cid:17) = lift p [ C ( X , {S x } , Γ ( X , S ))] Hence one can apply to Γ ( X , S ) and Γ (cid:16) e X , e S (cid:17) given by Definition 4.5.56 opera-tions of the lift and the descent. Let X be a locally compact Hausdorff space. Let F be a continuity structurefor X and the { A x } x ∈X , and let C def = C ( X , { A x } , F ) be the sheaf of the continuitystructure F (cf. Definition 4.7.1). From the Lemma D.8.30 for any open subset U ⊂ X there is the natural action C b ( X ) × C ( U ) → C ( U ) . If S ⊂ C is a subsheafof C then it is possible that C b ( X ) S ( U ) S ( U ) . However it is possible thatthere is a subring R ⊂ C b ( X ) such that R S ( U ) ⊂ S ( U ) . Definition 4.7.7.
Let M be a Riemannian manifold Let F be a continuity structurefor M and the { A x } x ∈ M , and let C def = C ( M , { A x } , F ) be the sheaf of the conti-nuity structure F (cf. Definition 4.7.1). Let U ⊂ M be an open subset, and let C ∞ ( M ) × C ( U ) → C ( U ) be the action induced by the action C b ( M ) × C ( U ) → C ( U ) . A subsheaf S ⊂ C is said to be smooth if C ∞ ( M ) S ( U ) ⊂ S ( U ) for allopen U ⊂ M . Lemma 4.7.8. subsheaf S ⊂ C is smooth if and only if for any x ∈ M there is a basis ofopen neighborhoods {U α } of x such that C ∞ ( M ) S ( U α ) ⊂ S ( U α ) .Proof. If the subsheaf S ⊂ C is smooth then for any x ∈ M there is a basis ofopen neighborhoods {U α } of x such that C ∞ ( M ) S ( U α ) ⊂ S ( U α ) . Converselylet U ⊂ M be an open subset. For any x ∈ U we select an open neighborhood U x ∈ {U α } such that U x ⊂ U , so one has U = ∪ x ∈U U x . From the Definition H.4.2 itfollows that any section s ∈ S ( U ) is uniquely defined by its restrictions s | U x suchthat s | U x ′ (cid:12)(cid:12)(cid:12) U x ′ ∩ U x ′′ = s | U x ′′ (cid:12)(cid:12)(cid:12) U x ′ ∩ U x ′′ .151f f ∈ C ∞ ( M ) then f s | U x ∈ S ( U x ) and f s | U x ′ (cid:12)(cid:12)(cid:12) U x ′ ∩ U x ′′ = f s | U x ′′ (cid:12)(cid:12)(cid:12) U x ′ ∩ U x ′′ .so from the Definition H.4.2 it follows that f s ∈ S ( U ) , i.e. S is smooth. Example 4.7.9.
Consider the Example 4.7.4. Suppose that X = M is a Riemannianmanifold, and S is a spinor bundle (cf. Section E.4.1). There is a C ∞ ( M ) submod-ule Γ ∞ ( M , S ) ⊂ Γ ( M , S ) of smooth sections (cf. Definition E.4.1), which inducedthe smooth subsheaf C ∞ ( M , S ) ⊂ C ( M , {S x } , Γ ( X , S )) . Lemma 4.7.10.
Consider the situation of the Definition 4.7.7 and suppose that p : e M → M be a transitive covering of where e M has given by the Proposition E.5.1 structure of aRiemannian manifold. If subsheaf S ⊂ C def = C ( M , { A x } , F ) is smooth then the inverseimage p − S ⊂ p − C is smooth.Proof. Let e x ∈ e M , select a basis n e U α o of open connected neighborhoods of e x suchthat the restriction p | e U α is injective. If U α def = p (cid:16) e U α (cid:17) then one has p − S (cid:16) e U α (cid:17) ∼ = S ( U α ) C ∞ (cid:16) e U α (cid:17) ∼ = C ∞ ( U α ) . (4.7.4)From (4.7.4) it follows that C ∞ (cid:16) e U α (cid:17) p − S (cid:16) e U α (cid:17) ⊂ p − S (cid:16) e U α (cid:17) and taking intoaccount the Lemma 4.7.8 we conclude that the sheaf S is smooth.If consider the situation of the Lemma 4.7.10 then from (4.7.1) it turns out thatthere is the following inclusion p − H om ( S , S ) : H om ( S , S ) ( M ) ⊂ H om (cid:16) p − S , p − S (cid:17) (cid:16) e M (cid:17) .where H om is the sheaf of local morphisms (cf. H.4.7). Denote by E nd ( S ) def = H om ( S , S ) , E nd (cid:16) p − S (cid:17) def = H om (cid:16) p − S , p − S (cid:17) , p − E nd ( S ) def = p − H om ( S , S ) , p − E nd ( S ) : E nd ( S ) ⊂ E nd (cid:16) p − S (cid:17) . (4.7.5)152 efinition 4.7.11. Let us consider the situation of the Lemma 4.7.10, and sup-pose that D ∈ E nd ( S ) ( M ) , ξ ∈ S ( M ) . We say that both p − E nd ( S ) ( D ) ∈ E nd (cid:0) p − S (cid:1) (cid:16) e M (cid:17) and p − S ( ξ ) ∈ S (cid:16) e M (cid:17) are the p - inverse images of D and ξ re-spectively. We write p − D def = p − E nd ( S ) ( D ) , p − ξ def = p − S ( ξ ) (4.7.6) Remark 4.7.12. If ξ ∈ S ( M ) and D ∈ E nd ( S ) ( M ) then one has p − ( D ξ ) = p − D p − ξ ∈ S (cid:16) e M (cid:17) . (4.7.7)If following conditions hold: • S ⊂ C (cid:16) e X , n e A e x o , e F (cid:17) , • e ξ ∈ S (cid:16) e M (cid:17) , • there is an open subset e U ⊂ e M homeomorphically mapped onto p (cid:16) e U (cid:17) suchthat supp e ξ ⊂ e U ,then one has p − D e ξ = lift p e U (cid:16) D desc p (cid:16) e ξ (cid:17)(cid:17) . (4.7.8)If both p : e M → M and p : e M → e M are coverings and p = p ◦ p then forevery D ∈ E nd ( S ) one has p − D = p − (cid:16) p − D (cid:17) . (4.7.9) Example 4.7.13.
Consider the situation of the Example 4.7.9, and let p : e M → M be a covering. If e S = p ∗ S is the inverse image of S by p (cf. Definition A.3.7) thenfrom the Lemma 4.5.59 it follows that there is the natural isomorphism Γ (cid:16) e M , e S (cid:17) ∼ = lift p [ C ( M , {S x } , Γ ( M , S ))] .If the subsheaf C ∞ (cid:16) e M , e S (cid:17) ⊂ C (cid:16) e M , e S (cid:17) is the p -inverse image of C ∞ ( M , S ) thenfrom the Lemma 4.7.10 it follows that the sheaf C ∞ (cid:16) e M , e S (cid:17) is smooth (cf. Defini-tion 4.7.7). So if Γ ∞ (cid:16) e M , e S (cid:17) def = C ∞ (cid:16) e M , e S (cid:17) (cid:16) e M (cid:17) Γ ∞ (cid:16) e M , e S (cid:17) is a C ∞ (cid:16) e M (cid:17) -module. The given by (E.4.5) Dirac operator can beregarded as an element of E nd ( C ∞ ( M , S )) ( M ) , i.e. D / ∈ E nd ( C ∞ ( M , S )) ( M ) .If e D / def = p − D / is the p -inverse image of D / (cf. Definition 4.7.11) then e D / = p − D / ∈ E nd (cid:16) C ∞ (cid:16) e M , e S (cid:17)(cid:17) (cid:16) e M (cid:17) . (4.7.10)Otherwise from the Equation (4.7.5) it turns out that e D / can be regarded as C -linearoperator, i.e. e D / : Γ ∞ (cid:16) e M , e S (cid:17) → Γ ∞ (cid:16) e M , e S (cid:17) .If Ω D def = { a [ D / , b ] ∈ E nd ( C ∞ ( M , S )) ( M ) | a , b ∈ C ∞ ( M ) } (4.7.11)then Ω D is a C ∞ ( M ) -bimodule and there is the natural inclusion Ω D ⊂ End C ∞ ( M ) ( Γ ∞ ( M , S )) ⊂ End C ( M ) ( Γ ( M , S )) .Any ω ∈ Ω D , is given by the family { ω x } x ∈ X where ω x ∈ End ( S x ) . There is thefamily of normed spaces { Ω ′ x } x ∈X such that any Ω ′ x is generated by the describedabove endomorphisms ω x ∈ End ( S x ) and the norm of Ω ′ x comes from the normof End ( S x ) . If Ω x is the norm completion of Ω ′ x then Ω x is the Banach space. The C ∞ ( M ) -module Ω D is a continuous structure for M and { Ω x } x ∈X and the space C ( M , { Ω x } , Ω D ) induces a sheaf C ( M , { Ω x } , Ω D ) . On the other hand form Ω D ⊂ E nd ( C ∞ ( M , S )) ( M ) the C ∞ ( M ) -module Ω D induces a smooth subsheaf C ∞ ( Ω D ) ⊂ E nd (cid:16) C ∞ (cid:16) e M , e S (cid:17)(cid:17) .If e U ⊂ e M is an open subset such that the restriction p e U is injective and a ∈ C ∞ ( M ) is such that supp a ⊂ U def = p (cid:16) e U (cid:17) then one has lift p e U ([ D / , a ]) = h p − D / , lift p e U ( a ) i ∈ p − C ∞ ( Ω D ) (cid:16) e M (cid:17) ⊂⊂ End C ∞ ( M ) (cid:16) Γ ∞ (cid:16) e M , e S (cid:17)(cid:17) . (4.7.12) Let X be a second-countable, locally compact, Hausdorff space. Let p : e X → X be a transitive covering (cf. Definition 4.3.1). It turns out that for any x ∈ X there154s a connected open neighborhood U x evenly covered by p . Since X be a second-countable there is a countable subset { x α } α ∈ A ⊂ X such that X = S α ∈ A U x α .Denote by U α def = U x α . From the Theorems A.1.29 and A.1.30 it turns out that X isparacompact, and from the Theorem A.1.25 it turns out that there is a partition ofunity ∑ α ∈ A a α = C b ( X ) (4.8.1)dominated by {U α } α ∈ A . Lemma 4.8.1. If X is a second countable, locally compact, Hausdorff space and C b ( X ) = ∑ α ∈ A a α (4.8.2) is a partition of unity (cf. Definition A.1.23 then the series (4.8.2) is convergent withrespect to the strict topology on the multiplier algebra M ( C ( X )) (cf. Definition D.1.12).Proof. Suppose the partition of unity is dominated by a countable covering X = S α ∈ A U α . If a ∈ C ( X ) and ε > U ⊂ X such that | a ( x ) | < ε /2 for every x ∈ X \ U .Since U is compact from the Corollary 4.2.5 it turns out that there is a coveringsum ∑ α ∈ A a α for U (cf. Definition 4.2.6) where the subset A ⊂ A if finite. For all x ∈ U one has ∑ α ∈ A a α ( x ) =
1, it follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − a ∑ α ∈ A a α !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − ∑ α ∈ A a α ! a (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε . Remark 4.8.2.
In the situation of the Lemma 4.2.3 if f n ∈ C ( X ) is given by (4.2.2)then from the Lemma 4.8.1 it follows that there is the limit1 C b ( X ) = β - lim n → ∞ f n . (4.8.3)with respect to the strict topology on the multiplier algebra M ( C ( X )) (cf. Defi-nition D.1.12). Lemma 4.8.3.
Let us consider the situation described in 4.3.12. For every α ∈ A weselect an open connected subset e U α of e X which is homeomorphically mapped onto U α . Onehas i) e X = [ g ∈ G ( e X | X ) , α ∈ A g e U α (4.8.4) (ii) If e a α = lift e U α ( a α ) ∈ C (cid:16) e X (cid:17) then the series ∑ ( g , α ) ∈ G ( e X | X ) × A g e a α = C b ( e X ) (4.8.5) is a partition of unity dominated by the family n g e U α o g ∈ G ( e X | X ) , α ∈ A . (4.8.6) Proof. (i) Let e x ∈ e X and let x = p ( e x ) . There is α ∈ A such that x ∈ U α . If e x ′ ∈ e U α is such that p ( e x ′ ) = x then there is g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such that e x = g e x ′ ,hence one has e x ∈ g e U α .(ii) The series (4.8.1) is a partition of unity it turns out there is an open connectedneighborhood U of x and a finite subset A ⊂ Λ such that U T supp a α = ∅ for any α ∈ A \ A and U T U α = ∅ for any α ∈ A . There is the unique connected openneighborhood e U of e x such that e U which is mapped homeomorphically onto U .For any α ∈ A there is U T supp a α = ∅ , so there is x α ∈ U T supp a α . Otherwisethere are e x α ∈ e U , e x ′ α ∈ e U α such that p ( e x α ) = p ( e x ′ α ) = x α . The covering p is atransitive, so there is the unique g α ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such that g α e x ′ α = e x α . It followsthat g α supp e a α T e U ∼ = supp a α T U . So there is a finite subset f A = n g α e U α o α ∈ A ⊂ n g e U α o g ∈ G ( e X | X ) , α ∈ A such that e U T g α supp e a α = ∅ if U T g α supp e a α ∈ e Λ and e U T g α e U α = ∅ if e U T g α supp e a α / ∈ e Λ . It follows that the family (4.8.6) is locally finite. From ∑ α ∈ A a ( x ) = ∑ α ∈ A g α e a ( e x ) = efinition 4.8.4. A partition of unity (4.8.5) given by the Lemma 4.8.3 is said to be compliant to the covering p : e X → X . The union (4.8.4), i.e. e X = [ g ∈ G ( e X | X ) , α ∈ A g e U α is also said to be compliant to the covering p : e X → X . Lemma 4.8.5.
Let e X be a second-countable, connected, locally connected, locally compact,Hausdorff space. Let p : e X → X be a transitive covering. Let us use the notation of theDefinition 4.6.9. Any e a ∈ A c (cid:16) e X (cid:17) can be represented by the following way e a = e a + ... + e a m where e a j ∈ A c (cid:16) e X (cid:17) and the set e U j = n e x ∈ e X (cid:12)(cid:12)(cid:12) e a j ( e x ) = o is homeomorphically mappedonto p (cid:16) e U j (cid:17) for all j =
1, ..., m. Moreover if e a is positive then one can select positive e a , ..., e a m .Proof. Consider the compliant to p C b ( e X ) = ∑ α ∈ A e a α partition of unity (cf. Definition 4.8.4). The set e U def = supp e a is compact so from theCorollary 4.2.5 it turns out that there is a covering sum ∑ α ∈ A U f α = m ∑ i = f α j for U (cf. Definition 4.2.6). So one has e a = f α e a + ... + f α m e a or, equivalently e a = e a + ... + e a m where e a j = f α j e a for every j =
1, ..., m . If e a is positive then e a j is positive for each j =
1, ..., m . From e U j ⊂ e U α j it follows that e U j is homeomorphically mapped onto p (cid:16) e U j (cid:17) .157 emma 4.8.6. If p : e X → X is a covering and the union e X = [ g ∈ G ( e X | X ) , α ∈ A g e U α is compliant to p : e X → X (cf. Definition 4.8.4) then there is the natural homeomorphism e X ∼ = G g ∈ G ( e X | X ) , α ∈ A g e U α / ∼ where e x ′ ∼ e x ′′ ⇔ e x ′ ∈ g ′ e U α ′ AND e x ′′ ∈ g ′′ e U α ′′ ANDAND g ′ e U α ′ ∩ g ′′ e U α ′′ = ∅ AND p (cid:0) e x ′ (cid:1) = p (cid:0) e x ′′ (cid:1) . (4.8.7) Proof. If e x ′ ∈ g ′ e U α ′ , e x ′′ ∈ g ′′ e U α ′′ and g ′ e U α ′ ∩ g ′′ e U α ′′ = ∅ then clearly e x ′ = e x ′′ .Otherwise if g ′ e U α ′ ∩ g ′′ e U α ′′ = ∅ then g ′ e U α ′ ∩ g ′′ e U α ′′ is homeomorphically mappedonto p (cid:16) g ′ e U α ′ ∩ g ′′ e U α ′′ (cid:17) , hence from p ( e x ′ ) = p ( e x ′′ ) it turns out e x ′ = e x ′′ . Remark 4.8.7.
If both X and e X are compact and G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is finite then one canselect a finite A . Moreover if f A = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × A and e e ( g , α ) = g √ e a α one has1 C ( e X ) = ∑ e α ∈ f A e e e α (4.8.8)If g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is not trivial then from g e U e α ∩ e U e α = ∅ it follows that e e e α ( g e e e α ) =
0. (4.8.9)From the equations (4.8.8) and (4.8.9) it follows that1 C ( e X ) = ∑ e α ∈ f A e e e α ih e e e α (4.8.10)where e e e α ih e e e α means a compact operator induced by the C ∗ -Hilbert module struc-ture given by (2.1.2). Let X be a compact, Hausdorff space; and for each x in X , let A x be a(complex) Banach space. Let us consider a continuity structure F for X . If p : e X → X is a finite-fold covering then C (cid:16) e X (cid:17) is a finitely generated C ( X ) -module(cf. Theorem 1.1.2). There is the map φ : C (cid:16) e X (cid:17) ⊗ C ( X ) C ( X , { A x } , F ) → lift p [ C ( X , { A x } , F )] ; n ∑ j = e a j ⊗ ξ j n ∑ j = e a j lift p (cid:0) ξ j (cid:1) . (4.8.11)158he map φ can be described by the following way. If for all j =
1, ..., n and ξ j is rep-resented by the family (cid:8) ξ jx (cid:9) x ∈X respectively then φ (cid:16) ∑ nj = e a j ⊗ ξ j e a j lift p (cid:0) ξ j (cid:1)(cid:17) isrepresented by the family ( n ∑ j = e a j ( e x ) c − e x (cid:0) ξ jx (cid:1)) e x ∈ e X where c e x is given by (4.5.41). (4.8.12) Lemma 4.8.9.
The given by (4.8.11) map φ is the isomorphism of left C (cid:16) e X (cid:17) -modules.Proof. For all e a ∈ C (cid:16) e X (cid:17) one has φ e a n ∑ j = e a j ⊗ ξ j ! = φ n ∑ j = e a e a j ⊗ ξ j ! == n ∑ j = e a e a j ⊗ lift p (cid:0) ξ j (cid:1) = e a n ∑ j = e a j ⊗ lift p (cid:0) ξ j (cid:1) = e a φ n ∑ j = e a j ⊗ ξ j ! ,i.e. φ is the homomorphism of left C (cid:16) e X (cid:17) -modules. From the equation (4.8.8) itfollows that n ∑ j = e a j ⊗ ξ j = ∑ e α ∈ f A n ∑ j = e e e α e e e α e a j ⊗ ξ j = ∑ e α ∈ f A n ∑ j = e e e α desc p (cid:0)e e e α e a j (cid:1) ⊗ ξ j = ∑ e α ∈ f A n ∑ j = e e e α ⊗ desc p (cid:0)e e e α e a j (cid:1) ξ j . (4.8.13)If φ (cid:16) ∑ nj = e a j ⊗ ξ j (cid:17) = ∑ nj = e a j ( e x ) c − e x (cid:0) ξ jx (cid:1) = e x ∈ e X . It turns out that ∑ nj = e e e α e a j ( e x ) c − e x (cid:0) ξ jx (cid:1) =
0, hence one has desc p (cid:0)e e e α e a j (cid:1) ξ j =
0, and taking into account (4.8.13) we conclude that ∑ nj = e a j ⊗ ξ j =
0, i.e. the map φ is injective.For all e ξ ∈ lift p [ C ( X , { A x } , F )] one has e ξ = φ ∑ e α ∈ f A e e e α ⊗ desc p (cid:16)e e e α e ξ (cid:17)! ,hence the map φ is surjective. 159 .8.10. Let X be a locally compact, paracompact, Hausdorff space; and for each x in X , let A x be a (complex) Banach space. Let us consider a continuity structure F for X . If p : e X → X is a covering then C c (cid:16) e X (cid:17) is a C ( X ) -module. There is themap φ : C c (cid:16) e X (cid:17) ⊗ C ( X ) C ( X , { A x } , F ) → C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) ; n ∑ j = e a j ⊗ ξ j e a j lift p (cid:0) ξ j (cid:1) (4.8.14)The map φ can be described by the following way. If for all j =
1, ..., n and ξ j is rep-resented by the family (cid:8) ξ jx (cid:9) x ∈X respectively then φ (cid:16) ∑ nj = e a j ⊗ ξ j e a j lift p (cid:0) ξ j (cid:1)(cid:17) isrepresented by the family ( n ∑ j = e a j ( e x ) c − e x (cid:0) ξ jx (cid:1)) e x ∈ e X where c e x is given by (4.5.41). (4.8.15) Lemma 4.8.11.
The given by (4.8.14) map φ is the isomorphism of left C (cid:16) e X (cid:17) -modules.Proof. Similarly the Lemma 4.8.9 one can proof that the map φ is a homomorphismof left C (cid:16) e X (cid:17) -modules. Denote by1 C b ( e X ) = ∑ e α ∈ f A e a e α = ∑ e α ∈ f A e e e α ; e e e α def = pe a e α ∀ e α ∈ f A (4.8.16)the compliant to the covering p : e X → X (cf. Definition 4.8.4) partition ofunity. For any finite sum ∑ nj = e a j ⊗ ξ j ∈ C c (cid:16) e X (cid:17) ⊗ C ( X ) C ( X , { A x } , F ) the set e U = ∪ nj = supp e a j is the finite union of compact sets, so that e U is compact. Fromthe Corollary 4.2.5 it follows that there is a covering sum (cf. Definition 4.2.6) for e U , i.e. a finite subset f A e U ⊂ f A such that ∑ e α ∈ f A e U e e e α ( e x ) = ∀ e x ∈ e U It follows that ∑ e α ∈ f A e U e e e α e a j = e a j ∀ j =
1, ..., n .160ence one has n ∑ j = e a j ⊗ ξ j = ∑ e α ∈ f A e U n ∑ j = e e e α e e e α e a j ⊗ ξ j = ∑ e α ∈ f A e U n ∑ j = e e e α desc p (cid:0)e e e α e a j (cid:1) ⊗ ξ j = ∑ e α ∈ f A e U n ∑ j = e e e α ⊗ desc p (cid:0)e e e α e a j (cid:1) ξ j . (4.8.17)If φ (cid:16) ∑ nj = e a j ⊗ ξ j (cid:17) = ∑ nj = e a j ( e x ) c − e x (cid:0) ξ jx (cid:1) = e x ∈ e X . It turns out that ∑ nj = e e e α e a j ( e x ) c − e x (cid:0) ξ jx (cid:1) =
0, hence one has desc p (cid:0)e e e α e a j (cid:1) ξ j =
0, and taking into account 4.8.17 we conclude ∑ nj = e a j ⊗ ξ j = φ is injective.For all e ξ ∈ C c ( X , { A x } , F ) → C c (cid:16) lift p [ C ( X , { A x } , F )] (cid:17) the set supp e ξ is com-pact. From the Corollary 4.2.5 it follows that there is a covering sum for supp e ξ , i.e.a finite subset f A supp e ξ ⊂ f A such that ∑ e α ∈ f A supp e ξ e e e α ( e x ) = ∀ e x ∈ supp e ξ ,hence one has ∑ e α ∈ f A supp e ξ e e e α e ξ = e ξ , e ξ = φ ∑ e α ∈ f A e U e e e α ⊗ desc p (cid:16)e e e α e ξ (cid:17) ,it turns out that the map φ is surjective. Let A be a homogeneous of order n C ∗ -algebra (cf. Definition D.8.20). If X is the spectrum of A then the Theorem D.2.26 yields an the inclusion f : C b ( X ) ֒ → M ( A ) . Taking into account that the C ∗ -algebra M n ( C ) is unital it is easy to provethat the inclusion f induces the inclusion C ( X ) ֒ → A (4.8.18)The C ∗ -algebra A can be represented by the following way A = C (cid:0) X , { M n ( C ) x } x ∈X , A (cid:1) B is a left A -module such that B = C (cid:0) X , { B x } x ∈X , B (cid:1) such thatthere is the action M n ( C ) x × B x → B x and the action A × B → B is given by ( { a x ∈ M n ( C ) x } , { ξ x ∈ B x } )
7→ { a x ξ x } If X is compact and p : e X → X is a finite-fold covering then there is the map φ : lift p (cid:2) C (cid:0) X , { M n ( C ) x } x ∈X , A (cid:1)(cid:3) ⊗ A B → lift p (cid:2) C (cid:0) X , { B x } x ∈X , B (cid:1)(cid:3) ; n ∑ j = (cid:8)e a j e x (cid:9) ⊗ (cid:8) ξ jx (cid:9) ( n ∑ j = e a j ( e x ) c − e x (cid:0) ξ jx (cid:1)) e x ∈ e X where c e x is given by (4.5.41) (4.8.19) Lemma 4.8.13.
The given by (4.8.19) map is the isomorphism of lift p [ C ( X , { M n ( C ) x } , A )] -modules.Proof. Similarly the Lemma 4.8.9 one can proof that the map φ is a homomorphismof left lift p [ C ( X , { M n ( C ) x } , A )] -modules. Since X is compact and p is a finite-fold covering the space e X is also compact. It turns out that C (cid:16) e X (cid:17) = C (cid:16) e X (cid:17) andfrom the Lemma 4.5.12 it follows that lift p (cid:2) C (cid:0) X , { M n ( C ) x } x ∈X , A (cid:1)(cid:3) = C (cid:16) lift p (cid:2) C (cid:0) X , { M n ( C ) x } x ∈X , A (cid:1)(cid:3)(cid:17) ; lift p (cid:2) C (cid:0) X , { B x } x ∈X , B (cid:1)(cid:3) = C (cid:16) lift p (cid:2) C (cid:0) X , { B x } x ∈X , B (cid:1)(cid:3)(cid:17) .If we consider the finite sum (4.8.8)1 C ( e X ) = ∑ e α ∈ f A e e e α then from the inclusion (4.8.18) it turns out that the elements e e e α can be regardedas elements of the C ∗ -algebra C (cid:16) lift p (cid:2) C (cid:0) X , { M n ( C ) x } x ∈X , A (cid:1)(cid:3)(cid:17) . The followingproof of this lemma is similar to the proof of the Lemma 4.8.9. Example 4.8.14. If M is compact Riemannian manifold with a spin bundle S (cf.E.4.1) and p : e M → M is a transitive covering. Suppose that C ( M ) -module B inthe equation (4.8.19) is the space of sections Γ ( M , S ) . Then one has A = [ C (cid:0) M , { M n ( C ) x } x ∈X , A (cid:1) = C ( M ) ⊗ M n ( C ) , lift p (cid:2) C (cid:0) M , { M n ( C ) x } x ∈X , A (cid:1)(cid:3) = C (cid:16) e M (cid:17) ⊗ M n ( C ) , C (cid:16) lift p (cid:2) C (cid:0) X , { B x } x ∈X , B (cid:1)(cid:3)(cid:17) = Γ (cid:16) e M , e S (cid:17) .162he specialization of the given by (4.8.19) isomorphism of C (cid:16) e M (cid:17) ⊗ M n ( C ) -modulessatisfies to the following equation C (cid:16) e M (cid:17) ⊗ M n ( C ) ⊗ C ( M ) ⊗ M n ( C ) Γ ( M , S ) ≈ −→ Γ (cid:16) e M , e S (cid:17) ; m ∑ j = e a j ⊗ ξ j e a j lift p (cid:0) ξ j (cid:1) . (4.8.20) Consider the similar to 4.8.12 situation. But now we suppose that X is alocally compact, paracompact, Hausdorff space. There is the map φ : C c (cid:16) lift p [ C ( X , { M n ( C ) x } , A )] (cid:17) ⊗ A B → C c (cid:16) lift p [ C ( X , { B x } , B )] (cid:17) ; n ∑ j = (cid:8)e a j e x (cid:9) ⊗ (cid:8) ξ jx (cid:9) ( n ∑ j = e a j ( e x ) c − e x (cid:0) ξ jx (cid:1)) e x ∈ e X where c e x is given by (4.5.41) (4.8.21) Lemma 4.8.16.
The given by (4.8.21) map is the isomorphism ofC (cid:16) lift p [ C ( X , { M n ( C ) x } , A )] (cid:17) -modules.Proof. Similarly the Lemma 4.8.9 one can proof that the map φ is a homomorphismof left C (cid:16) lift p [ C ( X , { M n ( C ) x } , A )] (cid:17) -modules. Denote by1 C b ( e X ) = ∑ e α ∈ f A e e e α the given by (4.8.16) compliant to the covering p : e X → X partition of unity (cf.Definition 4.8.4). From the inclusion (4.8.18) it turns out that the elements e e e α can beregarded as elements of the involutive algebra C c (cid:16) lift p (cid:2) C (cid:0) X , { M n ( C ) x } x ∈X , A (cid:1)(cid:3)(cid:17) .The following proof of this lemma is similar to the proof of the Lemma 4.8.11. Example 4.8.17. If M is compact Riemannian manifold with a spin bundle S (cf.E.4.1) and p : e M → M is a transitive covering. Suppose that C ( M ) -module B inthe equation (4.8.19) is the space of sections Γ ( M , S ) . Then one has A = [ C (cid:0) M , { M n ( C ) x } x ∈X , A (cid:1) = C ( M ) ⊗ M n ( C ) , C c (cid:16) lift p (cid:2) C (cid:0) M , { M n ( C ) x } x ∈X , A (cid:1)(cid:3)(cid:17) = C c (cid:16) e M (cid:17) ⊗ M n ( C ) , C (cid:16) lift p (cid:2) C (cid:0) X , { B x } x ∈X , B (cid:1)(cid:3)(cid:17) = Γ c (cid:16) e M , e S (cid:17) .163he specialization of the given by (4.8.19) isomorphism of C (cid:16) e M (cid:17) ⊗ M n ( C ) -modules satisfies to the following equation C c (cid:16) e M (cid:17) ⊗ M n ( C ) ⊗ C ( M ) ⊗ M n ( C ) Γ (cid:16) M , S k (cid:17) ≈ −→ Γ c (cid:16) e M , e S k (cid:17) ; m ∑ j = e a j ⊗ ξ j e a j lift p (cid:0) ξ j (cid:1) . (4.8.22) Let (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) be a commutative spectral triple (cf. Equation (E.4.6)),and let p : e M → M be a finite-fold covering . From the Proposition E.5.1 it followsthat e M has natural structure of the Riemannian manifold. If T M is a tangent bun-dle then from the Lemma 4.5.4 it follows that Γ ( M , T M ) is a continuity structurefor M and the family { T x M } x ∈ M of tangent spaces (cf. Definition D.8.27) such that Γ ( M , T M ) = C ( M , { T x M } , Γ ( M , T M )) .It is known [46] that the tangent bundle T e M of e M is the inverse image of T M (cf.Definition A.3.7). From the Theorem 4.5.59 it follows that Γ (cid:16) e M , T e M (cid:17) is the p -liftof C ( M , { T x M } , Γ ( M , T M )) i.e. Γ (cid:16) e M , T e M (cid:17) ∼ = lift p [ C ( M , { T x M } , Γ ( M , T M ))] (4.9.1)The described in E.4.1 tensor fields on e M are lifts of corresponding tensor fields on M . In particular the metric tensor g on M corresponds to an element of the module Γ ( M , T M ) ⊗ C ( M ) Γ ( M , T M ) . The metric tensor e g of e M is given by e g = lift p ( g ) .Similarly the given by (E.4.1) volume elements v and e v on M and e M satisfy to theequation e v = lift p ( v ) . (4.9.2)If both C ℓ ( M ) → M and C ℓ ( e M ) → e M are described in E.4.1 bundles of C ∗ alge-bras then Γ (cid:16) e M , C ℓ ( e M ) (cid:17) ∼ = lift p [ C ( M , { C ℓ ( M ) x } , Γ ( M , C ℓ ( M )))] . If e S = p ∗ S theinverse image of the spinor bundle S (cf. Definition A.3.7) then Γ (cid:16) e M , e S (cid:17) ∼ = lift p [ C ( M , { S x } , Γ ( M , S ))] x ∈ M there is the natural isomorphism C ℓ ( M ) x ∼ = End C ( S x ) , so takinginto account C ℓ ( e M ) e x ∼ = C ℓ ( M ) p ( e x ) and End C (cid:16) e S e x (cid:17) ∼ = End C (cid:16) S p ( e x ) (cid:17) for any e x ∈ e M there is the natural isomorphism C ℓ ( e M ) e x ∼ = End C (cid:16) e S e x (cid:17) . It follows that the bundle e S yields the spin c structure on T e M . (cf. Definition E.4.2). The given by (4.9.2)volume form yields the measure e µ on e M , in result one has the Hilbert space e H def = L (cid:16) e M , e S , e µ (cid:17) .Using the Definition E.4.1 one can construct the C ∞ ( M ) -module Γ ∞ ( M , S ) and C ∞ (cid:16) e M (cid:17) -module Γ ∞ (cid:16) e M , e S (cid:17) of smooth sections. Also there is the spin c connection ∇ S (cf. (E.4.4)). Similarly to the Equation (E.4.5) one can define the Dirac operator e D / : Γ ∞ (cid:16) e M , e S (cid:17) → Γ ∞ (cid:16) e M , e S (cid:17) which can be regarded us an unbounded operator on e H . So one has the spectraltriple (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S , e µ (cid:17) , e D / (cid:17) . Definition 4.9.1.
The constructed above spectral triple (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S , e µ (cid:17) , e D / (cid:17) is said to be the geometrical p - lift of the spectral triple (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) .The Dirac operator e D / can be defined as the section of a sheaf. Let C ( M , { S x } , Γ ( M , S )) be sheaf of the continuity structure Γ ( M , S ) (cf. Definition4.7.1). According to the Example 4.7.13 the C ∞ ( M ) -module Γ ∞ ( M , S ) induces thesmooth subsheaf C ∞ (cid:0) M , {S x } x ∈ M , Γ ( M , S ) (cid:1) ⊂ C (cid:0) M , {S x } x ∈ M , Γ ( M , S ) (cid:1) (4.9.3)(cf. Definition 4.7.7). The given by C ∞ (cid:16) e M , {S e x } e x ∈ e M , Γ (cid:16) e M , e S (cid:17)(cid:17) def = p − C ∞ (cid:0) M , {S x } x ∈ M , Γ ( M , S ) (cid:1) (4.9.4)inverse image is such that Γ ∞ (cid:16) e M , e S (cid:17) = C ∞ (cid:16) e M , {S e x } e x ∈ e M , Γ (cid:16) e M , e S (cid:17)(cid:17) (cid:16) e M (cid:17) . (4.9.5)If e M is locally compact then denote by Γ ∞ c (cid:16) e M , e S (cid:17) def = C ∞ (cid:16) e M , {S e x } e x ∈ e M , Γ (cid:16) e M , e S (cid:17)(cid:17) (cid:16) e M (cid:17) ∩ Γ c (cid:16) e M , e S (cid:17) . (4.9.6)165rom the Example 4.7.13 it follows that/ D ∈ E nd (cid:0) C ∞ (cid:0) M , {S x } x ∈ M , Γ ( M , S ) (cid:1)(cid:1) ( M ) .The operator e D / is the p -inverse image p − / D of / D (cf. Definition 4.7.11), i.e. e D / = p − / D ∈ E nd (cid:16) p − C ∞ ( M , {S x } , Γ ( M , S )) (cid:17) (cid:16) e M (cid:17) == E nd (cid:16) C ∞ (cid:16) e M , n e S e x o , Γ (cid:16) e M , e S (cid:17)(cid:17)(cid:17) (cid:16) e M (cid:17) . (4.9.7) Let (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) be a commutative spectral triple (cf. Equation (E.4.6)),and let p : e M → M be an infinite covering . From the Proposition E.5.1 it followsthat e M has natural structure of the Riemannian manifold. Similarly to the Section4.9.1 on can define the bundle e S → e M which corresponds to the C (cid:16) e M (cid:17) -module Γ (cid:16) e M , e S (cid:17) = lift p [ C ( M , { S x } , Γ ( M , S ))] . (4.9.8)Similarly to the Equation (4.9.2) there is the natural volume form e v = lift p ( v ) which induces the measure e µ on e M . If Γ c (cid:16) e M , e S (cid:17) def = C c (cid:16) lift p [ C ( M , { S x } , Γ ( M , S ))] (cid:17) (4.9.9)then any e ξ ∈ Γ c (cid:16) e M , e S (cid:17) has compact support, hence ∀ e ξ ∈ Γ c (cid:16) e M , e S (cid:17) (cid:13)(cid:13)(cid:13) e ξ (cid:13)(cid:13)(cid:13) def = r Z e M (cid:13)(cid:13)(cid:13) e ξ e x (cid:13)(cid:13)(cid:13) d e µ < ∞ . (4.9.10)The completion of Γ c (cid:16) e M , e S (cid:17) with respect to the given by (4.9.10) norm is a Hilbertspace which is denoted by L (cid:16) e M , e S , e µ (cid:17) , or simply L (cid:16) e M , e S (cid:17) . Similarly to theEquations (4.9.3) and (4.9.4) we define following sheaves C ∞ (cid:0) M , {S x } x ∈ M , Γ ( M , S ) (cid:1) ⊂ C (cid:0) M , {S x } x ∈ M , Γ ( M , S ) (cid:1) C ∞ (cid:16) e M , n e S e x o e x ∈ e M , Γ (cid:16) e M , e S (cid:17)(cid:17) def = p − C ∞ (cid:0) M , {S x } x ∈ M , Γ ( M , S ) (cid:1) . (4.9.11)and the given by (4.9.7) Dirac operator e D / = p − / D ∈ E nd (cid:16) p − C ∞ ( M , {S x } , Γ ( M , S )) (cid:17) (cid:16) e M (cid:17) == E nd (cid:16) C ∞ (cid:16) e M , n e S e x o , Γ (cid:16) e M , e S (cid:17)(cid:17)(cid:17) (cid:16) e M (cid:17) . (4.9.12)166f Γ ∞ c (cid:16) e M , e S (cid:17) def = Γ c (cid:16) e M , e S (cid:17) ∩ C ∞ (cid:16) e M , n e S e x o e x ∈ e M , Γ (cid:16) e M , e S (cid:17)(cid:17) (cid:16) e M (cid:17) (4.9.13)then e D / can be regarded as a C -linear map e D / : Γ ∞ c (cid:16) e M , e S (cid:17) → Γ ∞ c (cid:16) e M , e S (cid:17) .However the subspace Γ ∞ c (cid:16) e M , e S (cid:17) ⊂ L (cid:16) e M , e S (cid:17) is dense with respect to the givenby (4.9.10) norm. It follows that e D / can be regarded as an unbounded operatoron L (cid:16) e M , e S (cid:17) . There is the family n End e S e x o e x ∈ e M of Banach spaces such that forany e a ∈ C ∞ (cid:16) e M (cid:17) the commutator h e D / , e a i corresponds to a family { e ω e x } e x ∈ e M , where e ω e x ∈ End e S e x for all e x ∈ e M . On the other hand for all e x there is the naturalisomorphism of algebras End e S e x ∼ = C ℓ e x (cf. Section E.4.1). So e ω e x can be regardedas element of C ℓ e x . Moreover the space Γ (cid:16) e M , C ℓ (cid:16) e M (cid:17)(cid:17) is the continuity structurefor e M and { C ℓ e x } (cf. Definition D.8.27) such that the { e ω e x } is continuous withrespect to Γ (cid:16) e M , C ℓ (cid:16) e M (cid:17)(cid:17) (cf. Definition D.8.28). So one can write h e D / , e a i ∈ C (cid:16) e M , Γ (cid:16) e M , { C ℓ e x } , C ℓ (cid:16) e M (cid:17)(cid:17)(cid:17) .Similarly to the Equations (E.2.1), (E.2.2) for any s ∈ N the operator e D / yieldsthe representation π s : C ∞ (cid:16) e M (cid:17) → B (cid:18) L (cid:16) e M , e S (cid:17) s (cid:19) .For any e a ∈ C ∞ (cid:16) e M (cid:17) the element π s ( e a ) corresponds to a continuous sectionof the family { M s ( C ℓ e x ) } e x ∈ e M , i.e. π s ( e a ) ∈ C (cid:16) e M , { M s ( C ℓ e x ) } e x ∈ e M , M s (cid:16) Γ (cid:16) e M , { C ℓ e x } , C ℓ (cid:16) e M (cid:17)(cid:17)(cid:17)(cid:17) .Denote by C ∞ (cid:16) e M (cid:17) def = def = ne a ∈ C ∞ (cid:16) e M (cid:17)(cid:12)(cid:12)(cid:12) π s ( e a ) ∈∈ C (cid:16) e M , { M s ( C ℓ e x ) } , M s (cid:16) Γ (cid:16) e M , { C ℓ e x } , C ℓ (cid:16) e M (cid:17)(cid:17)(cid:17)(cid:17) ∀ s ∈ N o . (4.9.14) Definition 4.9.2.
The triple (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S , e µ (cid:17) , e D / (cid:17) is said to be the geometri-cal p - lift of the spectral triple (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) .167 .10 Finite-fold coverings This section contains an algebraic version of the Proposition 1.1.2 in case offinite-fold coverings. C ∗ -algebras Lemma 4.10.1.
Let A be a commutative C ∗ -algebra. If (cid:16) A , e A , G , π (cid:17) is a noncommutativefinite-fold covering with compactification (cf. Definition 2.1.13) then the dimension of anyirreducible representation of e A is finite. Moreover if e X is the spectrum of e A then there isN ∈ N such that for any e x ∈ e X there is m ≤ N which satisfies to the following condition rep e x (cid:16) e A (cid:17) ≈ M m ( C ) . Proof.
Any irreducible representation e ρ : e A → B (cid:16) e H (cid:17) can be uniquely extendedup to e ρ : M (cid:16) e A (cid:17) → B (cid:16) e H (cid:17) . From (2.2.6) it follows that there are e b , ..., e b n ∈ e M (cid:16) e A (cid:17) such that e A = e b A + ... + e b n A ,so, one has rep e x (cid:16) e A (cid:17) = rep e x (cid:16)e b (cid:17) rep e x ( A ) + ... + rep e x (cid:16)e b n (cid:17) rep e x ( A ) If p : e X → X is given by the Lemma 2.3.7 then rep e x | A ∼ = rep p ( e x ) ( A ) ∼ = C , it turnsout that rep e x (cid:16) e A (cid:17) = rep e x (cid:16)e b (cid:17) C + ... + rep e x (cid:16)e b n (cid:17) C ,or equivalently ∀ e a ∈ e A ∃ { k , ..., k n } ⊂ C rep e x ( e a ) = rep e x (cid:16)e b (cid:17) k + ... + rep e x (cid:16)e b n (cid:17) k n , (4.10.1)i.e. rep e x ( e a ) linearly depends on n or less than n complex-valued parameters.Hence if N ≥ √ n then rep e x (cid:16) e A (cid:17) ≈ M m ( C ) where m ≤ N . Lemma 4.10.2.
Let A be commutative C ∗ -algebra If (cid:16) A , e A , G , π (cid:17) be a noncommutativefinite-fold covering with unitization (cf. Definition 2.1.13) then e A is commutative. roof. If e A red is the reduced algebra of (cid:16) A , e A , G , π (cid:17) (cf. Definition 2.3.12) thenfrom the Equation (2.3.20) it follows that e A red is commutative. From the Theorem1.1.1 it follows that the spectrum e X of e A red is Hausdorff. From the Remark 2.3.16it turns out that e X coincides with the spectrum of e A as a topological space. If e A is not commutative then there is e x ∈ e X such that dim rep e x (cid:16) e A (cid:17) >
1. Denote by N the maximal dimension of rep e x . From the Lemma 4.10.1 it turns out that there is N ∈ N such that the dimension of rep e x (cid:16) e A (cid:17) is less or equal to N for all e x ∈ e X .From the Proposition D.2.28 it turns out that the set N e X = ne x ∈ e X | dim rep e x = N o is open subset of e X . If N e A ⊂ e A is a closed ideal which corresponds to N e X , i.e. N e A = \ e x ∈ e X \ N e X ker rep e x then from Theorem D.8.21 it turns out N e A corresponds to a fibre bundle with thebase space N e X and and fibre space M N ( C ) . It follow that there is an open subset e U ⊂ N e X such that N e A (cid:12)(cid:12)(cid:12) e U = e A (cid:12)(cid:12)(cid:12) e U ∼ = C (cid:16) e U (cid:17) ⊗ M N ( C ) . (4.10.2)where N e A (cid:12)(cid:12)(cid:12) e U is given by the equation (D.2.7). There is e f ∈ C (cid:16) N e X (cid:17) which satisfiesto the Lemma 4.1.5, i.e. there is an open subset e V ⊂ e U such that e f (cid:16) e V (cid:17) = { } .Since e U is an open subset of e V the function e f can be regarded as element of C (cid:16) e X (cid:17) . Any e a ∈ e A (cid:12)(cid:12)(cid:12) e U can be represented by continuous function e U → M N ( C ) .Let ψ , ϑ : e U → M N ( C ) be given by ψ ( e x ) = f ( e x ) . ϑ ( e x ) = f ( e x ) . . . 00 0 . . . 0... ... . . . ...0 0 . . . 0 .169et both e p k , e b ∈ e A (cid:12)(cid:12)(cid:12) e U be represented by both ψ , ϑ respectively. From e A (cid:12)(cid:12)(cid:12) e U ⊂ e A itturns out that both e p k and e b can be regarded as elements of e A , i.e. e p k , e b ∈ e A .Since e V is a locally compact Hausdorff open subspace of e U there is x ∈ e V acontinuous positive ϕ ∈ C (cid:16) e X (cid:17) + such that ϕ ( e x ) = supp ϕ ⊂ e V . Denoteby e p ⊥ def = M ( e A ) − e p k ∈ e A . If ǫ ∈ R and e u ǫ def = p k e i ǫϕ + p ⊥ ∈ e A then for any e x ∈ e U following condition holds rep e x ( e u ǫ ) = e i ǫϕ ( e x ) e x ∈ e V e x ∈ e U \ e V . (4.10.3)Otherwise rep e x ( e u ǫ ) rep e x ( e u ∗ ǫ ) = rep e x ( e u ∗ ǫ ) rep e x ( e u ǫ ) = e x ∈ e V , it follows that rep e x ( e u ǫ ) rep e x ( e u ∗ ǫ ) = rep e x ( e u ∗ ǫ ) rep e x ( e u ǫ ) = e x ∈ e X . Hence one has e u ∗ ǫ e u ǫ = e u ǫ e u ∗ ǫ = e A i.e. e u ǫ is unitary. There is an internal *-automorphism χ ǫ ∈ Aut (cid:16) e A (cid:17) given by e a e u ∗ ǫ e a e u ǫ . From rep e x ( e u ∗ ǫ ) ϑ ( x ) rep e x ( e u ǫ ) = e − i ǫ . . . 00 0 . . . 0... ... . . . ...0 0 . . . 0 .it turns out that ǫ , ǫ ∈ [
0, 2 π ) & ǫ = ǫ ⇒ rep e x (cid:16) χ ǫ (cid:16)e b (cid:17)(cid:17) = rep e x (cid:16) χ ǫ (cid:16)e b (cid:17)(cid:17) ⇒ χ ǫ = χ ǫ .The C ∗ -algebra e A red is commutative, it follows that e a red ∈ e A red ⇒ rep e x ( e a red ) = f red ( e x ) f red ( e x ) . . . 0... ... . . . ...0 0 . . . f red ( e x ) ∀ e x ∈ e X rep e x ( e u ǫ ) rep e x ( e a red ) rep e x ( e u ∗ ǫ ) = rep e x ( e a red ) for every e x ∈ e X . It means that χ ǫ ( e a red ) = e a red for all e a red ∈ e A red hencethe group G red def = n g ∈ Aut (cid:16) e A (cid:17) (cid:12)(cid:12)(cid:12) g e a red = e a red ; ∀ e a red ∈ e A red o is not finite. From the inclusion G red ⊂ G it follows that G is not finite , i.e. one hasthe contradiction with (a) of the Definition 2.1.5. From this contradiction it turnsout that dim rep e x (cid:16) e A (cid:17) = e x ∈ e X , it means that e A is commutative. Corollary 4.10.3.
Let A be an unital commutative C ∗ -algebra. Let (cid:16) A , e A , G , π (cid:17) be anunital noncommutative finite-fold covering (cf. Definition 2.1.9). If p : e X → X is thecontinuous map from the spectrum of e A to the spectrum of A given by the Lemma 2.3.7then p is a finite-fold transitive topological covering such that G ∼ = G (cid:16) e X | X (cid:17) . More-over the quadruple (cid:16) A , e A , G , π (cid:17) is equivalent to (cid:16) C ( X ) , C (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , C ( p ) (cid:17) where C ( p ) : C ( X ) ֒ → C (cid:16) e X (cid:17) is the finite covering algebraic functor (cf. Definition4.6.12).Proof. From the Lemma 4.10.2 it follows that e A is commutative. From the Theorem1.1.1 it turns out that A ∼ = C ( X ) ∼ = C ( X ) , e A ∼ = C (cid:16) e X (cid:17) ∼ = C (cid:16) e X (cid:17) and π isequivalent to C ( p ) . According to the Definition 2.1.9 it turns out that C (cid:16) e X (cid:17) isa finitely generated Hilbert C ( X ) -module, so from the Lemma D.4.13 it turns outthat C (cid:16) e X (cid:17) is a projective C ( X ) -module. Hence from the Theorem 1.1.2 it turnsout that p is a covering. From the Lemma 2.3.7 it turns out that the covering p istransitive and there is the natural isomorphism G ∼ = G (cid:16) e X | X (cid:17) .The algebraic construction of finite-fold coverings requires the following defi-nition.
Definition 4.10.4.
A covering p : e X → X is said to be a covering with compactifica-tion if there are compactifications X ֒ → Y and e X ֒ → e Y such that: • There is a covering p : e Y → Y , • The covering p is the restriction of p , i.e. p = p | e X .171 xample 4.10.5. Let g : S → S be an n -fold covering of a circle. Let X ∼ = e X ∼ = S × [
0, 1 ) . The map p : e X → X , p = g × Id [ ) is an n -fold covering. If Y ∼ = e Y ∼ = S × [
0, 1 ] then a compactification [
0, 1 ) ֒ → [
0, 1 ] induces compactifications X ֒ → Y , e X ֒ → e Y . The map p : e Y → Y , p = g × Id [ ] is a covering such that p | e X = p . So if n > p is a nontrivial covering withcompactification. Example 4.10.6.
Let X = C \ { } be a complex plane with punctured 0, which isparametrized by the complex variable z . Let X ֒ → Y be any compactification. Ifboth { z ′ α ∈ X } , { z ′′ α ∈ X } are nets such that lim α | z ′ α | = lim α | z ′′ α | = α | z ′ α − z ′′ α | = x = lim α z ′ α = lim α z ′′ α ∈ Y . (4.10.4)If e X ∼ = X then for any n ∈ N there is a finite-fold covering p : e X → X , z z n .If both X ֒ → Y , e X ֒ → e Y are compactifications, and p : e Y → Y is a covering suchthat p | e X = p then from (4.10.4) it turns out p − ( x ) = { e x } where e x is the uniquepoint such that following conditions hold: e x = lim α e z α ∈ e Y ,lim α | e z α | = (cid:12)(cid:12) p − ( x ) (cid:12)(cid:12) =
1. However p is an n -fold covering and if n > (cid:12)(cid:12) p − ( x ) (cid:12)(cid:12) = n >
1. It contradicts with (cid:12)(cid:12) p − ( x ) (cid:12)(cid:12) =
1, and from the contradictionit turns out that for any n > p is not a covering with compactification.172he following lemma supplies the quantization of coverings with compactifi-cation. Lemma 4.10.7.
Let X be a locally compact space and there is an injective *-homomorphism π : C ( X ) ֒ → e A of C ∗ -algebras then following conditions are equivalent:(i) The quadruple (cid:16) C ( X ) , e A , G , π (cid:17) is a noncommutative finite-fold covering withunitization.(ii) The C ∗ -algebra e A is commutative. Moreover if X (resp. e X ) is the spectrum ofA (resp. e A) then the given by the Lemma 2.3.7 map p : e X → X is a transi-tive topological finite-fold covering with compactification such that the quadruple (cid:16) C ( X ) , e A , G , π (cid:17) is naturally equivalent to (cid:16) C ( X ) , C (cid:16) e X (cid:17) , G (cid:16) e X | X (cid:17) , C ( p ) (cid:17) where *-homomorphism C ( p ) : C ( X ) → C (cid:16) e X (cid:17) is C is a finite coveringalgebraic functor (cf. Definition 4.6.12).Proof. (i) ⇒ (ii) From the Lemma 4.10.2 it follows that the C ∗ -algebra e A is commu-tative, taking into account the Theorem 1.1.1 it follows that e A ∼ = C (cid:16) e X (cid:17) . Fromthe Definition 2.1.13 it turns out that following conditions hold:(a) There are unital C ∗ -algebras B , e B and inclusions C ( X ) ⊂ B , C (cid:16) e X (cid:17) ∼ = e A ⊂ e B such that both C ( X ) (resp. C (cid:16) e X (cid:17) ∼ = e A ) are an essential ideals of both B (resp. e B ),(b) There is a unital noncommutative finite-fold covering (cid:16) B , e B , G , e π (cid:17) such that π = e π | C ( X ) and the action G × e A → e A is induced by G × e B → e B .Since both C ( X ) and C (cid:16) e X (cid:17) are an essential ideals of B and e B the C ∗ -algebras B and e B are commutative. Since both B and e B are unital the spectra Y and e Y of these C ∗ -algebras are is a compact Hausdorff spaces, hence from the Theorem1.1.1 it follows that B ∼ = C ( Y ) and e B ∼ = C (cid:16) e Y (cid:17) . From the Corollary 4.10.3 it turnsout that there is a transitive finite-fold covering e p : e Y → Y such that the inclusion B ֒ → e B corresponds to C ( e p ) : C ( Y ) ֒ → C (cid:16) e Y (cid:17) . Both C ( X ) and C (cid:16) e X (cid:17) areessential ideals, of C ( Y ) and C (cid:16) e Y (cid:17) , hence from the Example D.1.5 it follows173hat both X and e X are open dense subsets of both Y and e Y . It means that bothinclusions X ֒ → Y and e X ֒ → e Y are compactifications. From π = e π | C ( X ) it followsthat e p (cid:16) e X (cid:17) = X so p = e p | e X : e X → X is a covering. From transitivity of action G × e Y → e Y and the condition G × C (cid:16) e X (cid:17) = C (cid:16) e X (cid:17) it turns out that the action G × e X → e X is transitive. From p = e p | e X it follows that p is a covering withcompactification.(ii) ⇒ (i) The map p : e X → X is a transitive topological finite-fold covering withcompactification, hence there are compactifications X ֒ → Y , e X ֒ → e Y and thetransitive finite-fold covering e p : e Y → Y such that p = e p | e X . Firstly we prove thatthe quadruple (cid:16) C ( X ) , C (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , π (cid:17) is a finite-fold noncommutativepre-covering, i.e. it satisfies to the conditions (a) and (b) of the Definition 2.1.5.(a) The group G = n g ∈ Aut (cid:16) C (cid:16) e X (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ C ( X ) o ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is finite.(b) C ( X ) ∼ = C (cid:16) e X (cid:17) G ( e X | X ) .Secondly we prove that (cid:16) C ( X ) , C (cid:16) e X (cid:17) , G (cid:16) e X | X (cid:17) , C ( p ) (cid:17) satisfies to theconditions (a), (b) of the Definition 2.1.13.(a) Both maps X ֒ → Y , e X ֒ → e Y are compactifications, hence both inclusions C ( X ) ֒ → C ( Y ) and C (cid:16) e X (cid:17) ֒ → C (cid:16) e Y (cid:17) correspond to essential ideals (cf.Example D.1.5). It means that the inclusions are unitizations.(b) From the Theorem 1.1.2 it follows that C (cid:16) e Y (cid:17) is a finitely generated projec-tive module, it follows that the quadruple (cid:16) C ( Y ) , C (cid:16) e Y (cid:17) , G , C ( e p ) (cid:17) is an unital noncommutative finite-fold covering (cf. Definition 2.1.9). From p = e p | e X it follows that e π | C ( X ) = π . The action G × e X → e X is induced bythe action G × e Y → e Y , hence the action G × C (cid:16) e X (cid:17) → C (cid:16) e X (cid:17) is inducedby the action G × C (cid:16) e Y (cid:17) → C (cid:16) e Y (cid:17) .174 heorem 4.10.8. Let X be a locally compact, connected, locally connected, second-countable,Hausdorff space. Following conditions hold:(i) If the triple quadruple (cid:16) C ( X ) , e A , G , π (cid:17) is a noncommutative finite-fold covering(cf. Definition 2.1.17), e X is the spectrum of e A and p : e X → X is given by theLemma 2.3.7, then p is a finite-fold transitive topological covering, e A ∼ = C (cid:16) e X (cid:17) and G ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) .(ii) If p : e X → X is a finite-fold transitive topological covering and π : C ( X ) ֒ → C (cid:16) e X (cid:17) is the *-homomorphism induced by p and the action G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × C (cid:16) e X (cid:17) → C (cid:16) e X (cid:17) is induced by G (cid:16) e X | X (cid:17) × e X → e X then there is the noncommutativefinite-fold covering (cid:16) C ( X ) , C (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , π (cid:17) (cf. Definition 2.1.17).Proof. (i) Let { u λ } λ ∈ Λ ⊂ M ( C ( X )) + ∼ = C b ( X ) be an increasing net which satis-fies to the conditions (a) and (b) of the Definition 2.1.17. If both A λ and e A λ are C ∗ -norm completions of u λ C ( X ) u λ and u λ e Au λ respectively then A λ and e A λ area hereditary subalbebras of C ( X ) and e A . If e X is the spectrum of e A then fromthe Proposition D.2.20 it follows that there are open subsets U λ and e U λ of X and e X which are spectra of both A λ and e A λ respectively. Since A λ is a subalgebraof a commutative subalgebra one has A λ ∼ = C ( U λ ) . From the condition (b) ofthe Definition 2.1.17 it follows that there is a noncommutative finite-fold coveringwith unitization (cid:16) C ( U λ ) , e A λ , G , π | C ( U λ ) : C ( U λ ) ֒ → e A λ (cid:17) From the Lemma 4.10.7 it follows that the C ∗ -algebra e A λ is commutative, i.e. e A λ ∼ = C (cid:16) e U λ (cid:17) and the restriction p | e U λ : e U λ → U λ is a transitive finite-fold covering.From the condition (a) of the Definition 2.1.17 it follows that β - lim λ ∈ Λ u λ = C b ( X ) so the union ∪ e A λ ∼ = ∪ C (cid:16) e U λ (cid:17) is dense in e A i.e. there is a dense commutativesubalgebra of e A . It turns out that the C ∗ -algebra e A is commutative, hence e A ∼ = C (cid:16) e X (cid:17) . If x ∈ X is any point then from the Exercise A.1.12 and the DefinitionA.1.10 it follows that there is f ∈ C ( X ) such that f ( x ) =
1. Since ∪ λ ∈ Λ C ( U λ ) is dense in C ( X ) there is λ ∈ Λ and f ∈ C ( U λ ) such that k f − f k < | f ( x ) | ≥ x ∈ U λ . So one concludes that X = ∪ λ ∈ Λ U λ .175imilarly one can prove that e X = ∪ λ ∈ Λ e U λ . If x ∈ X is any point then thereis λ ∈ Λ such that x ∈ U λ . Since p | e U λ : e U λ → U λ is a transitive finite-foldcovering there is an open neighborhood V of x such that V ⊂ U λ and V is evenlycovered by p | e U λ (cf. Definition A.2.1). Clearly V is evenly covered by p , and G transitively acts on p − ( x ) . Since π is injective, the map p is surjective, so p is atransitive covering. The following equation G (cid:16) C (cid:16) e X (cid:17) (cid:12)(cid:12)(cid:12) C ( X ) (cid:17) ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is clear.(ii) Firstly we prove that the quadruple (cid:16) C ( X ) , C (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , π (cid:17) is afinite-fold noncommutative pre-covering, i.e. it satisfies to the conditions (a) and(b) of the Definition 2.1.5.(a) The group G = n g ∈ Aut (cid:16) C (cid:16) e X (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ ( C ( X ) o ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is finite.(b) C ( X ) ∼ = C (cid:16) e X (cid:17) G .The space X is second-countable hence from the Theorem A.1.29 it follows that X is a Lindelöf space. Let us consider a finite or countable sequence U $ ... $ U n $ ... of connected open subsets of X given by the Lemma 4.3.33. In particularfollowing condition hold: • For any n ∈ N the closure V n of U n is compact. • ∪ U n = X . • The space p − ( U n ) is connected for any n ∈ N .From the Remark 4.8.2 it turns out that there are f n ∈ C ( U n ) ⊂ C ( X ) such thatthere is the limit 1 C b ( X ) = β - lim n → ∞ f n (4.10.5)with respect to the strict topology on the multiplier algebra M ( C ( X )) ∼ = C b ( X ) (cf. Definition D.1.12). Let us check conditions (a), (b) of the Definition 2.1.17.(a) Follows from the Equation (4.10.5).176b) If both A n and e A n are the C ∗ -norm completions of f n C ( X ) f n and f n C (cid:16) e X (cid:17) f n respectively then both A n and e A n are hereditary subalgebras of C ( X ) and C (cid:16) e X (cid:17) . From the Proposition D.2.20 it follows that both spectra U n and e U n of A n and e A n are open subsets of X and e X respectively. Otherwise theclosure V n of U n is compact, and taking into account that p is a finite-foldcovering the closure e V n of e U n is also compact, so from the Theorem 1.1.2 itturns out that C (cid:16) e V n (cid:17) is a finitely generated C ( V n ) -module. Clearly the cov-ering p e V n : e V n → V n is transitive, and taking into account the Lemma 4.3.34for every n ∈ N there is the natural isomorphism of groups G (cid:16) e V n (cid:12)(cid:12)(cid:12) V n (cid:17) ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , hence (cid:16) C ( V n ) , C (cid:16) e V n (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , π | C ( e V n ) (cid:17) is a unital non-commutative finite-fold covering (cf. Definition 2.1.9). On the other handboth C ( U n ) and C (cid:16) e U n (cid:17) are essential ideals of C ( V n ) and C (cid:16) e V n (cid:17) respec-tively (cf. Example D.1.5), so for every n ∈ N the quadruple (cid:16) C ( U n ) , C (cid:16) e U n (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , π | C ( e U n ) (cid:17) is a noncommutative finite-fold covering with unitization (cf. Definition2.1.13). Remark 4.10.9.
The Theorem 4.10.8 yields a pure algebraic definition of the fun-damental group (cf. the Theorem 4.11.39 and the Corollary 4.11.40). Alternativetheories (e.g. described in [16]) which allow coverings of commutative C ∗ -algebrasby noncommutative ones do not have this property. Lemma 4.10.10.
Let X be a connected, locally compact, second-countable, Hausdorffspace. Denote by FinCov -C ( X ) a category of noncommutative finite-fold coverings ofC ( X ) , i. e. • Objects of
FinCov -C ( X ) are noncommutative finite-fold coverings of C ( X ) . • If π ′ : C ( X ) → C (cid:16) e X ′ (cid:17) and π ′′ : C ( X ) → C (cid:16) e X ′′ (cid:17) are objects of FinCov -C ( X ) then the morphism from π ′ to π ′′ is an injective *-homomorphism π : C (cid:16) e X ′ (cid:17) → C (cid:16) e X ′′ (cid:17) such that the following diagram commutative. ( X ′ ) C ( X ′′ ) C ( X ) ππ ′ π ′′ Following conditions hold:(i) If the category
FinCov - X is given by the Definition 4.3.25 then there is the naturalequivalence of the categories C : FinCov - X ∼ = FinCov -C ( X ) .(ii) Any unordered pair of *-homomorphisms ( π ′ , π ′′ ) from the above diagram is com-pliant (cf. Definition 3.1.1).Proof. (i) From the Theorem 4.10.8 it follows that there is a one-to-one correspon-dence between objects of FinCov - X and objects of FinCov - C ( X ) given by (cid:16) p : e X → X (cid:17) C ( p ) : C ( X ) ֒ → C (cid:16) e X (cid:17) where C ( p ) is given by the Definition 4.6.12. If p ′ : e X ′ → X and p ′′ : e X ′′ → X are objects of FinCov - X and p : e X ′ → e X ′′ is a morphism of FinCov - X then fromthe Corollary 4.3.8 it turns out that p is a transitive finite-fold covering. From theequation (4.6.10) it follows that p corresponds to the injective *-homomorphism C ( p ) : C (cid:16) e X ′′ (cid:17) → C (cid:16) e X ′ (cid:17) . (4.10.6)So one has a functor from FinCov - X to FinCov - C ( X ) . Let us define the inversefunctor. From the Theorem 4.10.8 it follows that any object π : C ( X ) ֒ → C (cid:16) e X (cid:17) of FinCov - C ( X ) corresponds to a transitive finite-fold covering p : e X → X suchthat π = C ( p ) . If C ( p ′ ) : C ( X ) → C (cid:16) e X ′ (cid:17) and C ( p ′′ ) : C ( X ) → C (cid:16) e X ′′ (cid:17) areobjects of FinCov - C ( X ) and π : C (cid:16) e X ′′ (cid:17) → C (cid:16) e X ′ (cid:17) is a *-homomorphism suchthat C ( p ′ ) = π ◦ C ( p ′′ ) then from the Lemma 4.6.19 it turns out that there is afinite-fold transitive covering p : e X ′ → e X ′′ such than π = C ( p ) and p ′ = p ◦ p ′′ .So one has the inverse functor from FinCov - C ( X ) to FinCov - X .(ii) The conditions (a)-(d) of the Definition 3.1.1 directly follow from (a)-(d) of theLemma 4.6.19. 178 .10.2 Induced representation Let us consider a compact Riemannian manifold M with a spin bundle S (cf.E.4.1). Denote by µ the Riemannian measure (cf. [26]) on M which correspondsto the given by the Equation (E.4.1) volume element. The bundle S is Hermitian(cf. Definition A.3.11) so there is a Hilbert space L ( M , S , µ ) (or L ( M , S ) in asimplified notation) (cf. A.3.10) with the given by (A.3.3) scalar product, i.e. ( ξ , η ) L ( M , S ) def = Z M ( ξ x , η x ) x d µ . (4.10.7)Moreover from (A.3.4) it follows that there is the natural representation C ( M ) → B (cid:0) L ( M , S ) (cid:1) . (4.10.8)From A.3.10 it follows that L ( M , S , µ ) is the Hilbert norm completion of Γ ( M , S ) .On the other hand from the Lemma 4.5.4 it follows that Γ ( M , S ) is continuitystructure for X and the {S x } x ∈X (cf. Definition D.8.27), such that Γ ( M , S ) ∼ = C ( M , {S x } , Γ ( M , S )) .If p : e M → M is a finite-fold transitive covering then from the Lemma 4.8.9 itfollows that there is the natural isomorphism φ : C (cid:16) e X (cid:17) ⊗ C ( M ) C ( M , {S x } , Γ ( M , S )) ≈ −→ lift p [ C ( M , {S x } , Γ ( M , S ))] (4.10.9)of C (cid:16) e M (cid:17) -modules. From the Lemma 4.5.59 it follows that lift p [ C ( M , {S x } , Γ ( M , S ))] ∼ = Γ (cid:16) e M , e S (cid:17) where e S is the inverse image of S by p (cf. Definition A.3.7). Hence theequation (4.10.9) can be rewritten by the following way φ : C (cid:16) e M (cid:17) ⊗ C ( M ) Γ ( M , S ) ≈ −→ Γ (cid:16) e M , e S (cid:17) (4.10.10) Lemma 4.10.11.
Following conditions hold:(i) The map (4.10.10) can be extended up to the following homomorphismC (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) ≈ −→ L (cid:16) e M , e S (cid:17) of left C (cid:16) e M (cid:17) -modules.(ii) The image of C (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) is dense in L (cid:16) e M , e S (cid:17) . roof. (i) Let ∑ nj = e a j ⊗ ξ j ∈ C (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) be any element. From A.3.10 L ( M , S , µ ) is the Hilbert norm completion of Γ ( M , S ) . Hence for any j =
1, ..., n there is a net (cid:8) ξ j α (cid:9) ⊂ Γ ( M , S ) such that ξ j = lim α ξ j α where we mean the con-vergence with respect to the Hilbert norm k·k L ( M , S ) . If C = max j = n (cid:13)(cid:13)e a j (cid:13)(cid:13) thenthere is α such that α ≥ α ⇒ (cid:13)(cid:13) ξ j − ξ j α (cid:13)(cid:13) L ( M , S ) < ε nC ,so one has α ≥ α ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = e a j ⊗ ξ j α − n ∑ j = e a j ⊗ ξ j α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( e M , e S ) < ε .The above equation it follows that the net n ∑ nj = e a j ⊗ ξ j α o α satisfies to the Cauchycondition, so it is convergent with respect to the topology of L (cid:16) e M , e S (cid:17) .(ii) From the Lemma 4.8.9 it follows that the image of C (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) in L (cid:16) e M , e S (cid:17) contains Γ (cid:16) e M , e S (cid:17) , however Γ (cid:16) e M , e S (cid:17) is dense in L (cid:16) e M , e S (cid:17) . It followsthat φ (cid:16) C (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) (cid:17) is dense in L (cid:16) e M , e S (cid:17) . Remark 4.10.12.
Indeed the image of C (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) coincides with L (cid:16) e M , e S (cid:17) .However this fact is not used here.Let both µ and e µ be Riemannian measures (cf. [26]) on both M and e M respec-tively which correspond to both the volume element (cf. (E.4.1)) and its p -lift (cf.(4.9.2)). If e a ⊗ ξ , e b ⊗ η ∈ C (cid:16) e M (cid:17) ⊗ C ( M ) Γ ( M , S ) ⊂ L (cid:16) e M , e S (cid:17) then the given by theEquation (2.3.1) scalar product ( · , · ) ind on Γ ( M , S ) ⊗ L ( M , S , µ ) satisfies to the180ollowing equation (cid:16)e a ⊗ ξ , e b ⊗ η (cid:17) ind = (cid:18) ξ , De a , e b E C ( e M ) η (cid:19) L ( M , S ) == ∑ e α ∈ f A (cid:18) ξ , De a e α e a , e b E C ( e M ) η (cid:19) L ( M , S ) = ∑ e α ∈ f A (cid:16) ξ , desc (cid:16)e a e α e a ∗ e b (cid:17) η (cid:17) L ( M , S ) == ∑ e α ∈ f A Z M (cid:16) ξ x , desc (cid:16)e a e α e a ∗ e b (cid:17) η x (cid:17) x d µ == ∑ e α ∈ f A Z M (cid:16) desc ( e e e α e a ) ξ x , desc (cid:16)e e e α e b (cid:17) η x (cid:17) x d µ == ∑ e α ∈ f A Z e M (cid:16)e a lift e U e α ( e e α ξ ) e x , e b lift e U e α ( e e α η ) e x (cid:17) e x d e µ == ∑ e α ∈ f A Z e M e a e α (cid:16)e a lift e U e α ( ξ ) e x , e b lift e U e α ( η ) e x (cid:17) e x d e µ == Z e M (cid:16)e a lift p ( ξ ) e x , e b lift p ( η ) e x (cid:17) e x d e µ = (cid:16)e a lift p ( ξ ) , e b lift p ( η ) (cid:17) L ( e M , e S ) == (cid:16) φ ( e a ⊗ ξ ) , φ (cid:16)e b ⊗ η (cid:17)(cid:17) L ( e M , e S ) (4.10.11)where φ is given by (4.8.11) and both finite families { e a e α } e α ∈ f A , { e e e α } e α ∈ f A ⊂ C (cid:16) e M (cid:17) are explained in the Remark 4.8.7. The equation (4.10.11) means that ( · , · ) ind =( · , · ) L ( e M , e S ) , and taking into account the dense inclusion C (cid:16) e M (cid:17) ⊗ C ( M ) Γ ( M , S ) ⊂ L (cid:16) e M , e S (cid:17) with respect to the Hilbert norm of L (cid:16) e M , e S (cid:17) one concludes that thespace of induced representation coincides with L (cid:16) e M , e S (cid:17) . It means that inducedrepresentation C (cid:16) e M (cid:17) × L (cid:16) e M , e S (cid:17) → L (cid:16) e M , e S (cid:17) is given by (A.3.4). So one hasthe following lemma. Lemma 4.10.13.
If the representation e ρ : C (cid:16) e M (cid:17) → B (cid:16) e H (cid:17) is induced by the pair (cid:16) C ( M ) → B (cid:0) L ( M , S ) (cid:1) , (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17)(cid:17) (cf. Definition 2.3.1) then following conditions holds(a) There is the homomorphism of Hilbert spaces e H ∼ = L (cid:16) e M , e S (cid:17) , b) The representation e ρ is given by the natural action of C (cid:16) e M (cid:17) on L (cid:16) e M , e S (cid:17) (cf. (A.3.4) ).Proof. (a) Follows from (4.10.11).(b) From the Lemma 4.10.11 it follows that the map C (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) ≈ −→ L (cid:16) e M , e S (cid:17) is the homomorphism of C (cid:16) e M (cid:17) modules, so the given by (A.3.4) C (cid:16) e M (cid:17) -action coincides with the C (cid:16) e M (cid:17) -action the given by (2.3.2). Remark 4.10.14.
If the spectral triple (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S , e µ (cid:17) , e D / (cid:17) is the geometri-cal p -lift of the spectral triple (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) (cf. Definition 4.9.1), thenclearly the corresponding to (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) representation of C ( M ) equals to the given by the Lemma 4.10.13 representation . Consider the situation of the Section 4.10.2. Suppose that ∑ g ∈ G ( e M | M ) , α ∈ A g e a α = C ( e M ) (4.10.12)is the compliant with the covering p : e M → M partition of unity (cf. Definition4.8.4). The space M is compact, hence one can suppose that the set A is finite.If we denote by f A def = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) × A then the above partition of unity is thefollowing finite sum ∑ e α ∈ f A e a e α = C ( e M ) .From the Proposition A.1.26 one can assume that e a e α ∈ C ∞ (cid:16) e M (cid:17) and e e e α def = pe a e α ∈ C ∞ (cid:16) e M (cid:17) , ∑ e α ∈ f A e e e α = ∑ e α ∈ f A e e e α desc p ( e e e α ) = C ( e M ) . (4.10.13) Lemma 4.10.15.
Following conditions hold:(i) The finite family { e e e α } e α ∈ f A satisfies to the Lemma 2.7.3. ii) C (cid:16) e M (cid:17) ∩ M | f A | ( C ∞ ( M )) = C ∞ (cid:16) e M (cid:17) . (4.10.14) (iii) The given by the Corollary 4.10.3 unital noncommutative finite-fold covering (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is smoothly invariant (cf. Definition 2.7.2).Proof. (i) For any e a ∈ C (cid:16) e M (cid:17) from (4.10.13) it follows that e a = ∑ e α ∈ f A e e e α h e e e α , e a i e A ∈ ∑ e α ∈ f A e e e α C ( M ) ,i.e. the right C ( M ) -module C (cid:16) e M (cid:17) is generated by the finite set { e e e α } ; that is C (cid:16) e M (cid:17) = ∑ e α ∈ f A e e e α C ( M ) .Let us prove that { e e e α } satisfies to conditions (a) and (b) of the Lemma 2.7.3.(a) One has (cid:10)e e e α ′ , e e f α ′′ (cid:11) C ( e M ) = ∑ g ∈ G ( e M | M ) g (cid:16)e e ∗ e α ′ , e e f α ′′ (cid:17) ,so from e e e α ′ , e e f α ′′ ∈ C ∞ (cid:16) e M (cid:17) it turns out that (cid:10)e e e α ′ , e e f α ′′ (cid:11) C ( e M ) ∈ C ∞ ( M ) .(b) The given by the Equation (4.10.12) set { g e a α } g ∈ G ( e M | M ) , α ∈ A is G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -invariant, hence from e e e α def = √ e a e α it turns out that the family { e e e α } is also G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -invariant.(ii) From (i) the Lemma 2.7.3 it follows that C (cid:16) e M (cid:17) ∩ M n ( C ∞ ( M )) == (cid:26)e a ∈ C ( M ) | (cid:10)e e e α ′ , e a e e f α ′′ (cid:11) C ( e M ) ∈ C ∞ ( M ) ; ∀ e α ′ , f α ′′ ∈ f A (cid:27) (4.10.15)From e e e α ′ , e e f α ′′ ∈ C ∞ (cid:16) e M (cid:17) it turns out that any e a ∈ C ∞ (cid:16) e M (cid:17) satisfies to the right partof (4.10.15). If e a ∈ C (cid:16) e M (cid:17) \ C ∞ (cid:16) e M (cid:17) then there is point e x ∈ e M such that e a is notsmooth at e x . There is e α ∈ f A such that e e e α ( e x ) =
0, so e e e α e a is not smooth at e x . On the183ther hand one has h e e e α , e a e e e α i C ( e M ) = desc p (cid:0)e a e e e α (cid:1) is not smooth at p ( e x ) so e α does notmatch to right part of (4.10.15), i.e. e a / ∈ C (cid:16) e M (cid:17) ∩ M n ( C ∞ ( M )) . In result one has C (cid:16) e M (cid:17) ∩ M n ( C ∞ ( M )) = C ∞ (cid:16) e M (cid:17) . (4.10.16)(iii) From (ii) the Lemma 2.7.3.For any e a ∈ C ∞ (cid:16) e M (cid:17) following condition holds e a = ∑ e α ∈ f A e γ e α = ∑ e α ∈ f A e φ e α e ψ e α ; where e φ e α = e e e α , e ψ e α = e e e α e a , e γ e α = e a e α e e e α . (4.10.17)Denote by Ω D the module of differential forms associated with the given by (E.4.6)spectral triple (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) (cf. Definition E.3.5). Denote by φ e α def = desc (cid:0) e φ e α (cid:1) , ψ e α def = desc (cid:0) e ψ e α (cid:1) , γ e α def = desc ( e γ e α ) ∈ C ∞ ( M ) (4.10.18)Let us define a C -linear map e / ∇ : C ∞ (cid:16) e M (cid:17) → C ∞ (cid:16) e M (cid:17) ⊗ C ∞ ( M ) Ω D , e a ∑ e α ∈ f A (cid:0) e φ e α ⊗ [ / D , ψ e α ] + e ψ e α ⊗ [ / D , φ e α ] (cid:1) . (4.10.19)For any a ∈ C ∞ ( M ) from [ / D , ψ e α a ] = ψ e α [ / D , a ] + [ / D , ψ e α ] a , e φ e α ⊗ ψ e α = e φ e α e ψ e α ⊗ C ∞ ( M ) and (E.4.7) it follows that e / ∇ ( e aa ) = ∑ e α ∈ f A (cid:0) e φ e α ⊗ [ / D , ψ e α a ] + e ψ e α a ⊗ [ / D , φ e α ] (cid:1) == ∑ e α ∈ f A (cid:0) e φ e α ⊗ [ / D , ψ e α ] a + e φ e α ⊗ ψ e α [ / D , a ] + e ψ e α ⊗ a [ / D , φ e α ] (cid:1) == ∑ e α ∈ f A (cid:0) e φ e α ⊗ [ / D , ψ e α ] a + e φ e α e ψ e α ⊗ [ / D , a ] + e ψ e α ⊗ [ / D , φ e α ] a (cid:1) == ∑ e α ∈ f A (cid:0) e φ e α ⊗ [ / D , ψ e α ] + e ψ e α ⊗ [ / D , φ e α ] (cid:1) a + ∑ e α ∈ f A e φ e α e ψ e α ⊗ [ / D , a ] = e / ∇ ( e a ) a + e a ⊗ [ / D , a ] .The comparison of the above equation and (E.3.8) states that e / ∇ is a connection. If g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) then from desc (cid:0) g e φ e α (cid:1) = φ e α and desc (cid:0) g e ψ e α (cid:1) = ψ e α it follows that e / ∇ ( g e a ) = e / ∇ ∑ e α ∈ f A (cid:0) g e φ e α (cid:1) (cid:0) g e ψ e α (cid:1)! == ∑ e α ∈ f A (cid:0) g e φ e α ⊗ [ / D , ψ e α ] + g e ψ e α ⊗ [ / D , φ e α ] (cid:1) = g (cid:16) e / ∇ ( e a ) (cid:17) ,184.e. e / ∇ is G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -equivariant (cf. (2.7.6)). Similarly to (4.8.11) one can define a C ∞ (cid:16) e M (cid:17) -module homomorphism φ ∞ : C ∞ (cid:16) e M (cid:17) ⊗ Γ ∞ ( M , S ) → Γ ∞ (cid:16) e M , e S (cid:17) , n ∑ j = e a j ⊗ ξ j n ∑ j = e a j lift p (cid:0) ξ j (cid:1) (4.10.20)where both Γ ∞ ( M , S ) and Γ ∞ (cid:16) e M , e S (cid:17) are defined by the Equations (E.4.2) and(4.9.4) respectively. Lemma 4.10.16.
The given by the Equation (4.10.20) homomorphism is an isomorphism.Proof.
The proof of the Lemma 4.8.9 uses the partition of unity. However from theProposition A.1.26 it turns out that there is a smooth partition of unity. Using itone can proof this lemma as well as the Lemma 4.8.9 has been proved.Now we have the G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -equivariant connection e / ∇ , so on can find thespecialization of explained in 2.7.5 construction of the operator e D : e A ⊗ A H ∞ → e A ⊗ A H ∞ . The following table reflects the mapping between general theory andthe commutative specializationGeneral theory Commutative specializationHilbert spaces H and e H L ( M , S ) and L (cid:16) e M , e S (cid:17) Pre- C ∗ -algebras A and e A C ∞ ( M ) and C ∞ (cid:16) e M (cid:17) Dirac operators
D D / e D ? H ∞ def = T ∞ n = Dom D n ⊂ H Γ ∞ ( M , S ) = T ∞ n = Dom D / n If / ∇ e a = ∑ mj = e a j ⊗ ω j ∈ C ∞ (cid:16) e M (cid:17) ⊗ C ∞ ( M ) Ω D / then for all ξ ∈ Γ ∞ ( M , S ) denoteby e a ξ = φ ∞ m ∑ j = e a j ⊗ ω j ( ξ ) ! = m ∑ j = e a j lift p (cid:0) ω j ( ξ ) (cid:1) ∈ Γ ∞ (cid:16) e M , e S (cid:17) φ ∞ is given by the Equation (4.10.20), and taking into account the Equation(4.10.19) one has e a ξ = ∑ e α ∈ f A (cid:16) e φ e α lift p ([ / D , ψ e α ] ξ ) + e ψ e α lift p ([ / D , φ e α ] ξ ) (cid:17) .The partition of unity (4.10.12) compliant with the covering p : e M → M , so forany e α ∈ f A there is an open subset e U e α ⊂ e M such that • supp e a e α ⊂ e U e α . • The restriction p | e U e α is injective.From (4.10.17) it follows that supp e φ e α , supp e ψ e α , supp e γ e α ⊂ e U e α . From the Equa-tions (4.5.72) and (4.10.18) it follows that e φ e α = lift p e U e α ( φ e α ) , e ψ e α = lift p e U e α ( ψ e α ) , e γ e α = lift p e U e α ( γ e α ) ,hence one has e a ξ = ∑ e α ∈ f A (cid:16) lift p e U e α ( φ e α ) lift p ([ / D , ψ e α ] ξ ) + lift p e U e α ( ψ e α ) lift p ([ / D , φ e α ] ξ ) (cid:17) == ∑ e α ∈ f A (cid:16) lift p e U e α ( φ e α [ / D , ψ e α ] ξ ) + lift p e U e α ( ψ e α [ / D , φ e α ] ξ ) (cid:17) .Otherwise from (E.4.7) it turns out that ψ e α [ / D , φ e α ] = [ / D , φ e α ] ψ e α , so it follows that e a ξ = ∑ e α ∈ f A (cid:16) lift p e U e α ( φ e α [ / D , ψ e α ] ξ ) + lift p e U e α ( ψ e α [ / D , φ e α ] ξ ) (cid:17) == ∑ e α ∈ f A (cid:16) lift p e U e α ( φ e α [ / D , ψ e α ] ξ + ψ e α [ / D , φ e α ] ξ ) (cid:17) == ∑ e α ∈ f A (cid:16) lift p e U e α ([ / D , φ e α ψ e α ] ξ ) (cid:17) = ∑ e α ∈ f A (cid:16) lift p e U e α ([ / D , γ e α ] ξ ) (cid:17) .If p − ( D / ) ∈ E nd (cid:16) C ∞ (cid:16) e M , e S (cid:17)(cid:17) is the p -inverse image of D / (cf. Definition 4.7.11)then taking into account (4.7.12) one has e a ξ = ∑ e α ∈ f A h p − ( D / ) , e γ e α i lift p ( ξ ) = " p − ( D / ) , ∑ e α ∈ f A e γ e α lift p ( ξ ) == h p − ( D / ) , e a i lift p ( ξ ) .186f e D is the specialization of the given by (2.7.7) operator then φ ∞ (cid:16) e D ( e a ⊗ ξ ) (cid:17) = e a ξ + φ ∞ ( e a ⊗ D ξ ) == h p − ( D / ) , e a i lift p ( ξ ) + e a lift p ( D / ξ ) == h p − ( D / ) , e a i lift p ( ξ ) + e ap − ( D / ) lift p ( ξ ) = p − D / e a lift p ( ξ ) == p − ( D / ) φ ∞ ( e a ⊗ ξ ) ,or, shortly φ ∞ (cid:16) e D ( e a ⊗ ξ ) (cid:17) = p − D / φ ∞ ( e a ⊗ ξ ) , (4.10.21)However φ ∞ is the isomorphism, it can be regarded as the identical map, so theequation (4.10.21) is equivalent to e D ( e a ⊗ ξ ) = p − ( D / ) ( e a ⊗ ξ ) , (4.10.22)in result one has e D = p − / D . (4.10.23)In more detail the above formula has the following meaning e / ∇ ( e a ) = m ∑ j = e a j ⊗ (cid:2) D / , a j (cid:3) ⇒ m ∑ j = e a j ⊗ (cid:2) D / , a j (cid:3) ξ + e a ⊗ D / ξ = p − / D ( e a ⊗ ξ ) (4.10.24)where p − D / is the p -inverse image of D / (cf. Definition 4.7.11 and Equation 4.7.10). Theorem 4.10.17.
Let (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) be a commutative spectral triple andlet p : e M → M be a transitive finite-fold covering. If (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is an unital noncommutative finite-fold covering given by the Corollary 4.10.3 then thegeometrical p-lift of (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) (cf. Definition 4.9.1) coincides with the (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) -lift of (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) (cf. Definition 2.7.6).Proof. From the given by Section 4.9 construction the the geometrical p -lift of (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) equals to (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S (cid:17) , e D / (cid:17) where • e S is the e S = p ∗ S the inverse image of the spin bundle S (cf. Definition A.3.7). • The given by (A.3.4) natural representation C (cid:16) e M (cid:17) → B (cid:16) L (cid:16) e M , e S (cid:17)(cid:17) is as-sumed. 187 Operator e D / can be regarded as e D / = p − / D ∈ E nd (cid:16) C ∞ (cid:16) e M , n e S e x o , Γ (cid:16) e M , e S (cid:17)(cid:17)(cid:17) (cid:16) e M (cid:17) . (4.10.25)Denote by (cid:16) e A , e H , e D (cid:17) the (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) , C ( p ) (cid:17) -lift of (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) . Consider the given by (4.10.13) finite set { e e e α } e α ∈ f A . From(iii) of the Lemma 4.10.15 it turns that (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is smoothlyinvariant. According to the Definition 2.7.6 and taking into account the Equation(4.10.16) one has e A = C (cid:16) e M (cid:17) ∩ M n ( C ∞ ( M )) = C ∞ (cid:16) e M (cid:17) . If ρ : C ( M ) → L ( M , S ) is the representation of the spectral triple (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) then the repre-sentation e ρ : C (cid:16) e M (cid:17) → B (cid:16) e H (cid:17) of the spectral triple (cid:16) e A , e H , e D (cid:17) is induced by thepair (cid:16) ρ , (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17)(cid:17) . From the Remark 4.10.14 it follows thatthe representation e ρ is equivalent to the representation, which corresponds to thespectral triple (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) . From (4.10.23) it follows that e D = p − / D and taking into account (4.10.25) one has e D = e D / . It result the (cid:16) e A , e H , e D (cid:17) spectraltriple is equivalent to the (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S (cid:17) , e D / (cid:17) one. Let M be an unoriented Riemannian manifold, and let p : e M → M be a twolisted regular covering by oriented Riemannian manifold e M which admits a spin c structure (cf. Definition E.4.2) given by a spinor bundle e S . From the Corollary4.10.3 it follows that the triple (cid:16) C ( M ) , C (cid:16) e M (cid:17) , Z (cid:17) is an unital finite-fold non-commutative covering (cf. Definition 2.1.9). If there is a bundle S → M issuch that e S is the p -inverse image of S (cf. Definition A.3.7) then S is a Her-mitian vector bundle (cf. Definition A.3.11). It follows that there is a Hilbertspace L ( M , S ) and the faithful representation ρ : C ( M ) → B (cid:0) L ( M , S ) (cid:1) . Simi-larly to the Lemma 4.10.13 one can proof that the given by (A.3.4) representation e ρ : C (cid:16) e M (cid:17) ֒ → B (cid:16) L (cid:16) e M , e S (cid:17)(cid:17) is induced by the pair (cid:16) C ( M ) → B (cid:0) L ( M , S ) (cid:1) , (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17)(cid:17) (cf. Definition 2.3.1). The manifold e M admits a spin c structure, so there is thespectral triple (cid:16) C ∞ (cid:16) e M , (cid:17) , L (cid:16) e M , e S (cid:17) , e D / (cid:17) . Since e S is Z -invariant the operator e D /188s also Z -invariant there is an unbounded operator D / def = e D / (cid:12)(cid:12)(cid:12) L ( e M , e S ) Z (4.10.26)where the the induced by the map (2.3.3) action Z × L (cid:16) e M , e S (cid:17) → L (cid:16) e M , e S (cid:17) isimplied. On the other hand L (cid:16) e M , e S (cid:17) Z ∼ = L ( M , S ) . Theorem 4.10.18.
The described above triple (cid:18) C ∞ ( M ) , L (cid:16) e M , e S (cid:17) Z = L ( M , S ) , / D (cid:19) . (4.10.27) a commutative unoriented spectral triple (cf. Definition 2.9.1).Proof. The following table shows the specialization of objects 1-5 required by theDefinition 2.9.1.Definition 2.9.1 Commutative specialization1 A C ∞ ( M ) (cid:16) A , e A , Z (cid:17) (cid:16) C ( M ) , C (cid:16) e M (cid:17) , Z (cid:17) ρ : A → B ( H ) C ( M ) → B (cid:0) L ( M , S ) (cid:1) D D /5 (cid:16) e A , e H , e D (cid:17) (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S (cid:17) , e D / (cid:17) Let us proof conditions (a)-(d) of the Definition 2.9.1. Conditions (a)-(c) followfrom the discussion at the beginning of the Section 4.10.4. Let us prove (d). Thefollowing evident formula C ∞ ( M ) = C ( M ) ∩ C ∞ (cid:16) e M (cid:17) is the specialization of the Equation (2.9.4). The Equation (4.10.26), is the special-ization of (2.9.5). 189 xample 4.10.19. If S is the 3-dimensional sphere then the tangent bundle TS istrivial. The follows that the bundle C ℓ (cid:0) S (cid:1) → S (cf. E.4.1) is also trivial. So thereis the trivial spinor bundle e S → S , and the spectral triple (cid:16) C ∞ (cid:0) S (cid:1) , L (cid:16) S , e S (cid:17) , e D / (cid:17) which satisfies to described in [39,73] axioms. Otherwise there is two listed regulartopological covering p : S → R P where R P is the 3-dimensional real projectivespace. The spinor bundle e S is trivial, hence there is the trivial bundle S → RP such that e S is the p -inverse image of S . From the Theorem 4.10.18 it follows thatthere is the given by (cid:0) R P , L (cid:0) R P , S (cid:1) , D / (cid:1) unoriented spectral triple. Let X be a connected, locally connected, locally compact, Hausdorff topo-logical space. Let { p λ : X λ → X } λ ∈ Λ be a family of transitive finite-fold coveringswhich is indexed by a countable set Λ such that X λ is connected for all λ ∈ Λ . Letus define an order on Λ such that µ ≥ ν if and only if there is a surjective coninuous map p : X µ → X ν such that p µ = p ν ◦ p . (4.11.1)From the Corollary A.2.6 it follows that the map p is a covering. Suppose thatthere is the minimal λ min ∈ Λ such that X λ min def = X and suppose that the orderedset Λ is directed (cf. Definition A.1.3). Suppose that for any λ ∈ Λ there is a basepoint x λ ∈ X λ such that ∀ µ , ν ∈ Λ p µ (cid:0) x µ (cid:1) = p ν ( x ν ) , µ ≥ ν ⇒ ∃ p : X µ → X ν p µ = p ν ◦ p AND p (cid:0) x µ (cid:1) = x ν .For any µ ≥ ν there is the unique pointed map p : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) whichsatisfies to the above equation we denote it by p µν def = p . From the Corollary A.2.6it turns out that p µν is a covering for all µ > ν . The spaces X λ and coverings p µν yield a category which is equivalent to the pre-order category Λ (cf. DefinitionH.1.1). So there is the inverse limit b X = lim ←− λ ∈ Λ X λ in the category of topologicalspaces and continuous maps. For all λ ∈ Λ a there is the natural continuous map b p λ : b X → X . 190 efinition 4.11.2.
Consider the situation 4.11.1 and suppose that Λ is a countabledirected set (cf. Definition A.1.3) such that there is the unique minimal element λ min ∈ Λ . Let X be a connected locally connected second-countable topologi-cal space. Consider a full subcategory S the finite covering category of X (cf.Definition 4.3.25). The set of objects is marked by Λ , i.e. the set of objects is afamily { p λ : X λ → X } λ ∈ Λ of transitive finite-fold coverings such that X λ min = X ,and p λ min = Id X . Morphism from p µ : X µ → X to p ν : X ν → X is a surjectivecontinuous map p : X µ → X ν such that p µ = p ν ◦ p . We say that S is a topologicalfinite covering category and we write S ∈ FinTop . Definition 4.11.3.
Let S X be a topological finite covering category with the setof objects { p λ : X λ → X } λ ∈ Λ . Suppose that there are base points x ∈ X and x λ ∈ X λ such that for any λ ∈ Λ the p λ is a pointed map (cid:0) X λ , x λ (cid:1) → ( X , x ) (cf.Definition 4.3.29). Let us consider a subcategory S ( X , x ) = (cid:8)(cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:1) } of S X such that any morphism p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) of S ( X , x ) is a pointedmap. We say that S ( X , x ) is a pointed topological finite covering category and we write S ( X , x ) def = def = (n p λ : (cid:16) X λ , x λ (cid:17) → ( X , x ) o λ ∈ Λ , (cid:8) p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) (cid:9) µ , ν ∈ Λ µ ≥ ν ) S ( X , x ) ∈ FinTop pt . (4.11.2) Remark 4.11.4.
The category S ( X , x ) is a subcategory of the category FinCov - ( X , x ) the pointed finite covering category of ( X , x ) Remark 4.11.5.
The category S ( X , x ) is equivalent to the pre-order category Λ (cf.Definition H.1.1). Definition 4.11.6.
Let S ( X , x ) be a pointed topological finite covering categorygiven by the Equation (4.11.2) and let b X def = lim ←− λ ∈ Λ X λ be the inverse limit oftopological spaces. Let b p λ : b X → X λ be the natural continuous map for all λ ∈ Λ .The subset b U ⊂ b X is said to be special if for all λ ∈ Λ following conditions hold: • The restriction b p λ | b U is injective. • The set b p λ (cid:16) b U (cid:17) is connected and open. Lemma 4.11.7.
Let b X = lim ←− λ ∈ Λ X λ be the inverse limit in the category of topologicalspaces and continuous maps. Any special set of b X is a Borel subset of b X . roof. If U λ ⊂ X λ is an open set then b p − λ ( U λ ) ⊂ b X is open. If b U is a special setthen b U = T λ ∈ Λ b p − λ ◦ b p λ (cid:16) b U (cid:17) , i.e. b U is a countable intersection of open sets. So b U is a Borel subset. Lemma 4.11.8.
Consider the situation of the Definition 4.11.3 and denote by n b U α ⊂ b X o α ∈ A the collection of all special sets (cf. Definition 4.11.6). Following condition holds:(i) The collection n b U α o is a basis for a topology on b X (cf. Definition A.1.1).(ii) Let X be the topological space which coincides with b X as a set and the topology of X is generated by n b U α o , and let b p : X → b X be the natural biective map. For all λ ∈ Λ the composition p λ def = b p λ ◦ b p : X → b X → X λ is a covering.(iii) The bijective map b p : X → b X is continuous.Proof. (i) One needs proof (a) and (b) of the Definition A.1.1.(a) Let b x ∈ b X , denote by x λ = b p λ ( b x ) . Let { λ n } n ∈ N ⊂ Λ be a cofinal subsetsuch that m > k ⇒ λ m > λ k . Let b p : b X → X be the natural continuousmap, and let x = b p ( b x ) . Since X is locally connected there exists an openconnected neighborhood U of x evenly covered by p λ : X λ → X . For any n there is the unique open connected neighborhood U λ n ⊂ X λ n of x λ n such that p λ n ( U λ n ) = U . For any λ ∈ Λ there is λ n such that λ n ≥ λ . If U λ def = p λ n λ ( U λ n ) then p λ ( U λ ) = U . If b U def = T λ ∈ Λ b p − λ ( U λ ) then the restriction b p λ | b U injectiveand maps b U onto the open set U λ . Thus b U is special and b x ∈ b U .(b) If both b U and b U are special sets then since b p λ | b U is injective the restriction b p λ | b U ∩ b U is also injective. Since b p λ (cid:16) b U ∩ b U (cid:17) = b p λ (cid:16) b U (cid:17) ∩ b p λ (cid:16) b U (cid:17) and both b p λ (cid:16) b U (cid:17) , b p λ (cid:16) b U (cid:17) are open, the set b p λ (cid:16) b U ∩ b U (cid:17) is open. If b p : b X → X is the natural continuous map then since X is locally connected there is aconnected open subset U ⊂ X such that U ⊂ p (cid:16) b U (cid:17) ∩ p (cid:16) b U (cid:17) . If b U = b U ∩ b U ∩ b p − ( U ) then clearly b U ⊂ b U ∩ b U . Moreover for all λ ∈ Λ the set b U is homeomorphically mapped onto open connected set b p λ (cid:16) b U (cid:17) ∩ b p λ (cid:16) b U (cid:17) ∩ p − λ ( U ) ∼ = U , i.e. b U is special.(ii) Let x λ ∈ X λ be a point, and let x = p λ (cid:0) x λ (cid:1) . There is a connected openneighborhood U of x evenly covered by p λ , and the connected neighborhood U λ x λ such that p λ ( U λ ) . Let { λ n } ⊂ Λ be an indexed by natural numbers finite orcountable linearly ordered cofinal subset such that m > k ⇒ λ m > λ k and λ n > λ for any n . Let b x ∈ b X be such that b p ( b x ) = x λ . For any n there is the uniqueopen connected neighborhood U λ n ⊂ X λ n of x λ n such that p λ n ( U λ n ) = U . For any λ ′ ∈ Λ there is λ n such that λ n ≥ λ . If U λ ′ def = p λ n λ ′ ( U λ n ) then p λ ′ ( U λ ′ ) = U . If b U b x def = T λ ′ ∈ Λ b p − λ ′ ( x λ ′ ) then the restriction b p λ ′ | b U injective and maps b U onto theopen set U λ ′ . Thus b U is special and b x ∈ b U . Similarly for any b x ∈ b p − λ (cid:0) x λ (cid:1) onecan construct b U b x such that b x ∈ b U b x and b U b x is mapped homeomorphically onto U λ .If b x , b x ∈ b p − λ (cid:0) U λ (cid:1) are such that b x = b x then b U b x ∩ b U b x = ∅ . So one has b p − λ ( U λ ) = G b x ∈ b p − λ ( x λ ) b U b x ,hence taking into account that both b p λ : b X → X λ and p λ : X → X λ are surjectivewe conclude that p λ is a covering.(iii) All maps p λ : X → X λ are continuous, hence from the definition of the inverseimage it follows that the natural map X → b X is continuous. Definition 4.11.9.
Consider the situation of the Definition 4.11.3, and let b X = lim ←− X λ be the inverse limit in the category of topological spaces and continuousmaps (cf. [70]). If b p : b X → X is the natural continuous map then a homeo-morphism g of the space b X is said to be a covering transformation if a followingcondition holds b p = b p ◦ g .The group b G of such homeomorphisms is said to be the group of covering transfor-mations of S . Denote by G (cid:16) b X (cid:12)(cid:12)(cid:12) X (cid:17) def = b G . Lemma 4.11.10.
Consider the situation of the Definition 4.11.3, and suppose that for all λ ∈ Λ the the covering is transitive (cf. Definition 4.3.1). Let (cid:16) b X , b x (cid:17) = lim ←− (cid:0) X λ , x λ (cid:1) be the inverse limit in the category of topological spaces and continuous maps. There is thenatural group isomorphism G (cid:16) b X (cid:12)(cid:12)(cid:12) X (cid:17) ∼ = lim ←− G ( X λ | X ) . For any λ ∈ Λ there is thenatural surjective homomorphism h λ : G (cid:16) b X (cid:12)(cid:12)(cid:12) X (cid:17) → G ( X λ | X ) and T λ ∈ Λ ker h λ is atrivial group.Proof. There is a continuous map b p : (cid:16) b X , b x (cid:17) → ( X , x ) and for any λ ∈ Λ thereis the natural continuous map b p λ : b p : (cid:16) b X , b x (cid:17) → (cid:0) X λ , x λ (cid:1) Let b x ′ ∈ b X be such193hat b p ( b x ′ ) = x . If x ′ λ = b p λ ( b x ′ ) and x λ = b p λ ( b x ) then p λ ( x λ ) = p λ ( x ′ λ ) , where p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) is the natural covering. Since p λ is transitive for any λ ∈ Λ there is the unique g λ ∈ G ( X λ | X ) such that x ′ λ = g λ x λ . In result there isa sequence { g λ ∈ G ( X λ | X ) } λ ∈ Λ which satisfies to the following condition g µ ◦ p λµ = p λµ ◦ g λ where λ > µ and p λµ : X λ → X µ is a morphism of S ( X , x ) . The sequence { g λ } naturally defines an element b g ∈ lim ←− G ( X λ |X ) . Let us define an homeomorphism ϕ b g : b X → b X by a following construction. If b x ′′ ∈ b X is any point then there is thefamily (cid:8) x ′′ λ ∈ X λ (cid:9) λ ∈ Λ such that x ′′ λ = b p λ (cid:0)b x ′′ (cid:1) .On the other hand there is the family n x ′′ b g λ ∈ X λ o λ ∈ Λ x ′′ b g λ = g λ x ′′ λ which for any n > m satisfies to the following condition p λµ (cid:16) x ′′ b g λ (cid:17) = x ′′ g µ .From the above equation and properties of inverse limits it follows that there isthe unique b x ′′ b g ∈ b X such that b p λ (cid:16)b x ′′ b g (cid:17) = x ′′ b g λ ; ∀ λ ∈ Λ .The required homeomorphism ϕ b g is given by ϕ b g (cid:0) b x ′′ (cid:1) = b x ′′ b g .From b p ◦ ϕ b g = b p it follows that ϕ b g corresponds to an element in G (cid:16) b X | X (cid:17) which mapped onto g λ for any λ ∈ Λ . Otherwise ϕ b g naturally correspondsto the element b g ∈ lim ←− G ( X λ |X ) , so one has the natural group isomorphism G (cid:16) b X (cid:12)(cid:12)(cid:12) X (cid:17) ∼ = lim ←− G ( X λ | X ) . From the above construction it turns out that anyhomeomorphism b g ∈ G (cid:16) b X | X (cid:17) uniquely depends on b x ′ = b g b x ∈ b p − ( x ) . Itfollows that there is the 1-1 map ϕ : b p − ( x ) ≈ −→ G (cid:16) b X (cid:12)(cid:12)(cid:12) X (cid:17) . Since the covering194 λ : X λ → X is transitive there is the 1-1 map ϕ λ : p − λ ( x ) ≈ −→ G ( X λ | X ) . Thenatural surjective map b p − ( x ) → p − λ ( x ) induces the surjective homomorphism G (cid:16) b X (cid:12)(cid:12)(cid:12) X (cid:17) → G ( X λ | X ) . If b g ∈ T Λ ∈ λ ker h λ is not trivial then b g b x = b x and there is n ∈ N such that b p λ ( b x ) = b p λ ( b g b x ) = h λ ( b g ) b p λ ( b x ) , so h λ ( b g ) ∈ G ( X λ | X ) is not trivial and b g / ∈ ker h λ . From thiscontradiction it follows that T λ ∈ Λ ker h λ is a trivial group. Lemma 4.11.11.
Consider the situation of the Definition 4.11.3. Let X be the topologicalspace given by the Lemma 4.11.8 with natural bicontinuous map X → b X . There is thenatural isomorphism G (cid:0) X | X (cid:1) ≈ −→ G (cid:16) b X | X (cid:17) induced by the map
X → b X .Proof. Since X coincides with b X as a set, and the topology of X is finer than thetopology of b X there is the natural injective map G (cid:0) X | X (cid:1) ֒ → G (cid:16) b X | X (cid:17) . If b g ∈ G (cid:16) b X | X (cid:17) and b U is a special set, then for any n ∈ N following conditionholds b p λ (cid:16)b g b U (cid:17) = h λ ( b g ) ◦ b p λ (cid:16) b U (cid:17) (4.11.3)where b p λ : b X → X λ is the natural map, and h λ : G (cid:16) b X | X (cid:17) → G ( X λ | X ) is givenby the Lemma 4.11.10. Clearly h λ ( b g ) is a homeomorphism of X λ , so from (4.11.3)it follows that b p λ (cid:16)b g b U (cid:17) is an open subset of X λ . Hence b g b U is special. So b g mapsspecial sets onto special sets. Since topology of X is generated by special sets themap b g is a homeomorphism of X , i.e. b g ∈ G (cid:0) X | X (cid:1) . Corollary 4.11.12.
In the situation of the Lemma 4.11.11, the natural coverings
X → X and
X → X λ are transitive. Definition 4.11.13.
Consider the situation of the Definition 4.11.3. A pair (cid:0) ( Y , y ) , p Y (cid:1) of a topological connected pointed space ( Y , y ) with and preservingbase-point continuous map p Y λ : ( Y , y ) → (cid:16) b X , b x (cid:17) is said to be a transitive coveringof S ( X , x ) if for any λ ∈ Λ the composition p Y λ def = b p λ ◦ p Y is a transitive covering.We write (cid:0) ( Y , y ) , p Y (cid:1) ↓ S ( X , x ) . Definition 4.11.14.
Let us consider the situation of the Definition 4.11.13. A mor-phism from (cid:16) Y ′ , p Y ′ (cid:17) ↓ S ( X , x ) to (cid:16) Y ′′ , p Y ′′ (cid:17) ↓ S ( X , x ) is a preserving base-point195ontinuous map f : (cid:16) Y ′ , p Y ′ (cid:17) → (cid:16) Y ′′ , p Y ′′ (cid:17) such that p Y ′′ ◦ f = p Y ′ . Remark 4.11.15. If f : (cid:16) Y ′ , p Y ′ (cid:17) → (cid:16) Y ′′ , p Y ′′ (cid:17) is a morphism from (cid:16) Y ′ , p Y ′ (cid:17) ↓ S ( X , x ) to (cid:16) Y ′′ , p Y ′′ (cid:17) ↓ S ( X , x ) then from the Lemma 4.3.7 it follows that f is atransitive covering. There is a category with objects and morphisms described by Definitions4.11.13, 4.11.14 respectively. Denote by ↓ S ( X , x ) this category. Lemma 4.11.17.
There is the final object of the category ↓ S ( X , x ) described in 4.11.16.Proof. Let b X = lim ←− λ ∈ Λ X λ be the inverse limit in the category of topological spacesand continuous maps and b p λ : b X → X λ , b p : b X → X be natural continuous maps.The family (cid:8) x λ (cid:9) uniquely defines the point b x ∈ b X . If X is the topological spacegiven by (ii) of the Lemma 4.11.8 then the point b x uniquely defines the point x ∈ X because X coincides with b X as the set. Let b p : X → b X be the given by (ii)of the Lemma continuous map. Denote by e X ⊂ X the connected component of x ,and denote by e x = x . From the Lemma 4.11.8 it follows that the compositions b p λ ◦ b p : (cid:0) X , x (cid:1) → ( X , x ) and b p ◦ b p : (cid:0) X , x (cid:1) → (cid:0) X λ , x λ (cid:1) are pointed coverings.From the Theorem A.2.7 it follows that restrictions b p λ ◦ b p (cid:12)(cid:12)(cid:12) e X : (cid:16) e X , x (cid:17) → ( X , x ) and b p ◦ b p (cid:12)(cid:12)(cid:12) e X : (cid:16) e X , x (cid:17) → (cid:0) X λ , x λ (cid:1) are also coverings. For all λ ∈ Λ the covering b p λ ◦ b p is transitive, so b p ◦ b p (cid:12)(cid:12)(cid:12) e X is also transitive because e X is a connected componentof X .Thus the pair (cid:16) e X , b p (cid:12)(cid:12)(cid:12) e X (cid:17) is an object of the category ↓ S ( X , x ) , let us prove thatit is the final object. Since the continuous pointed map b p : (cid:0) X , x (cid:1) → (cid:16) b X , b x (cid:17) is bijective there is the natural map p Y : ( Y , y ) → (cid:0) X , x (cid:1) . Denote by p Y : ( Y , y ) → ( X , x ) the natural pointed covering. Suppose that y ∈ Y is any point,and let x = p Y ( y ) ∈ X and x = p ( x ) = p Y ( y ) ∈ X . Let both U and U be openconnected neighborhoods of x evenly covered by both p Y and p respectively. Let U be an open connected neighborhood of x such that U ∈ U ∩ U . One has p − ( U ) = G x ′ ∈ p − ( x ) U x ′ , p − Y ( U ) = G y ′ ∈ p − Y ( x ′ ) V y ′ U x ′ and V y ′ are connected neighborhoods of x ′ and y ′ respectively. Itfollows that p Y (cid:0) U x (cid:1) = G y ′ ∈ p − Y ( x ′ ) p Y ( y ′ )= x V y ′ It turns out that the map p Y is continuous. Since Y is connected one has p Y ( Y ) ⊂ e X , or there is the natural continuous map e p Y : ( Y , y ) → (cid:16) e X , e x (cid:17) . Consider the situation of the Lemma 4.11.17. Let G ⊂ G (cid:0) X (cid:12)(cid:12) X (cid:1) be themaximal among subgroups G ′ ⊂ G (cid:0) X (cid:12)(cid:12) X (cid:1) such that G ′ e X = e X . For all x ∈ X there is e x ∈ e X such that p ( x ) = p ( e x ) . It follows that there is g ∈ G (cid:0) X (cid:12)(cid:12) X (cid:1) suchthat x = g e x . Since g is a homeomorphism it homeomorphically maps e x onto theconnected component of x . It follows that X = G g ∈ J g e X (4.11.4)where J ⊂ G (cid:0) X (cid:12)(cid:12) X (cid:1) is a set of representatives of G (cid:0) X (cid:12)(cid:12) X (cid:1) / G . Otherwise thereis the group isomorphism G ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) (4.11.5)and the covering e X → X is transitive.
Definition 4.11.19.
The given by the Lemma 4.11.17 final object (cid:16)(cid:16) e X , e x (cid:17) , p e X (cid:17) of the category ↓ S ( X , x ) is said to be the topological inverse limit of S ( X , x ) . Thenotation (cid:16)(cid:16) e X , e x (cid:17) , p e X (cid:17) = lim ←− S ( X , x ) or simply e X = lim ←− S ( X , x ) will be used.The given by (ii) of the Lemma 4.11.8 space X is said to be the disconnected inverselimit of S ( X , x ) . Lemma 4.11.20.
Let S ( X , x ) = (cid:8)(cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt be a pointedtopological finite covering category, and let X be the disconnected inverse limit of S ( X , x ) .For any compact U ⊂ X there is λ U ∈ Λ such that for any λ ≥ λ U ∈ Λ such that for any λ ≥ λ U the set U is mapped homeomorphically onto p λ (cid:0) U (cid:1) ⊂ X λ where p λ : X → X λ is the natural covering.Proof. Let us select any λ ∈ Λ . For any x ∈ U there is an open there is a connectedopen neigborhood U x which is mapped homeomorphically onto p λ (cid:0) U x (cid:1) . One has U ⊂ ∪ x ∈U U x . Since U is compact there is a finite set (cid:8) U x , ..., U x n (cid:9) ⊂ (cid:8) U x (cid:9) x ∈U such197hat U ⊂ ∪ nj = U x j . Write (cid:8) U , ..., U n (cid:9) def = (cid:8) U x , ..., U x n (cid:9) . Suppose that j , k =
1, ..., n be such that U j ∩ U k = ∅ and p λ (cid:0) U j (cid:1) ∩ p λ (cid:0) U k (cid:1) = ∅ . Then there are x j ∈ U j and x k ∈ U k such that p λ (cid:0) x j (cid:1) = p λ ( x k ) . Since x j = x k there is λ jk ∈ Λ such that p λ jk (cid:0) x j (cid:1) = p λ jk ( x k ) . It turns out that p λ jk (cid:0) U j (cid:1) ∩ p λ jk (cid:0) U k (cid:1) = ∅ . If λ U = max jk λ jk then for all λ ≥ λ U the set U is mapped homeomorphically onto p λ (cid:0) U (cid:1) ⊂ X λ . Corollary 4.11.21.
Let S ( X , x ) = (cid:8)(cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt be a pointedtopological finite covering category, and let X be the disconnected inverse limit of S ( X , x ) .If the space X is normal then for any compact U ⊂ X there is λ U ∈ Λ and a contin-uous function f λ U : X λ U → [
0, 1 ] such that f λ U (cid:16) p λ U (cid:0) U (cid:1)(cid:17) = { } and g supp f λ U ∩ supp f λ U = ∅ for all nontrivial g ∈ G (cid:16) X λ U (cid:12)(cid:12)(cid:12) X (cid:17) .Proof. From the Lemma 4.11.20 one can suppose that gp λ U (cid:0) U (cid:1) ∩ p λ U (cid:0) U (cid:1) = ∅ forall nontrivial g ∈ G (cid:16) X λ U (cid:12)(cid:12)(cid:12) X (cid:17) , so one has p λ U (cid:0) U (cid:1) \ [ g ∈ G (cid:16) X λ U (cid:12)(cid:12)(cid:12) X (cid:17) \{ e } gp λ U (cid:0) U (cid:1) = ∅ where e is the trivial element of G (cid:16) X λ U (cid:12)(cid:12)(cid:12) X (cid:17) . It turns out that there is f ′ : X λ U → [
0, 1 ] such that f ′ (cid:16) p λ U (cid:0) U (cid:1)(cid:17) = { } and f ′ [ g ∈ G (cid:16) X λ U (cid:12)(cid:12)(cid:12) X (cid:17) \{ e } gp λ U (cid:0) U (cid:1) = { } .Clearly g supp f ′ ∩ p λ U (cid:0) U (cid:1) = ∅ for any nontrivial g ∈ G (cid:16) X λ U (cid:12)(cid:12)(cid:12) X (cid:17) , so one has p λ U (cid:0) U (cid:1) \ supp (cid:0) − f ′ (cid:1) [ [ g ∈ G (cid:16) X λ U (cid:12)(cid:12)(cid:12) X (cid:17) \{ e } g supp f ′ = ∅ It turns out that there is f : X λ U → [
0, 1 ] such that f (cid:16) p λ U (cid:0) U (cid:1)(cid:17) = { } and f supp (cid:0) − f ′ (cid:1) [ [ g ∈ G (cid:16) X λ U (cid:12)(cid:12)(cid:12) X (cid:17) \{ e } g supp f ′ = { } .198rom f ( supp ( − f ′ )) = { } it follows that supp f ⊂ supp f ′ and taking intoaccount that supp f ∩ g supp f ′ = ∅ for all nontrivial g ∈ G (cid:16) X λ U (cid:12)(cid:12)(cid:12) X (cid:17) one has supp f ∩ g supp f = ∅ . Lemma 4.11.22.
Let ( X , x ) be a connected, locally connected, locally compact, Hausdorffpointed space, and let e p : (cid:16) e X , e x (cid:17) → ( X , x ) be a pointed transitive covering such that e X is connected and the covering group G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) (cf. Definition A.2.3) is residually finite(cf. Definition B.3.2). Let us consider a family S ( X , x ) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ of all transitive finite-fold pointed coverings such for every λ ∈ Λ there is a covering e p λ : (cid:16) e X , e x (cid:17) → (cid:0) X λ , x λ (cid:1) such that e p = p λ ◦ e p λ . Following conditions hold:(i) The family S ( X , x ) is a pointed a topological finite covering category.(ii) The pointed space (cid:16) e X , e x (cid:17) is the inverse noncommutative limit of S ( X , x ) . Proof.
For any finite group H with surjective group homomorphism φ : G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) → H there the space X H = e X / ker φ . There base-point e x naturally defines the basepoint x H ∈ X H . There are natural pointed coverings p H : (cid:0) X H , x H (cid:1) → ( X , x ) and e p H : (cid:16) e X , e x (cid:17) → (cid:0) X H , x H (cid:1) , p H : (cid:0) X H , x H (cid:1) → ( X , x ) such that e p = p H ◦ e p H . Itturns out that p H belongs to the family S ( X , x ) . Otherwise since e p is a transitivecovering for any λ ∈ Λ there is the surjective homomorphism φ λ : G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) → G ( X λ | X ) such that the following condition holds X λ = e X / ker φ λ . So there is aone-to-one correspondence between objects of S ( X , x ) and finite groups which areepimorphic images of G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) . The group G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is residually finite so themap φ = ∏ λ ∈ Λ φ λ : G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) ֒ → ∏ λ ∈ Λ G ( X λ | X ) (4.11.6)is an injective homomorphism.(i) Let us consider the order on Λ given by (4.11.1), i.e. µ ≥ ν if and only if there is a surjective coninuous map p : X µ → X ν such that p µ = p ν ◦ p . (4.11.7)199or any λ ∈ Λ the kernel ker φ λ ⊂ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is a subgroup of finite index. Forany µ , ν ∈ Λ the intersection ker φ µ ∩ ker φ ν ⊂ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is also a subgroup offinite index, so there is the group G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) / (cid:0) ker φ µ ∩ ker φ ν (cid:1) is finite. It followsthat there is the space e X / (cid:0) ker φ µ ∩ ker φ ν (cid:1) with the natural finite-fold covering p µν : e X / (cid:0) ker φ µ ∩ ker φ ν (cid:1) → X and the covering e p µν : e X → e X / (cid:0) ker φ µ ∩ ker φ ν (cid:1) such that e p = p µν ◦ e p µν . From the definition of S ( X , x ) there is λ ∈ Λ such thatthe covering p µν : e X / (cid:0) ker φ µ ∩ ker φ ν (cid:1) → X is equivalent to p λ : X λ → X . Fromthe natural coverings e X / (cid:0) ker φ µ ∩ ker φ ν (cid:1) → e X / ker φ µ , e X / (cid:0) ker φ µ ∩ ker φ ν (cid:1) → e X / ker φ ν it follows that there are coverings X λ → X µ and X λ → X µ and from(4.11.7) it turns out that λ ≥ µ and λ ≥ ν . Hence Λ is a directed set.(ii) The pointed pair (cid:16)(cid:16) e X , e x (cid:17) , (cid:8) e p µν (cid:9) µ , ν ∈ Λ (cid:17) is an object of the category ↓ S ( X , x ) .If (cid:16) e X ′ , e x ′ (cid:17) is the inverse limit of S ( X , x ) then there is the natural continuous map p : (cid:16) e X , e x (cid:17) → (cid:16) e X ′ , e x ′ (cid:17) . If e x , e x ∈ e X are such that e x = e x and p ( e x ) = p ( e x ) then there is g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) such that e x = e g e x . If e p ′ λ : (cid:16) e X ′ , e x ′ (cid:17) → (cid:0) X λ , x λ (cid:1) thenfrom p ( e x ) = p ( e x ) it turns out that e p λ ( e x ) = e p λ ( e x ) for all λ ∈ Λ . It turns outthat φ λ ( g ) is trivial for every λ ∈ Λ . Taking into account that the homomorphism(4.11.6) is injective we conclude that g is trivial, so e x = e x . Thus the map p isinjective, and since p is a covering (cf. Remark 4.11.15) it is a homeomorphism.From the natural homeomorphism (cid:16) e X , e x (cid:17) ∼ = (cid:16) e X ′ , e x ′ (cid:17) it turns out that (cid:16) e X , e x (cid:17) is the inverse noncommutative limit of S ( X , x ) . The inverse limit of coverings e X is obtained from inverse limit of topologicalspaces b X by a change of a topology. The topology of e X is finer then topology of b X , it means that C (cid:16) b X (cid:17) is a subalgebra of C b (cid:16) e X (cid:17) . The topology of e X is obtainedfrom topology of b X by addition of special subsets. Addition of new sets to atopology is equivalent to addition of new elements to C (cid:16) b X (cid:17) . To obtain C b (cid:16) e X (cid:17) we will extend C (cid:16) b X (cid:17) by special elements (cf. Definition 3.1.19). If e U ⊂ e X isa special set and e a ∈ C c (cid:16) e X (cid:17) is positive element such that e a | e X \ e U = { } , and200 ∈ C c ( X ) is given by a = ∑ b g ∈ b G b g e a , then following condition holds a ( e p λ ( e x )) = ∑ b g ∈ b G b g e a ( e p λ ( e x )) = ( e a ( e x ) e x ∈ e U e p λ ( e x ) / ∈ e p λ (cid:16) e U (cid:17) .From above equation it follows that ∑ b g ∈ b G b g e a = ∑ b g ∈ b G b g e a . (4.11.8)The equation (4.11.8) is purely algebraic and related to special subsets. From theTheorem 4.11.37 it follows that the algebraic condition (4.11.8) is sufficient forconstruction of C (cid:16) e X (cid:17) . Thus noncommutative inverse limits of coverings can beconstructed by purely algebraic methods. C ∗ -algebras This section supplies a purely algebraic analog of the topological constructiongiven by the Subsection 4.11.1.
Let S X = { p λ : X λ → X , } λ ∈ Λ ∈ FinTop be a pointed topological finitecovering category. Let us consider a continuity structure F for X and the { A x } x ∈X where A x is a C ∗ -algebra for any x ∈ X . There is the contravariant finite coveringfunctor A associated with A = (cid:0) X , { A x } x ∈X , F (cid:1) from the category the category FinCov - X to the category of C ∗ - algebras and *-homomorphisms (cf. Definition4.6.8). The restriction of A on S X is also denoted by A . So one has the category S A ( X ) = { A ( p λ ) : A ֒ → A ( X λ ) } λ ∈ Λ of C ∗ -algebras and *-homomorphisms. Lemma 4.11.24.
Let us consider a specialization the situation 4.11.23 such that A = (cid:0) X , { C x } x ∈X , C ( X ) (cid:1) and A = C (cf. Definition 4.6.12). Following conditions hold:(i) If S X = { p λ : X λ → X } λ ∈ Λ ∈ FinTop is a topological finite covering category(cf. Definition 4.11.2) then there is the natural algebraical finite covering category(cf. Definition 3.1.4) S C ( X ) = { π λ = C ( p λ ) : C ( X ) ֒ → C ( X λ ) } λ ∈ Λ ∈ FinAlg . (ii) Conversely any algebraical finite covering category S C ( X ) = { π λ : C ( X ) ֒ → C ( X λ ) } λ ∈ Λ ∈ FinAlg . (4.11.9)201 aturally induces topological finite covering category S X = { p λ : X λ → X } λ ∈ Λ ∈ FinTop .Proof. (i) One needs check (a), (b) of the Definition 3.1.4.(a) Follows from (ii) of the Lemma 4.10.10.(b) If µ , ν ∈ Λ are such that µ > ν then there is the continuous map p : X µ → X ν such that p µ = p ν ◦ p (4.11.10)From the Corollary 4.3.8 it follows that p is a finite-fold transitive cov-ering, and from the Lemma 4.5.43 it turns out that p corresponds to *-homomorphism C ( p ) : C ( X ν ) ֒ → C (cid:0) X µ (cid:1) (cf. Definition 4.6.8). Takinginto account (4.11.10) one has C (cid:0) p µ (cid:1) = C ( p ) ◦ C ( p ν ) . (4.11.11)The equation (4.11.11) is a specialization of (3.1.4).(ii) From the Theorem 4.10.8 it turns out that any object π λ : C ( X ) ֒ → C ( X λ ) ofthe category S C ( X ) corresponds to the object p λ : X λ → X of the category S X ,i.e. p λ is a transitive finite-fold covering. If µ , ν ∈ Λ are such that µ > ν then from(b) of the Definition 3.1.4 it follows that there is an injective *-homomorphism π : C ( X ν ) → C (cid:0) X µ (cid:1) such that C (cid:0) p µ (cid:1) = π ◦ C (cid:0) p µ (cid:1) . (4.11.12)From the condition (a) of the Definition 3.1.4 and the condition (a) of the Defini-tion 3.1.1 it turns out that the homomorphism π is a noncommutative finite foldcovering, so taking into account Theorem 4.10.8 one has the topological finite-fold covering p : X µ → X ν such that π = C ( p ) . From (4.11.12) it follows that p µ = p ◦ p ν . Lemma 4.11.25. If S ( X , x ) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt is a pointedtopological finite covering category such that for any µ > ν there is the unique pointedcovering p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) then S C ( X ) == (cid:0) { C ( p λ ) : C ( X ) ֒ → C ( X λ ) } , (cid:8) C (cid:0) p µν (cid:1) : C ( X ν ) ֒ → C (cid:0) X µ (cid:1)(cid:9)(cid:1) (4.11.13) is a pointed algebraical finite covering category (cf. Definition 3.1.6). roof. According to our construction for any µ > ν the category S C ( X ) containsthe unique injective *-homomorphism from C ( X ν ) to C (cid:0) X µ (cid:1) . This lemma fol-lows from the Remark 3.1.7. Let S ( X , x ) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt be a pointed topo-logical finite covering category (cf. Definition 4.11.2). Similarly to 4.11.23 considera continuity structure F for X and the { A x } x ∈X where A x is a simple C ∗ -algebra(cf. Definition D.1.31) for any x ∈ X . There is the contravariant finite coveringfunctor A associated with A = (cid:0) X , { A x } x ∈X , F (cid:1) from the category the category S ( X , x ) to the category of C ∗ - algebras and *-homomorphisms. There is the cate-gory S A == (cid:0) { A ( p λ ) : A ֒ → A ( X λ ) } , (cid:8) A (cid:0) p µν (cid:1) : A ( X ν ) ֒ → A (cid:0) X µ (cid:1)(cid:9)(cid:1) (4.11.14) Lemma 4.11.27. If S A given by (4.11.14) is a pointed algebraical finite covering category(cf. Definition 3.1.6) and Λ is countable then(i) The spectrum of the C ∗ -inductive limit b A def = C ∗ - lim −→ Λ A ( X λ ) is the projectivelimit lim ←− X λ .(ii) Let X be the disconnected inverse limit of S ( X , x ) (cf. Definition 4.11.19), let p : X → X be the natural covering. Suppose thatA (cid:0) X (cid:1) = C (cid:16) lift p [ A ( X )] (cid:17) . If H a is the space of the atomic representation π a : A (cid:0) X (cid:1) ֒ → B (cid:0) H a (cid:1) (cf. Defi-nition D.2.33), b X is the projective limit lim ←− X λ of topological spaces and b H a is thespace of the atomic representation b π a : C ∗ - lim −→ Λ A ( X λ ) ֒ → B (cid:16) b H a (cid:17) then there isthe natural isomorphism H a ∼ = b H a . .(iii) If n a λ ∈ b A o λ ∈ Λ is a bounded strongly convergent in B (cid:16) b H a (cid:17) net then the strong imit a = lim b π a ( a λ ) is given bya ξ = a ... , ξ b x |{z} b x th place , ... = ... , a b x ξ b x |{z} b x th place , ... ; ξ = ... , ξ b x |{z} b x th place , ... ∈ b H a = M b x ∈ b X b H b x (4.11.15) where a b x ∈ B (cid:16) b H b x (cid:17) is the strong limit a b x = lim λ ∈ Λ rep b x ( a λ ) .Proof. (i) The spectrum of A ( X λ ) is X λ , and taking into account the Lemma 3.1.27one concludes that spectrum of the C ∗ -inductive limit C ∗ - lim −→ Λ A ( X λ ) is the pro-jective limit lim ←− X λ .(ii) Denote by b X def = lim ←− X λ and b A def = C ∗ - lim −→ Λ A ( X λ ) . The space X is the spec-trum of A (cid:0) X (cid:1) . From (ii) of the Lemma 4.11.8 it follows that there is the bi-jective continuous map b p : X → b X . If x ∈ X , b x ∈ b X , x ∈ X are points and rep x : A ( X ) → B ( H x ) , rep x : A (cid:0) X (cid:1) → B (cid:0) H x (cid:1) , rep b x : b A → B ( H b x ) are corre-sponding irreducible representations then there are natural isomorphisms H x ∼ = H p ( x ) , b H b x ∼ = H b p ( b x ) where b p : b X → X is the natural surjective map. It follows that H x ∼ = b H b p ( x ) and taking into account H a = M x ∈X H x , b H a = M b x ∈ b X b H b x ∼ = M x ∈X H b p ( x ) one has H a ∼ = b H a .(iii) If a b x ∈ B (cid:16) b H b x (cid:17) is not the strong limit of lim λ ∈ Λ rep b x ( a λ ) then there ex-ists ε > ξ b x ∈ b H b x such that for any λ ∈ Λ there is λ > λ such that204 ( rep b x ( a λ ) − rep b x ( a )) ξ b x k > ε . If ξ a = ... , ξ b x |{z} b x th place , ... ∈ b H a then one has k ( a λ − a ) ξ a k > ε , i.e. a is not a strong limit of { a λ } . Converselyselect any C > k a λ k < C for all λ ∈ Λ . There is a finite subset b X ⊂ b X and η a ∈ b H a such that η a = ... , η b x |{z} b x th place , ... ; η b x = ( ξ b x b x ∈ b X b x / ∈ b X ; k ξ a − η a k < ε C .There is λ ∈ Λ such that k ( rep b x ( a λ ) − rep b x ( a )) ξ b x k < ε (cid:12)(cid:12)(cid:12) b X (cid:12)(cid:12)(cid:12) ∀ b x ∈ b X ∀ λ > λ From the above equations one has k ( a λ − a ) ξ k < ε ,i.e. there is the strong limit a = lim λ ∈ Λ rep b x ( a λ ) . Consider the specialization of the Lemma 4.11.27 such that A = (cid:0) X , { C x } x ∈X , F (cid:1) = C ( X ) , A = C , and the given by (4.11.14) categoryequals to S C ( X ) == (cid:0) { C ( p λ ) : C ( X ) ֒ → C ( X λ ) } , (cid:8) C (cid:0) p µν (cid:1) : C ( X ν ) ֒ → C (cid:0) X µ (cid:1)(cid:9)(cid:1) . (4.11.16)For any b x ∈ b X the corresponding irreducible representation is the C -linear map rep b x : b A → C ; b a b a b x = b a ( b x ) a of (4.11.16)corresponds to a family { a b x ∈ C } b x ∈ b X . Similarly any weakly special element b = b c a b d with b c , b d ∈ \ C ( X ) corresponds to a family { b b x ∈ C } b x ∈ b X . If Ξ (cid:16) S C ( X ) (cid:17) is \ C ( X ) -bimodule of weakly special elements and D ∗ (cid:16) b X (cid:17) the C ∗ - algebra of all(possibly discontinuous) bounded maps b X → C then the above construction givesis the C -linear map Ξ (cid:16) S C ( X ) (cid:17) → D ∗ (cid:16) b X (cid:17) .Taking into account the bijective map X → b X one has the C -linear map φ : Ξ (cid:16) S C ( X ) (cid:17) → D ∗ (cid:0) X (cid:1) . (4.11.17)Indeed the image of φ is contained in the *-algebra of bounded D-Borel functions(cf. D.2). Lemma 4.11.29.
If A is the disconnected infinite noncommutative covering of S C ( X ) then the map (4.11.17) yields the natural injective *-homomorphism ϕ : A ֒ → D ∗ (cid:0) X (cid:1) . (4.11.18) Proof.
Since A is the C ∗ -norm completion of f Ξ (cid:16) Ξ (cid:16) S C ( X ) (cid:17)(cid:17) the map (4.11.17)uniquely defines the map ϕ . From the Lemma 4.11.27 it turns out that ϕ is injec-tive. Lemma 4.11.30.
If C (cid:0) X (cid:1) ⊂ D ∗ (cid:0) X (cid:1) is the natural inclusion then in the situation otthe Lemma 4.11.29 one has C (cid:0) X (cid:1) ⊂ ϕ (cid:0) A (cid:1) . Proof.
Denote by p : X → X and p λ : X → X λ the natural coverings. Let a ∈ C c (cid:0) X (cid:1) + be a positive function such that the open set U = (cid:8) x ∈ X (cid:12)(cid:12) a ( x ) > (cid:9) If U ⊂ X is homeomorphically mapped onto U = p (cid:0) U (cid:1) ⊂ X then U . Let us provethat a is special, i.e. it satisfies to the conditions (a) and (b) of the Definition 3.1.19(a) If f ε is given by (3.1.19) then for each ε > f ε ( a ) ∈ C c (cid:0) X (cid:1) + , supp f ε ( a ) ⊂ X .206rom the Lemma 4.6.18 it turns out that ∑ g ∈ ker ( G ( e X | X ) → G ( X λ |X ) ) π a ( g f ε ( e a )) = π a (cid:0) desc p λ ( f ε ( e a )) (cid:1) .Otherwise desc p λ ( f ε ( e a )) ∈ C ( X λ ) for any λ ∈ Λ .(b) For every λ ∈ Λ the set U is mapped homeomorhically onto X λ , it turnsout that if if a λ ∈ C ( X λ ) is given by (3.1.23) then according to the Lemma4.6.18 one has a λ = desc p λ ( a ) . Similarly for all λ ∈ Λ and for every z ∈ A + λ following condition holds supp z ∗ e az ⊂ U so from 4.6.18 one has λ > λ ⇒ ∑ g ∈ ker ( G ( e X | X ) → G ( X λ |X ) ) π a ( g ( z ∗ e az )) = π a (cid:0) desc p λ ( z ∗ e az ) (cid:1) .From the above equation it turns out that ∀ µ , ν ∈ Λ ν ≥ µ ≥ λ ⇒ ∑ g ∈ G ( X ν |X µ ) g ( z ∗ a ν z ) = desc p µν ( z ∗ a ν z ) ANDAND ∑ g ∈ G ( X ν |X µ ) g ( z ∗ a ν z ) = desc p µν ( z ∗ a ν z ) (4.11.19)where p µν : X µ → X ν is the natural covering. The equation (4.11.19) yieldsthe following ∀ µ , ν ∈ Λ ν ≥ µ ≥ λ ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:0) z ∗ a µ z (cid:1) − ∑ g ∈ G ( X ν |X µ ) g ( z ∗ a ν z ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = C c (cid:0) X (cid:1) + ⊂ ϕ (cid:0) A (cid:1) . Since C c (cid:0) X (cid:1) is dense in C (cid:0) X (cid:1) (cf. Definition A.1.20) and A is C ∗ -normclosed we have C (cid:0) X (cid:1) ⊂ ϕ (cid:0) A (cid:1) . Denote by G λ = G ( X λ | X ) groups of covering transformations and b G = lim ←− G λ . Denote by p : X → X , p λ : X → X λ , p λ : X λ → X , p µλ : X µ → X λ ( µ > λ )the natural covering projections. Lemma 4.11.32.
Suppose that X is a locally compact, locally connected, Hausdorff space.Let f ∈ D ∗ (cid:0) X (cid:1) + be a positive bounded discontinuous function such that following con-ditions hold: a) If f ε is given by (3.1.19) then for any ε ≥ , λ ∈ Λ there is a pont-wise convergentseries f ε = ∑ g ∈ ker ( b G → G ( X λ | X ) ) (cid:16) g f ε (cid:16) f (cid:17)(cid:17) ( x ) such that the function f ελ : X λ → C given by p λ ( x ) f ε ( x ) lies in C ( X λ ) .(b) Denote by f λ def = f λ . For all δ > there is λ ∈ Λ such that if λ > λ and U λ = { x λ ∈ X λ | f λ ( x λ ) ≥ δ } , U λ = { x λ ∈ X λ | f λ ( x λ ) ≥ δ } (4.11.21) then the restriction p λλ (cid:12)(cid:12)(cid:12) U λ : U λ ≈ −→ U λ is bijective.Then one has f ∈ C (cid:0) X (cid:1) .Proof. Suppose λ satisfies to the condition (b) of this lemma. If U = \ λ ≥ λ p − λ ( U λ ) (4.11.22)then the restriction p λ | U : U → U λ is a bijection for any λ ≥ λ . Really if there are x , x ∈ U such that p λ ( x ) = p λ ( x ) and x = x then there is nontrivial g ∈ b G such that x = gx . So there is λ > λ such that h λ ( g ) ∈ G ( X λ | X λ ) is not trivial(where h λ is the natural surjective homomorphism b G → G ( X λ | X λ ) ). It followsthat p λ ( x ) = p λ ( x ) but from (4.11.22) it follows that p λ ( x ) , p λ ( x ) ∈ U λ .Otherwise p λλ ( p λ ( x )) = p λλ ( p λ ( x )) so p λλ (cid:12)(cid:12)(cid:12) U λ : U λ ≈ −→ U λ is not bijective, itis a contradiction with (b). Let x ∈ U be a point and x = p ( x ) . There is openconnected neighborhood U of x such that U is evenly covered by p and U ⊂ p λ ( U λ ) . For any λ ∈ Λ we select a connected neighborhood U ′ λ of p λ ( x ) ∈ X λ which is homeomorphically mapped onto U . The set U ′ def = \ λ ∈ Λ p − λ (cid:0) U ′ λ (cid:1) is open with respect to the given by the Lemma 4.11.8 topology, so from U ′ ⊂ U itturns out that U is an open subset of X . From U = n x ∈ X (cid:12)(cid:12) f ( x ) ≥ δ o , it followsthat for any ε > δ following conditions holds V = n x ∈ X (cid:12)(cid:12) f ε (cid:16) f (cid:17) ( x ) > o ⊂ U f ε (cid:16) f (cid:17) = lift p λ U ( f ελ ) (cf. Definition 4.5.56). It follows that f ε (cid:16) f (cid:17) ∈ C c (cid:0) X (cid:1) . There is the C ∗ -normlimit f = lim ε → f ε (cid:16) f (cid:17) and since and from the Definition A.1.20 it turns out that f ∈ C (cid:0) X (cid:1) . Theorem 4.11.33.
Suppose that X is a locally compact, locally connected, Hausdorffspace. Let f ∈ D ∗ (cid:0) X (cid:1) + be a positive bounded discontinuous function such that followingconditions hold:(a) If f ε is given by (3.1.19) and ε > , λ ∈ Λ and x ∈ X then there are a pont-wiseconvergent series f λ = ∑ g ∈ ker ( b G → G ( X λ | X ) ) g f , f ελ = ∑ g ∈ ker ( b G → G ( X λ | X ) ) g f ε (cid:16) f (cid:17) , h λ = ∑ g ∈ ker ( b G → G ( X λ | X ) ) g f . (4.11.23) Moreover there are f λ , f ελ , h λ ∈ C ( X λ ) are such that for all x ∈ X one hasf λ = lift p λ ( f λ ) , f ελ = lift p λ ( f ε ) , h λ = lift p λ ( h λ ) . (4.11.24) (b) For all ε > there is µ ∈ Λ such that µ ≥ λ and ∀ λ ∈ Λ λ ≥ µ ⇒ (cid:13)(cid:13) f λ − g λ (cid:13)(cid:13) < ε . (4.11.25) Then one has f ∈ C (cid:0) X (cid:1) .Proof. Let µ ∈ Λ be such that for any λ ≥ µ ≥ λ following condition holds λ ≥ µ ≥ λ ⇒ (cid:13)(cid:13) f λ − g λ (cid:13)(cid:13) < ε . (4.11.26)209or all λ > µ denote by p λ def = p λµ : X λ → X µ . Let e x , e x ∈ X λ be such that e x = e x , p λ ( e x ) = p λ ( e x ) = x , f λ ( e x ) ≥ ε ; f λ ( e x ) ≥ ε . (4.11.27)From the equations (4.11.23), (4.11.24) it turns out that f µ ( x ) = ∑ e x ∈ ( p λ ) − ( x ) f λ ( e x ) . h µ ( x ) = ∑ e x ∈ ( p λ ) − ( x ) h λ ( e x ) , f λ ( e x ) = ∑ x ∈ ( p λ ) − ( e x ) f λ ( x ) . h λ ( e x ) = ∑ x ∈ ( p λ ) − ( e x ) f ( x ) . (4.11.28)From the above equation it turns out that f λ ( e x ) = ∑ x ∈ ( p λ ) − ( e x ) f ( x ) + ∑ ( x ′ , x ′′ ) ∈ p − λ ( e x ) × p − λ ( e x ) x ′ = x ′′ f (cid:0) x ′ (cid:1) f (cid:0) x ′′ (cid:1) ≥≥ ∑ x ∈ ( p λ ) − ( e x ) f λ ( x ) = h λ ( e x ) . f µ ( x ) = ∑ e x ∈ p − λ ( x ) f λ ( e x ) + ∑ ( e x ′ , e x ′′ ) ∈ p − λ ( x ) × p − λ ( x ) e x ′ = e x ′′ f λ (cid:0) e x ′ (cid:1) f λ (cid:0)e x ′′ (cid:1) ≥≥ ∑ e x ∈ p − λ ( x ) f λ ( e x ) + f λ ( e x ) f λ ( e x ) + f λ ( e x ) f λ ( e x ) ≥≥ ∑ e x ∈ p − λ ( x ) h λ ( e x ) + f λ ( e x ) f λ ( e x ) ≥ h µ ( x ) + ε ,hence one has (cid:13)(cid:13)(cid:13) f µ − g µ (cid:13)(cid:13)(cid:13) ≥ ε However the above equation contradicts with (4.11.26), so the situation (4.11.27) isimpossible. It follows that for all λ > µ one has p λ ( e x ) = p λ ( e x ) = x AND f λ ( e x ) ≥ ε AND f λ ( e x ) ≥ ε ⇒ e x = e x . (4.11.29)210he equation (4.11.29) is equivalent to the condition (b) of the Lemma 4.11.32, so f satisfies conditions (a), (b) of the Lemma 4.11.32, hence one has f ∈ C (cid:0) X (cid:1) . Remark 4.11.34.
Below the given by the Equation (4.11.18) injecive homomor-phism ϕ : A ֒ → D ∗ (cid:0) X (cid:1) will by replaced with inclusion A ⊂ D ∗ (cid:0) X (cid:1) . Similarlyto the Remark 3.1.38 for all λ ∈ Λ we implicitly assume that C ( X λ ) ⊂ D ∗ (cid:0) X (cid:1) .Similarly the following natural inclusion C ∗ - lim −→ λ ∈ Λ C ( X λ ) ⊂ D ∗ (cid:0) X (cid:1) will be implicitly used. These inclusions enable us replace the Equations 3.1.20with the following equivalent system of equations ∑ g ∈ G λ z ∗ ( ga ) z ∈ C ( X λ ) , ∑ g ∈ G λ f ε ( z ∗ ( ga ) z ) ∈ C ( X λ ) , ∑ g ∈ G λ ( z ∗ ( ga ) z ) ∈ C ( X λ ) , (4.11.30)where G λ def = ker (cid:0) G (cid:0) X (cid:12)(cid:12) X (cid:1) → G ( X λ | X ) (cid:1) . Corollary 4.11.35.
Suppose that S pt C ( X ) == (cid:0) { C ( p λ ) : C ( X ) ֒ → C ( X λ ) } , (cid:8) C (cid:0) p µν (cid:1) : C ( X ν ) ֒ → C (cid:0) X µ (cid:1)(cid:9)(cid:1) is a pointed algebraical finite covering category given by the Lemma 4.11.25. If A is adisconnected inverse noncommutative limit of S C ( X ) then ( A ⊂ C (cid:0) X (cid:1) (cf. Equation4.11.18).Proof. Suppose that a ∈ D ∗ (cid:0) X (cid:1) corresponds to a special element of S pt C ( X ) . If weselect any λ ∈ Λ and set z = C ( X λ ) ∼ ∈ C ( X λ ) ∼ then from the conditions (a)and (b) of the Definition 3.1.19 it turns out that f def = z ∗ az satisfies to the conditions(a) and (b) of the Theorem 4.11.33. It follows that f ∈ C (cid:0) X (cid:1) . However f = C ( X λ ) ∼ a C ( X λ ) ∼ = a , hence a ∈ C (cid:0) X (cid:1) . If b x , b y ∈ C (cid:16) b X (cid:17) then because thenatural map X → b X is continuous one can assume b x , b y ∈ C b (cid:0) X (cid:1) . It follows that b = b xa b y ∈ C (cid:0) X (cid:1) , i.e. any weakly special element b (cf. Definition 3.1.23) lies211n C (cid:0) X (cid:1) , i.e. b ⊂ C (cid:0) X (cid:1) . On the other hand A is the generated by weaklyspecial elements C ∗ -algebra, and C (cid:0) X (cid:1) is C ∗ -norm closed, it turns out that A ⊂ C (cid:0) X (cid:1) . From the Lemma 4.11.30 and the Corollary 4.11.35 it follows that discon-nected inverse noncommutative limit of S C ( X ) is isomorphic to C (cid:0) X (cid:1) . Accord-ing to the Definition 3.1.25 one has G (cid:0) C (cid:0) X (cid:1)(cid:12)(cid:12) C ( X ) (cid:1) = lim ←− λ ∈ Λ G ( C ( X λ ) | C ( X )) and taking into account G ( C ( X λ ) | C ( X )) ∼ = G ( X λ | X ) and G (cid:0) X (cid:12)(cid:12) X (cid:1) = lim ←− λ ∈ Λ G ( X λ | X ) we conclude that G (cid:0) C (cid:0) X (cid:1)(cid:12)(cid:12) C ( X ) (cid:1) = G (cid:0) X (cid:12)(cid:12) X (cid:1) . (4.11.31)Let e X ⊂ X be a connected component of X then the closed ideal C (cid:16) e X (cid:17) ⊂ C (cid:0) X (cid:1) is a maximal connected C ∗ -subalgebra of C (cid:0) X (cid:1) . If G ⊂ G (cid:0) C (cid:0) X (cid:1)(cid:12)(cid:12) C ( X ) (cid:1) is the maximal among subgroups G ′ ⊂ G (cid:0) C (cid:0) X (cid:1)(cid:12)(cid:12) C ( X ) (cid:1) such that G ′ C (cid:16) e X (cid:17) = C (cid:16) e X (cid:17) then G is the maximal among subgroups G ′′ ⊂ G (cid:0) X (cid:12)(cid:12) X (cid:1) such that G ′′ e X = e X (cf. (4.11.31)), or equivalently G = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) . (4.11.32)If J ⊂ G (cid:0) C (cid:0) X (cid:1)(cid:12)(cid:12) C ( X ) (cid:1) is a set of representatives of G (cid:0) C (cid:0) X (cid:1)(cid:12)(cid:12) C ( X ) (cid:1) / G (cid:16) C (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) C ( X ) (cid:17) then from the (4.11.4) it follows that X = G g ∈ J g e X and C (cid:0) X (cid:1) is a C ∗ -norm completion of the algebraic direct sum M g ∈ J C (cid:16) g e X (cid:17) . (4.11.33)212 heorem 4.11.37. Let S ( X , x ) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt be apointed topological finite covering category such that for any µ > ν there is the uniquepointed covering p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) . is the unique pointed covering p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) . If S pt C ( X ) == (cid:0) { C ( p λ ) : C ( X ) ֒ → C ( X λ ) } , (cid:8) C (cid:0) p µν (cid:1) : C ( X ν ) ֒ → C (cid:0) X µ (cid:1)(cid:9)(cid:1) is a given by the Lemma 4.11.25 pointed algebraical finite covering category (cf. Definition3.1.6) then following conditions hold:(i) S pt C ( X ) is good and the triple (cid:16) C ( X ) , C (cid:16) lim ←− S ( X , x ) (cid:17) , G (cid:16) lim ←− S ( X , x ) (cid:12)(cid:12)(cid:12) X (cid:17)(cid:17) is the infinite noncommutative covering of S pt C ( X ) (cf. Definition 3.1.34).(ii) There are isomorphisms: • lim ←− S pt C ( X ) ≈ C (cid:16) lim ←− S ( X , x ) (cid:17) . • G (cid:16) lim ←− S pt C ( X ) (cid:12)(cid:12)(cid:12) C ( X ) (cid:17) ≈ G (cid:16) lim ←− S ( X , x ) (cid:12)(cid:12)(cid:12) X (cid:17) .Proof. From the Lemma 4.11.30 and the Corollary 4.11.35 it follows that discon-nected inverse noncommutative limit of S C ( X ) is isomorphic to C (cid:0) X (cid:1) where X is the disconnected inverse limit of S ( X , x ) (cf. Definition 4.11.19). If e X ⊂ X is theconnected component of x then C (cid:16) e X (cid:17) is the connected component of C (cid:0) X (cid:1) .(i) To prove that S pt C ( X ) is good one needs check conditions (a)-(c) of the Definition3.1.33.(a) For any λ ∈ Λ the covering e X → X λ induces the inclusion C ( X λ ) ֒ → C b (cid:16) e X (cid:17) = M (cid:16) C b (cid:16) e X (cid:17)(cid:17) .(b) We already know that the direct sum (4.11.33) is dense in C (cid:0) X (cid:1) .(c) The homomorphism G (cid:16) C (cid:16) e X (cid:17) (cid:12)(cid:12)(cid:12) C ( X ) (cid:17) → G ( C ( X λ ) | C ( X )) is equiv-alent to G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) → G ( X λ | X ) . However both coverings e X → X and X λ → X are transitive, hence from the equation (4.3.8) it follows that thehomomorphism G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) → G ( X λ | X ) is surjective.(ii) Follows from (i) of this lemma. 213 emma 4.11.38. If S ( X , x ) = (cid:8)(cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt , S C ( X ) ( X , x ) = (cid:0) { C ( X ) ֒ → C ( X λ ) } , (cid:8) C (cid:0) X µ (cid:1) ֒ → C ( X ν ) (cid:9)(cid:1) then S pt C ( X ) allowsinner product (cf. Definition 3.4.1).Proof. Denote by p : (cid:0) X , x (cid:1) → ( X , x ) , p λ : (cid:0) X , x (cid:1) → (cid:0) X λ , x λ (cid:1) , p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) , p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) ( µ > ν ) the natural pointed coverings. If e X = lim ←− S ( X , x ) then from the Theorem 4.11.37 it turns out (cid:16) C ( X ) , C (cid:16) e X (cid:17) , G (cid:16) e X | X (cid:17)(cid:17) is the infinite noncommutative covering of S C ( X ) (cf. Definition 3.1.34). It isshown in [58] that the Pedersen’s ideal K (cid:16) C (cid:16) e X (cid:17)(cid:17) of C (cid:16) e X (cid:17) coincides with C c (cid:16) e X (cid:17) . If e a , e b ∈ C c (cid:16) e X (cid:17) then supp a is compact. From the Lemma 4.11.20 itfollows that there is λ ∈ Λ such that for all λ ≥ λ the restriction p λ | supp a isinjective. From (ii) of the Lemma 4.6.16 it follows that c λ = ∑ g ∈ ker ( G ( e X | X ) → G ( X λ | X ) ) g (cid:16)e a ∗ e b (cid:17) = desc p λ (cid:16)e a e b (cid:17) where notation desc p λ means the descent (cf. Definition 4.5.56). From the proper-ties of the descent it turns out that desc p λ (cid:16)e a e b (cid:17) ∈ C ( X λ ) , i.e. c λ ∈ C ( X λ ) . Takinginto account the Remark 3.4.2 we conclude that c λ ∈ C ( X λ ) for all λ ∈ Λ . Following theorem gives universal coverings of commutative C ∗ -algebras. Theorem 4.11.39.
Let X be a connected, locally connected, locally compact, second-countable, Hausdorff space. If there is the universal topological covering e p : e X → X (cf. Definition A.2.20) such that G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is residually finite group (cf. DefinitionB.3.2) then C (cid:16) e X (cid:17) is the universal covering of C ( X ) (cf. Definition 3.3.1).Proof. If { p λ : X λ → X } λ ∈ Λ is a family of all transitive finite-fold coverings of X then from the Theorem 4.10.8 it follows that the family of all noncommutativefinite-fold coverings of C ( X ) is given by { C ( p λ ) : C ( X ) ֒ → C ( X λ ) } λ ∈ Λ .214ince e X is the universal covering for any λ ∈ Λ there is a covering e p λ : e X →X λ such that e p = e p λ ◦ p λ . From the Lemma 4.11.24 it turns out that S C ( X ) = { C ( p λ ) : C ( X ) ֒ → C ( X λ ) } λ ∈ Λ is an algebraical finite covering category. Fromthe Lemma 4.11.22 it follows that any e x ∈ e X yields a pointed a topological finitecovering category S ( X , e p ( e x )) = n p λ : (cid:16) X λ , x λ (cid:17) → ( X , x ) o such that (cid:16) e X , e x (cid:17) is the topological inverse noncommutative limit of S ( X , e p ( e x )) .From the Lemma 4.11.25 the category S ( X , e p ( e x )) yields the pointed algebraicalfinite covering category (cid:18) { C ( p λ ) : C ( X ) ֒ → C ( X λ ) } λ ∈ Λ , (cid:8) C (cid:0) p µν (cid:1)(cid:9) µ , ν ∈ Λ µ > ν (cid:19) (4.11.34)(cf. Definition 3.1.6). From the Theorem 4.11.37 it follows that C (cid:16) e X (cid:17) is theinverse noncommutative limit of the pointed noncommutative finite covering cat-egory (4.11.34). Taking into account the Definition 3.3.1 we conclude that C (cid:16) e X (cid:17) is the universal covering of C ( X ) . Corollary 4.11.40. If ( X , x ) is a connected, locally path connected, semilocally 1-connected,locally compact, second-countable Hausdorff space such that the fundamental group π ( X , x ) is residually finite then following conditions hold:(i) C ( X ) has the algebraical universal covering.(ii) There is a pointed algebraical finite covering category (cid:16) S C ( X ) , (cid:8) C (cid:0) p µν (cid:1)(cid:9)(cid:17) and thenatural group isomorphism π ( X , x ) ∼ = π (cid:0) C ( X ) , (cid:8) C (cid:0) p µν (cid:1)(cid:9)(cid:1) . (4.11.35) Proof. (i) From the Lemmas A.2.21 and A.2.22 it turns out that there is the uni-versal covering e X → X . From the Corollary A.2.19 it turns out that π ( X , x ) ≈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , hence G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is residually finite. From the Theorem 4.11.39 it fol-lows that C (cid:16) e X (cid:17) is the universal covering of C ( X ) .(ii) Let e p : e X → X be the universal covering of X and let e x ∈ e X be such that e p ( e x ) = x . If e p λ : e X → X λ are natural coverings and x λ = e p λ ( e x ) then there is apointed topological finite covering category S ( X , e p ( e x )) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) .215rom the Lemma 4.11.25 there is the associated to S ( X , e p ( e x )) pointed algebraical fi-nite covering category (cid:18) { C ( p λ ) : C ( X ) → C ( X λ ) } , (cid:8) C (cid:0) p µν (cid:1)(cid:9) µ , ν ∈ Λ µ > ν (cid:19) such that C (cid:16) e X (cid:17) is the inverse limit of it (cf. Definition 3.1.34). From (3.3.1) it follows that π (cid:0) C ( X ) , (cid:8) C (cid:0) p µν (cid:1)(cid:9)(cid:1) = G (cid:16) C (cid:16) e X (cid:17) (cid:12)(cid:12)(cid:12) C ( X ) (cid:17) .Otherwise from the Corollary A.2.19 it follows that π ( X , x ) ≈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) andtaking into account G (cid:16) C (cid:16) e X (cid:17) (cid:12)(cid:12)(cid:12) C ( X ) (cid:17) ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) one has π ( X , x ) ∼ = π (cid:0) C ( X ) , (cid:8) C (cid:0) p µν (cid:1)(cid:9)(cid:1) . Similarly to the section 4.10.2 consider a compact Riemannian manifold M witha spinor bundle S (cf. E.4). Denote by µ the Riemannian measure (cf. [26]) on M which corresponds to the given by (E.4.1) volume element. The bundle S is Her-mitian (cf. Definition A.3.11) so there is a Hilbert space L ( M , S , µ ) (or L ( M , S ) in a simplified notation) (cf. A.3.10) with the given by (A.3.3) scalar product, i.e. ( ξ , η ) L ( M , S ) def = Z M ( ξ x , η x ) x d µ (4.11.36)Moreover from (A.3.4) it follows that there is the natural representation ρ : C ( M ) → B (cid:0) L ( M , S ) (cid:1) . (4.11.37)From A.3.10 it follows that L ( M , S , µ ) is the Hilbert norm completion of Γ ( M , S ) .Let p : e M → M be an infinite transitive covering, such that the group G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) is residually finite. From the Lemma 4.11.22 and the Theorem 4.11.37 it turns outthat the triple (cid:16) C ( M ) , C (cid:16) e M (cid:17) G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is an infinite noncommutative cover-ing (cf. Definition 3.1.34). From the Lemma 4.11.38 it follows that (cid:16) C ( M ) , C (cid:16) e M (cid:17) G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) allows inner product (cf. Definition 3.4.1). So thereis the induced by (cid:16) ρ , (cid:16) C ( M ) , C (cid:16) e M (cid:17) G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17)(cid:17) representation e ρ : C (cid:16) e M (cid:17) ֒ → B (cid:16) e H (cid:17) (cf. Definition 3.4.5), such that e H is the Hilbert norm completion of K (cid:16) C (cid:16) e M (cid:17)(cid:17) ⊗ C ( M ) L ( M , S ) K (cid:16) C (cid:16) e M (cid:17)(cid:17) is Pedersen’s ideal of C (cid:16) e M (cid:17) (cf. Definition D.1.33 and Re-mark 3.4.8). On the other hand it is proven in [58] that the Pedersen’s ideal K (cid:16) C (cid:16) e X (cid:17)(cid:17) of C (cid:16) e X (cid:17) coincides with C c (cid:16) e X (cid:17) . If Γ c (cid:16) e M , e S (cid:17) def = C c (cid:16) lift p [ C ( M , { S x } , Γ ( M , S ))] (cid:17) then from the Lemma 4.8.11 it follows that there is the given by (4.8.14) isomor-phism C c (cid:16) e M (cid:17) ⊗ C ( M ) Γ ( M , S ) ≈ −→ Γ c (cid:16) e M , e S (cid:17) of left C (cid:16) e M (cid:17) -modules. On the other hand then from (4.9.10) it follows that thereis the inclusion Γ c (cid:16) e M , e S (cid:17) ⊂ L (cid:16) e M , e S (cid:17) , hence there is the homomorphism φ : C c (cid:16) e M (cid:17) ⊗ C ( M ) Γ ( M , S ) → L (cid:16) e M , e S (cid:17) (4.11.38)of left C (cid:16) e M (cid:17) -modules. Lemma 4.11.41.
Following conditions hold:(i) The map (4.11.38) can be extended up to the following homomorphismC c (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) ≈ −→ L (cid:16) e M , e S (cid:17) of left C (cid:16) e M (cid:17) -modules.(ii) The image of C c (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) is dense in L (cid:16) e M , e S (cid:17) .Proof. (i) Let ∑ nj = e a j ⊗ ξ j ∈ C c (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) be any element. From A.3.10it turns out that L ( M , S , µ ) is the Hilbert norm completion of Γ ( M , S ) . Hencefor any j =
1, ..., n there is a net (cid:8) ξ j α (cid:9) ⊂ Γ ( M , S ) such that ξ j = lim α ξ j α wherewe mean the convergence with respect to the Hilbert norm k·k L ( M , S ) . If C = max j = n (cid:13)(cid:13)e a j (cid:13)(cid:13) then there is α such that α ≥ α ⇒ (cid:13)(cid:13) ξ j − ξ j α (cid:13)(cid:13) L ( M , S ) < ε nC ,so one has α ≥ α ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n ∑ j = e a j ⊗ ξ j α − n ∑ j = e a j ⊗ ξ j α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( e M , e S ) < ε .217he above equation it follows that the net n ∑ nj = e a j ⊗ ξ j α o α satisfies to the Cauchycondition, so it is convergent with respect to the topology of L (cid:16) e M , e S (cid:17) .(ii) From the Lemma (4.8.11) it follows that the image of C c (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) in L (cid:16) e M , e S (cid:17) contains Γ c (cid:16) e M , e S (cid:17) , however Γ c (cid:16) e M , e S (cid:17) is dense in L (cid:16) e M , e S (cid:17) . Itfollows that φ (cid:16) C c (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) (cid:17) is dense in L (cid:16) e M , e S (cid:17) .Let both µ and e µ be Riemannian measure (cf. [26]) on both M and e M respec-tively which correspond to both the volume element (cf. (E.4.1)) and its p -lift (cf.(4.9.2)). Let 1 C b ( e X ) = ∑ e α ∈ f A e a e α = ∑ e α ∈ f A e e e α ; e e e α def = pe a e α ∀ e α ∈ f A be the given by (4.8.16) partition of unity, compliant to the covering p : e M → M (cf. Definition 4.8.4). If e a ⊗ ξ , e b ⊗ η ∈ C c (cid:16) e M (cid:17) ⊗ C ( M ) Γ ( M , S ) ⊂ L (cid:16) e M , e S (cid:17) then supp e a is compact, so there exists covering sum for supp e a (cf. Definition 4.2.6), i.e.a finite subset f A supp e a ⊂ f A such that ∑ α ∈ A supp e a e a α ( e x ) = ∀ e x ∈ supp e a .The given by the Equation (3.4.4) scalar product ( · , · ) ind on C c (cid:16) e M (cid:17) ⊗ C ( M ) Γ ( M , S ) (cid:16)e a ⊗ ξ , e b ⊗ η (cid:17) ind = (cid:18) ξ , De a , e b E C ( e M ) η (cid:19) L ( M , S ) == ∑ e α ∈ f A supp e a (cid:18) ξ , De a e α e a , e b E C ( e M ) η (cid:19) L ( M , S ) == ∑ e α ∈ f A supp e a (cid:16) ξ , desc (cid:16)e a e α e a ∗ e b (cid:17) η (cid:17) L ( M , S ) == ∑ e α ∈ f A supp e a Z M (cid:16) ξ x , desc (cid:16)e a e α e a ∗ e b (cid:17) η x (cid:17) x d µ == ∑ e α ∈ f A supp e a Z M (cid:16) desc ( e e e α e a ) ξ x , desc (cid:16)e e e α e b (cid:17) η x (cid:17) x d µ == ∑ e α ∈ f A supp e a Z e M (cid:16)e a lift e U e α ( e e α ξ ) e x , e b lift e U e α ( e e α η ) e x (cid:17) e x d e µ == ∑ e α ∈ f A supp e a Z e M e a e α (cid:16)e a lift e U e α ( ξ ) e x , e b lift e U e α ( η ) e x (cid:17) e x d e µ == Z e M (cid:16)e a lift p ( ξ ) e x , e b lift p ( η ) e x (cid:17) e x d e µ = (cid:16)e a lift p ( ξ ) , e b lift p ( η ) (cid:17) L ( e M , e S ) == (cid:16) φ ( e a ⊗ ξ ) , φ (cid:16)e b ⊗ η (cid:17)(cid:17) L ( e M , e S ) (4.11.39)where φ is given by (4.8.14). The equation (4.11.39) means that ( · , · ) ind = ( · , · ) L ( e M , e S ) ,and taking into account the dense inclusion C (cid:16) e M (cid:17) ⊗ C ( M ) Γ ( M , S ) ⊂ L (cid:16) e M , e S (cid:17) with respect to the Hilbert norm of L (cid:16) e M , e S (cid:17) one concludes that the space ofinduced representation coincides with L (cid:16) e M , e S (cid:17) . It means that induced repre-sentation C (cid:16) e M (cid:17) × L (cid:16) e M , e S (cid:17) → L (cid:16) e M , e S (cid:17) is given by (A.3.4). So one has thefollowing lemma. Lemma 4.11.42.
If the representation e ρ : C (cid:16) e M (cid:17) → B (cid:16) e H (cid:17) is induced by the pair (cid:16) C ( M ) → B (cid:0) L ( M , S ) (cid:1) , (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17)(cid:17) (cf. Definition 3.4.5) then following conditions hold:(a) There is the homomorphism of Hilbert spaces e H ∼ = L (cid:16) e M , e S (cid:17) , b) The representation e ρ is given by the natural action of C (cid:16) e M (cid:17) on L (cid:16) e M , e S (cid:17) (cf.A.3.4).Proof. (a) Follows from (4.11.39),(b) From the Lemma 4.10.11 it follows that the map C c (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) ≈ −→ L (cid:16) e M , e S (cid:17) is the homomorphism of C (cid:16) e M (cid:17) modules, so the given by (A.3.4) C (cid:16) e M (cid:17) -action coincides with the C (cid:16) e M (cid:17) -action the given by (3.4.5). Remark 4.11.43.
If the spectral triple (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S , e µ (cid:17) , e D / (cid:17) is the geometri-cal p -lift of the spectral triple (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) (cf. Definition 4.9.2), thenclearly the corresponding to (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) representation of C ( M ) equals to the given by the Lemma 4.11.42 representation . Let (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) be a commutative spectral triple (cf. Equation (E.4.6)),and let p : e M → M be an infinite regular covering such that the covering group G (cid:16) e M | M (cid:17) (cf. Definition A.2.3) is residually finite (cf. Definition B.3.2). Fromthe Lemma 4.11.22 it it turns out that there is a (topological) finite covering cate-gory S = { p λ : M λ → M } (cf. 4.11.2) such that the (topological) inverse limit of S is naturally homeomorphic to e M . From the Proposition E.5.1 it follows that e M and M λ ( ∀ λ ∈ Λ ) have natural structure of the Riemannian manifolds. Accord-ing to the Example 4.7.5 denote by e S def = C (cid:16) lift p ( S ) (cid:17) and S λ def = C b (cid:16) lift p λ ( S ) (cid:17) (for all λ ∈ Λ ) the lifts of the spin bundle (cf. A.3). Similarly to the Section4.9 one can prove that for any λ ∈ Λ that there is the p λ -inverse image p − λ / D of / D (cf. Definition 4.7.11) and the operator p − λ / D can be regarded as an un-bounded operator on L ( M λ , S λ ) . Moreover there is the e p -inverse image e p − / D of / D (cf. Definition 4.7.11) and the operator e p − / D is an unbounded operator on L (cid:16) e M , e S (cid:17) . From the Lemma 4.11.24 and the Theorem 4.11.37 it turns out that S C ( M ) = { C ( M ) ֒ → C ( M λ ) } λ ∈ Λ is a good (algebraical) finite covering category,and the triple (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is an infinite noncommutative covering of S C ( M ) . Lemma 4.11.44.
The set S ( C ∞ ( M ) , L ( M , S ) , / D ) = n(cid:16) C ∞ ( M λ ) , L ( M λ , S λ ) , p − λ / D (cid:17)o λ ∈ Λ ∈ CohTriple (4.11.40)220 s a coherent set of spectral triples.Proof.
Consider the atomic representation π a : C ∗ -lim −→ λ ∈ Λ C ( M λ ) → B (cid:16) b H (cid:17) (cf.Definition D.2.33). Any coherent set of spectral triples is weakly coherent. So oneneeds check that S ( C ∞ ( M ) , L ( M , S ) , / D ) satisfies to conditions (a)-(d) of the Definition3.5.1.(a) Follows from the Theorem 4.10.17.(b) For all λ ∈ Λ the C ∗ -algebra C ( M λ ) is the C ∗ -norm completion of C ∞ ( M λ ) .(c) Follows from the Corollary 4.11.37.(d) Follows from the Theorem 4.10.17.From the Lemma 4.11.38 it follows that the pointed algebraical finite coveringcategory S pt C ( M ) allows inner product, hence from the Definition 3.5.2 it turns outthat S ( C ∞ ( M ) , L ( M , S ) , / D ) is a coherent set of spectral triples.For any λ ∈ Λ denote by e p λ : e M → M λ the natural covering. Lemma 4.11.45. If e W ∞ is the space of S ( C ∞ ( M ) , L ( M , S ) , / D ) -smooth elements (cf. Definition3.5.3) then e W ∞ ⊂ C ∞ c (cid:16) e M (cid:17) def = C ∞ (cid:16) e M (cid:17) \ C c (cid:16) e M (cid:17) . Proof.
It is shown in [58] that Pedersen’s ideal of C (cid:16) e M (cid:17) coincides with C c (cid:16) e M (cid:17) ,and taking into account the condition (a) of the Definition 3.5.3 one has e W ∞ ⊂ C c (cid:16) e M (cid:17) where the sum of the series implies the strict topology (cf. Definition D.1.12). If e a ∈ e W ∞ then from the condition (b) of the Definition 3.5.3 it follows that a λ = ∑ g ∈ ker ( b G → G λ ) g e a ∈ C ∞ ( M λ ) , ∀ λ ∈ Λ .From the Lemma 4.11.20 it turns out that there is µ ∈ Λ and an open set e U ⊂ e M such that supp e a ⊂ e U there for any λ ≥ µ the restriction e p λ | e U is injective. From theLemma 4.6.16 it follows that λ ≥ µ ⇒ e a = lift e p λ e U ( a λ ) (4.11.41)From (4.11.41) and a µ ∈ C ∞ (cid:0) M µ (cid:1) it turns out e a ∈ C ∞ (cid:16) e M (cid:17) .221 emma 4.11.46. If e W ∞ is the space of S ( C ∞ ( M ) , L ( M , S ) , / D ) -smooth elements (cf. Definition3.5.3) then C ∞ c (cid:16) e M (cid:17) ⊂ e W ∞ . Proof.
Let e a ∈ C ∞ c (cid:16) e M (cid:17) . From the Lemma 4.11.20 it turns out that there is µ ∈ Λ and an open set e U ⊂ e M such that supp e a ⊂ e U there for any λ ≥ µ the restriction e p λ | e U is injective. For any nontrivial g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) one has g e U ∩ e U = ∅ . (4.11.42)If ρ : C ( M ) → B (cid:0) L ( M , S ) (cid:1) is the given by A.3.4 representation then from theLemma 4.11.42 it follows that the induced by the pair (cid:16) ρ , (cid:16) A , e A π , G (cid:16) e A π (cid:12)(cid:12)(cid:12) A (cid:17)(cid:17)(cid:17) is given by the natural action e ρ : C (cid:16) e M (cid:17) × L (cid:16) e M , e S (cid:17) → L (cid:16) e M , e S (cid:17) explained inA.3.4). The action e ρ can be extended up to the action L ∞ (cid:16) e M (cid:17) × L (cid:16) e M , e S (cid:17) → L (cid:16) e M , e S (cid:17) . If g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) and χ e U ∈ L ∞ (cid:16) e M (cid:17) is the characteristic function of e U then g χ e U corresponds to a projector in B (cid:16) L (cid:16) e M , e S (cid:17)(cid:17) . From (4.11.42) it followsthat g ′ = g ′′ ⇒ g ′ χ e U ⊥ g ′′ χ e U If χ λ def = ∑ g ∈ ker ( G ( e M | M ) → G ( e M | M λ )) g χ e U and e ξ ∈ L (cid:16) e M , e S (cid:17) then k χ λ ξ k = ∑ g ∈ ker ( G ( e M | M ) → G ( e M | M λ )) (cid:13)(cid:13)(cid:0) g χ e U (cid:1) ξ (cid:13)(cid:13) .Since the intersection ∩ λ ∈ Λ ker (cid:16) G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) → G (cid:16) e M (cid:12)(cid:12)(cid:12) M λ (cid:17)(cid:17) is trivial one haslim λ ∈ Λ k χ λ ξ k = (cid:13)(cid:13)(cid:0) χ e U (cid:1) ξ (cid:13)(cid:13) ,lim λ ∈ Λ ∑ g ∈ ker ( G ( e M | M ) → G ( e M | M λ )) g is not trivalal (cid:13)(cid:13)(cid:0) g χ e U (cid:1) ξ (cid:13)(cid:13) = λ ∈ Λ ∑ g ∈ ker ( G ( e M | M ) → G ( e M | M λ )) g is not trivalal (cid:0) g χ e U (cid:1) ξ = λ ∈ Λ χ λ ξ = χ e U ξ ,222ence there is the limit lim λ ∈ Λ χ λ = χ e U (4.11.43)in the sense of the strong topology of B (cid:16) L (cid:16) e M , e S (cid:17)(cid:17) . Let us check that e a satisfiesto (a)-(d) of the Definition 3.5.3.(a) Follows from K (cid:16) C (cid:16) e M (cid:17)(cid:17) = C c (cid:16) e M (cid:17) .(b) If a λ ∈ C ( M λ ) is the sum of series ∑ g ∈ ker ( G ( e M | M ) → G ( e M | M λ )) g e a ∈ C ∞ ( M λ ) .is the sense of strict topology, then from supp e a ⊂ e U if follows that e a satisfies to theEquation (4.11.41). From the Lemma 4.6.16 it follows that λ ≥ µ ⇒ a λ = desc e p λ ( e a ) and taking into account e a ∈ C ∞ (cid:16) e M (cid:17) we conclude that a λ ∈ C ∞ ( M λ ) . For any ν < µ one has a ν = ∑ g ∈ G ( M λ | M ) ga µ .The above sum is finite and ga µ ∈ C ∞ (cid:0) M µ (cid:1) for all g ∈ G ( M λ | M ) so a ν ∈ C ∞ ( M ν ) .(c) The action L ∞ (cid:16) e M (cid:17) × L (cid:16) e M , e S (cid:17) → L (cid:16) e M , e S (cid:17) naturally yields the diagonal ac-tion L ∞ (cid:16) e M (cid:17) × L (cid:16) e M , e S (cid:17) s → L (cid:16) e M , e S (cid:17) s such that the defined above projectors χ e U and χ λ become elements of B (cid:18) L (cid:16) e M , e S (cid:17) s (cid:19) . The given by (4.11.43) limit canbe regarded as limit in the sense of the strong topology of B (cid:18) L (cid:16) e M , e S (cid:17) s (cid:19) . From supp e a ⊂ e U it follows that e a = χ e U e a , ∀ λ ≥ µ a λ = ∑ g ∈ ker ( G ( e M | M ) → G ( e M | M λ )) g e a == ∑ g ∈ ker ( G ( e M | M ) → G ( e M | M λ )) g χ e U a λ = χ λ a µ , e a = χ e U a µ (4.11.44)223f π s λ : C ∞ ( M λ ) → B (cid:18) L (cid:16) e M , e S (cid:17) s (cid:19) is a representation given by (E.2.1) and (E.2.2)then similarly to (4.11.44) one has ∀ λ ≥ µ π s λ ( a λ ) = χ λ π s µ (cid:0) a µ (cid:1) , π s ( e a ) = χ e U π s µ (cid:0) a µ (cid:1) .and taking into account (4.11.43) there is the limitlim λ ∈ Λ C b ( e M ) ⊗ π s λ ( a λ ) = (cid:18) lim λ ∈ Λ χ λ (cid:19) (cid:16) C b ( e M ) ⊗ π s λ (cid:0) a µ (cid:1)(cid:17) == χ e U (cid:16) C b ( e M ) ⊗ π s λ (cid:0) a µ (cid:1)(cid:17) = π s ( e a ) in the sense of the strong topology of B (cid:18) L (cid:16) e M , e S (cid:17) s (cid:19) .(d) From (2.7.7) it follows that the commutator [ D / λ , a λ ] corresponds to / ∇ λ ( a λ ) ∈ C ∞ ( M λ ) ⊗ C ∞ ( M ) Ω D / where / ∇ λ : C ∞ ( M λ ) → C ∞ ( M λ ) ⊗ C ∞ ( M ) Ω D / is the unique G ( M λ | M ) -equvariant connection (cf. Lemma 2.7.4). On the other hand from theexplicit equation (4.10.19) it follows that [ D / λ , a λ ] = / ∇ λ ( a λ ) = m ∑ j = a j λ ⊗ ω j .where ω j ∈ Ω D / and a j λ ∈ C ∞ ( M λ ) is such that supp a j λ ⊂ supp a λ . From D / λ = p − λ ( D / ) (cf. (4.10.23)) and a λ = lift p λµ e p λ ( e U ) (cid:0) a µ (cid:1) for all λ ≥ µ one has [ D / λ , a λ ] = m ∑ j = lift p λµ e p λ ( e U ) (cid:16) a j µ (cid:17) ⊗ ω j If e a j def = lift e p µ e U (cid:16) a j µ (cid:17) then e a j ∈ C c (cid:16) e M (cid:17) On the other hand similarly to (4.11.44) forany j =
1, ..., m one has e a j = χ e U e a j , ∀ λ ≥ µ a j λ = χ λ a j µ , e a j = χ e U a j µ it follows that lim λ ∈ Λ a j λ = lim λ ∈ Λ χ λ a j µ = (cid:18) lim λ ∈ Λ χ λ (cid:19) a j µ = χ e U a j µ = e a j .224t follows thatlim λ ∈ Λ [ D / λ , a λ ] = lim λ ∈ Λ m ∑ j = a j λ ⊗ ω j = m ∑ j = (cid:18) lim λ ∈ Λ a j λ (cid:19) ⊗ ω j = m ∑ j = a j λ ⊗ ω j == m ∑ j = e a j ⊗ ω j = m ∑ j = f lift e p µ e U (cid:16) a j µ (cid:17) ⊗ ω j . (4.11.45)From lift e p µ e U (cid:16) a j µ (cid:17) ∈ C c (cid:16) e M (cid:17) it follows thatlim λ ∈ Λ [ D / λ , a λ ] ∈ C c (cid:16) e M (cid:17) ⊗ Ω D / = K (cid:16) C (cid:16) e M (cid:17)(cid:17) ⊗ Ω D / Similarly to (4.8.14) one can define a C ∞ c (cid:16) e M (cid:17) -module homomorphism e φ ∞ : C ∞ c (cid:16) e M (cid:17) ⊗ C ∞ ( M ) Γ ∞ ( M , S ) → Γ ∞ c (cid:16) e M , e S (cid:17) , n ∑ j = e a j ⊗ ξ j n ∑ j = e a j lift p (cid:0) ξ j (cid:1) (4.11.46)where both Γ ∞ ( M , S ) and Γ ∞ c (cid:16) e M , e S (cid:17) are defined by the Equations (E.4.2) and(4.9.6) respectively. Lemma 4.11.47.
The given by the Equation (4.11.46) homomorphism is an isomorphism.Proof.
The proof of the Lemma 4.8.11 uses the partition of unity. However fromthe Proposition A.1.26 it turns out that there is a smooth partition of unity. Usingit one can proof this lemma as well as the Lemma 4.8.11 has been proved.
Let us explicitly find the commutative specialization of the explained in3.5.6 construction. Following table reflects the mapping between general theoryand the commutative specialization. 225eneral theory Commutative specializationHilbert spaces H and e H L ( M , S ) and L (cid:16) e M , e S (cid:17) Pre- C ∗ -algebra A C ∞ ( M ) Pedersen’s ideal K (cid:16) e A (cid:17) C c (cid:16) e M (cid:17) The space of smooth e W ∞ C ∞ c (cid:16) e M (cid:17) elementsDirac operators D D / e D ? H ∞ def = T ∞ n = Dom D n ⊂ H Γ ∞ ( M , S ) = T ∞ n = Dom D / n Select an element ξ ∈ H ∞ = Γ ∞ ( M , S ) and for all λ ∈ Λ denote by φ λ ∞ : C ∞ ( M λ ) ⊗ Γ ∞ ( M , S ) → Γ ∞ ( M λ , S λ ) , n ∑ j = e a j ⊗ ξ j n ∑ j = e a j lift p λ (cid:0) ξ j (cid:1) the given by the Lemma 4.10.16 isomorphism of C ∞ ( M λ ) -modules. If e a D / is thespecialization of the given by (3.5.8) operator then from the equation (4.11.45) itfollows that e a D / = m ∑ j = f lift e p µ e U (cid:16) a j µ (cid:17) ⊗ ω j (4.11.47)where parameters of the Equation 4.11.47 satisfy to the following condition h D / µ , desc e p µ ( e a ) i lift p µ ( ξ ) = φ µ ∞ m ∑ j = a j µ ⊗ ω j ( ξ ) ! e D is given by (3.5.9) then one has e D ( e a ⊗ ξ ) = m ∑ j = f lift e p µ e U (cid:16) a j µ (cid:17) ⊗ ω j ( ξ ) + e a ⊗ D / ξ = m ∑ j = f lift e p µ e U (cid:16) a j µ (cid:17) ⊗ ω j ( ξ ) + f lift e p µ e U (cid:0) a µ (cid:1) ⊗ D / ξ ∈ C c (cid:16) e M (cid:17) ⊗ C ∞ ( M ) Γ ∞ ( M , S ) . (4.11.48)If we consider the given by (4.11.46) and isomorphism e φ ∞ then e D can be regardedas the map e D : Γ ∞ c (cid:16) e M , e S (cid:17) → Γ ∞ c (cid:16) e M , e S (cid:17) and from the equation (4.11.48) it followsthat e φ ∞ ( e a ⊗ ξ ) = e a lift p ( ξ ) , e D (cid:0) e φ ∞ ( e a ⊗ ξ ) (cid:1) = e φ ∞ m ∑ j = f lift e p µ e U (cid:16) a j µ (cid:17) ⊗ ω j ( ξ ) + f lift e p µ e U (cid:0) a µ (cid:1) ⊗ D / ξ ! == m ∑ j = f lift e p µ e U (cid:16) a j µ (cid:17) lift p (cid:0) ω j ( ξ ) (cid:1) + f lift e p µ e U (cid:0) a µ (cid:1) lift p ( D / ξ ) == m ∑ j = f lift e p µ e U (cid:16) a j µ (cid:17) lift p (cid:0) ω j ( ξ ) (cid:1) + lift e p µ e U (cid:16) a µ lift p µ ( D / ξ ) (cid:17) == lift e p µ e U e φ ∞ m ∑ j = a j µ ⊗ ω j ( ξ ) + e a µ ⊗ D / ξ !! . (4.11.49)On the other hand from (2.7.7) it turns out that m ∑ j = a j µ ⊗ ω j ( ξ ) + e a µ ⊗ D / ξ = D / µ (cid:0) a µ ⊗ ξ (cid:1) ,hence e D ( e a ⊗ ξ ) = lift e p µ e U (cid:0) D / µ (cid:0) a µ ⊗ ξ (cid:1)(cid:1) or equivalently e D (cid:0) e φ ∞ ( e a ⊗ ξ ) (cid:1) = lift e p µ e U (cid:16) D / µ (cid:16) φ λ ∞ (cid:16) desc e p µ ( e a ) ⊗ ξ (cid:17)(cid:17)(cid:17) , e D e a lift p ( ξ ) = lift e p µ e U (cid:16) D / µ desc e p µ ( e a ) (cid:17) .On the other hand from (4.7.8) it follows that e p − µ D / µ e a lift p ( ξ ) = lift e p µ e U (cid:16) D / µ desc e p µ ( e a ) (cid:17) , e D e a lift p ( ξ ) = e p − µ D / µ e a lift p ( ξ ) .227owever from the Theorem 4.10.17 it follows that D / µ = p − µ D / , so from p = e p µ p µ it follows that e p − µ D / µ = e p − µ p − µ D / = p − D / , hence one has e D e a lift p ( ξ ) = e p − D / e a lift p ( ξ ) ∀ a ∈ C ∞ (cid:16) e M (cid:17) ξ ∈ Γ ∞ ( M , S ) (4.11.50)But from the Lemma 4.11.47 it follows that the Γ ∞ c (cid:16) e M , e S (cid:17) is the linear span of e a lift p ( ξ ) ∀ e a ∈ C ∞ c (cid:16) e M (cid:17) ∀ ξ ∈ Γ ∞ ( M , S ) so from (4.11.50) it follows that e D e ξ = p − D / e ξ ∀ e ξ ∈ Γ ∞ c (cid:16) e M , e S (cid:17) ,or, equivalently e D = p − D / . (4.11.51) Theorem 4.11.49.
In the described in this section situation following conditions hold:(i) The given by 4.11.40 coherent set S ( C ∞ ( M ) , L ( M , S ) , / D ) = (cid:8)(cid:0) C ∞ ( M λ ) , L ( M λ , S λ ) , / D λ (cid:1)(cid:9) λ ∈ Λ ∈ CohTriple of spectral triples is good (cf. Definition 3.5.5).(ii) The (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) -lift of the spectral triple (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) (cf. Definition 3.5.7) coincides with the geometricalp-lift of (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) (cf. Definition 4.9.2).Proof. (i) One has an inclusion C ∞ c (cid:16) e M (cid:17) ⊂ C (cid:16) e M (cid:17) of involutive algebras. Thegiven by the seminorms (3.5.7) topology T on C ∞ c (cid:16) e M (cid:17) is finer than the C ∗ -normtopology, so if e A is the T -completion of C ∞ c (cid:16) e M (cid:17) with respect to the C ∗ -norm then e A ⊂ C (cid:16) e M (cid:17) . Otherwise C ∞ c (cid:16) e M (cid:17) is dense in C (cid:16) e M (cid:17) with respect to the C ∗ -normtopology, hence e A is dense in C (cid:16) e M (cid:17) with respect to the the C ∗ -norm topology.(ii) Consider the given by (4.10.12) finite partition of unity1 C ( M ) = ∑ α ∈ A a α ; where a α ∈ C ∞ ( M ) .For any s ∈ N denote by C ′ s = max α ∈ A k a α k s , C s = max (cid:0) C ′ s (cid:1) .228or any α ∈ A there is e a α ∈ C ∞ c (cid:16) e M (cid:17) such that desc p ( e a α ) = a α . Moreover there isthe given by (4.8.5) partition of unity, i.e. ∑ g ∈ G ( e M | M ) , α ∈ A g e a α = C b ( e M ) If e a ∈ C ∞ (cid:16) e M (cid:17) then from (4.9.14) it follows that for all ε > s ∈ N there is acompact set e U ⊂ e M such that | π s ( e a ) ( e x ) | < ε C s | A | ∀ e x ∈ e M \ e U .On the other hand there is a covering sum for e U (cf. Definition 4.2.6), i.e. the finitesubset f A e U ⊂ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) × A such that ∑ ( g , α ) ∈ A e U ( g e a α ) ( e x ) = ∀ e x ∈ e U .If e a ′ = e a ∑ ( g , α ) ∈ A e U ( g e a α ) then e a ′ ∈ C ∞ c (cid:16) e M (cid:17) because the set is f A e U finite. Otherwisefrom the above inequalities one has (cid:13)(cid:13)e a − e a ′ (cid:13)(cid:13) s < ε ,it follows that the completion of C ∞ c (cid:16) e M (cid:17) with respect to the seminorms (3.5.7) is C ∞ (cid:16) e M (cid:17) . From the Definition 3.5.7 it follows that the (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) -lift of the spectral triple (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) is given by (cid:16) C ∞ (cid:16) e M (cid:17) , e H , e D (cid:17) .On the other hand the geometrical p -lift of (cid:0) C ∞ ( M ) , L ( M , S , µ ) , D / (cid:1) equals to (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S , e µ (cid:17) , e D / (cid:17) . From the Remark 4.11.43 the representation C (cid:16) e M (cid:17) ֒ → B (cid:16) e H (cid:17) is equivalent to the representation C (cid:16) e M (cid:17) ֒ → B (cid:16)e L (cid:16) e M , e S , e µ (cid:17)(cid:17) . Compari-son of the equations (4.9.7) and (4.11.51) yields the following e D = e D / . The material of this section is not relevant of coverings of commutative C ∗ -algebras. It contains common technical results which will be used in the Chapters5 and 7. 229 .12.1 Families of operator spaces If X is a locally compact Hausdorff space, and X C ( X ) is a full C ∗ -Hilbert C ( X ) -module, then similarly to (4.5.26) and (4.5.27) for any x ∈ X one can define anormed space C x . If H x is the norm completion of C x then there is a family {H x } x ∈X such that X C ( X ) is a continuity structure for X and the {H x } (cf. Defini-tion D.8.27). Let us define the product on H x given by (D.4.4), thus H x becomes aHilbert space for each x ∈ X . On the other hand the space X C ( X ) is norm closed,so from the Corollary 4.5.19 one has the C ( X ) -isomorphism X C ( X ) = C (cid:16) X , {H x } , X C ( X ) . (cid:17) (4.12.1)If D ∗ ( X ) is the C ∗ -algebra of complex-valued bounded (discontinuous) maps X → C . The D ∗ ( X ) -module of bounded families { ξ x ∈ H x } is a full C ∗ -Hilbert D ∗ ( X ) -module with given by h{ ξ x } , { η x }i D ∗ ( X ) def = (cid:16) x ( ξ x , η x ) H x (cid:17) (4.12.2)scalar product. Definition 4.12.1.
The described above C ∗ -Hilbert D ∗ ( X ) -module is said to be the discontinuous extension of X C ( X ) . We denote it by X D ∗ ( X ) . Remark 4.12.2.
There is the natural inclusion X C ( X ) ⊂ X D ∗ ( X ) . Lemma 4.12.3.
In the above situation one has(i) If X C ( X ) a Hilbert C ( X ) -module, then X C b ( X ) def = C b (cid:16) X , {H x } , X C ( X ) (cid:17) is aHilbert C b ( X ) -module, such that the Hilbert module product on X C ( X ) the restric-tion of the product on C b (cid:16) X , {H x } , X C ( X ) (cid:17) .(ii) X C ( X ) = C (cid:16) C b (cid:16) X , {H x } , X C ( X ) (cid:17)(cid:17) .Proof. (i) For any ξ , η ∈ C b (cid:16) X , {H x } , X C ( X ) (cid:17) we define h ξ , η i C b ( X ) ∈ C b ( X ) , x ∑ k = i k (cid:13)(cid:13)(cid:13) ξ x + i k η x (cid:13)(cid:13)(cid:13) . (4.12.3)230rom (D.4.3) it turns out that (4.12.3) supplies a Hilbert module product on X C b ( X ) which is the restriction of the product on C b (cid:16) X , {H x } , X C ( X ) (cid:17) . Since C b (cid:16) X , {H x } , X C b ( X ) (cid:17) is norm closed C b (cid:16) X , {H x } , X C ( X ) (cid:17) is a Hilbert C b ( X ) -module.(ii) Follows from the Corollary 4.5.19. Lemma 4.12.4.
If X C b ( X ) is a Hilbert C b ( X ) -module then one has(i) X C ( X ) = C (cid:16) X , {H x } , X C b ( X ) (cid:17) is a Hilbert C ( X ) -module, such that the prod-uct on X C ( X ) the restriction of the product on X C b ( X ) .(ii) X C b ( X ) = C b (cid:16) X C ( X ) (cid:17) .Proof. (i) From the Lemma 4.5.11 it turns out that norm ξ ∈ C ( X ) for any ξ ∈ C (cid:16) X , {H x } , X C b ( X ) (cid:17) .From (4.12.3) it follows that h ξ , η i C b ( X ) ∈ C ( X ) so C (cid:16) X , {H x } , X C b ( X ) (cid:17) is aHilbert C ( X ) -premodule, with respect to restriction of the Hilbert product on C b (cid:16) C (cid:16) X , {H x } , X C ( X ) (cid:17)(cid:17) . However from the Definition 4.5.8 it follows that C (cid:16) X , {H x } , X C b ( X ) (cid:17) is closed with respect to norm, hence C (cid:16) X , {H x } , X C b ( X ) (cid:17) is a Hilbert C ( X ) -module.(ii) Follows from (4.5.21). Lemma 4.12.5.
Consider the situation of the Lemma 4.12.4. If {K ( H x ) } x ∈X is a familyof compact operators then K (cid:16) X C ( X ) (cid:17) is a continuity structure for X and the {K ( H x ) } x ∈X ,such that K (cid:16) X C ( X ) (cid:17) ∼ = C (cid:16) X , {K ( H x ) } , K (cid:16) X C ( X ) (cid:17)(cid:17) . Proof.
Similarly to (4.5.25) for any x ∈ X there is the C -linear map φ x : X C ( X ) → H x .such that φ x (cid:16) X C ( X ) (cid:17) is dense in H x . Let us check (a) and (b) of the definitionD.8.27. 231a) If ξ , η ∈ X C ( X ) and ξ ih η is an elementary operator then one hasnorm ξ ih η = norm ξ norm η ∈ C ( X ) .Since K (cid:16) X C ( X ) (cid:17) is norm closure of the algebraic C -linear span of elemen-tary operators norm a ∈ C ( X ) for any a ∈ K (cid:16) X C ( X ) (cid:17) .(b) If a ∈ K ( H x ) then for any ε > a ′ = n ∑ j = α j ih β j α j , β j ∈ H x such that (cid:13)(cid:13) a − a ′ (cid:13)(cid:13) < ε φ x (cid:16) C b (cid:16) X , {H x } , X C ( X ) (cid:17)(cid:17) is dense in H x so for any j =
1, ..., n there are ξ j , η j ∈ X C ( X ) such that (cid:13)(cid:13) α j ih β j − φ x (cid:0) ξ j (cid:1) ih φ x (cid:0) η j (cid:1)(cid:13)(cid:13) < ε n it turns out (cid:13)(cid:13) a − φ x (cid:0) ξ j (cid:1) ih φ x (cid:0) η j (cid:1)(cid:13)(cid:13) < ε .From the Corollary 4.5.19 it turns out that there is the following isomorphism K (cid:16) X C ( X ) (cid:17) ∼ = C (cid:16) X , {K ( H x ) } , K (cid:16) X C ( X ) (cid:17)(cid:17) . Lemma 4.12.6.
Let X C b ( X ) def = C b (cid:16) X , {H x } , X C ( X ) (cid:17) be a Hilbert C b ( X ) -module givenby the Lemma 4.12.3. The is the natural inclusion K (cid:16) X C b ( X ) (cid:17) ⊂ C b (cid:16) X , {K ( H x ) } , K (cid:16) X C ( X ) (cid:17)(cid:17) . (4.12.4) Proof.
Let a ∈ K (cid:16) X C b ( X ) (cid:17) , x ∈ X and ε >
0. There is a ′ = n ∑ j = ξ j ih η j ∈ K (cid:16) X C b ( X ) (cid:17) ξ j , η j ∈ X C b ( X ) k a − a ′ k < ε . There is f x ∈ C c ( X ) such that f x ( X ) = [
0, 1 ] and thereis an open neighborhood U of x which satisfy to the condition f x ( X ) =
1. From f x ξ j , f x η j ∈ X C ( X ) it follows that a ′′ = n ∑ j = f x ξ j ih f x η j ∈ K (cid:16) X C ( X ) (cid:17) From k a − a ′ k < ε it turns out k a ( x ) − a ′′ ( x ) k < ε for each x ∈ U , i.e. a iscontinuous with respect to K (cid:16) X C ( X ) (cid:17) at x (cf. Definition D.8.28). Since x ∈ X is an arbitrary point a is continuous and taking into account that a is bounded onehas a ∈ C b (cid:16) X , {K ( H x ) } , K (cid:16) X C ( X ) (cid:17)(cid:17) .If X is locally compact Hausdorff space then it is completely regular (cf. Ex-ercise A.1.12 and Definition A.1.10). Let β X be the Stone- ˇCech compactificationof X and let (cid:8) β H β x (cid:9) β x ∈ β X be the Stone- ˇCech extension of {H x } (cf. Definition4.5.22). The C b ( X ) -module C b (cid:16) X , {H x } , X C ( X ) (cid:17) induces the continuity structure β X C ( X ) for β X and (cid:8) β H β x (cid:9) such that there is the natural C b ( X ) -isomorphism C b (cid:16) X , {H x } , X C ( X ) (cid:17) ∼ = C b (cid:16) β X , (cid:8) β H β x (cid:9) , β X C ( X ) (cid:17) ∼ = β X C ( X ) . (4.12.5)By the definition one has X C b ( X ) def = C b (cid:16) X , {H x } , X C ( X ) (cid:17) , so from (4.12.5) one canregard X C b ( X ) = X C ( β X ) = C b (cid:16) β X , (cid:8) β H β x (cid:9) , β X C ( X ) (cid:17) .There is the family (cid:8) K (cid:0) H β x (cid:1)(cid:9) β x ∈ β X . Otherwise from K (cid:16) X C b ( X ) (cid:17) ∼ = K (cid:16) X C ( β X ) (cid:17) and taking into account the lemma 4.12.5 one can proof that K (cid:16) X C ( β X ) (cid:17) is a con-tinuity structure for β X and (cid:8) K (cid:0) H β x (cid:1)(cid:9) . Otherwise K (cid:16) X C ( β X ) (cid:17) is norm closed C ( β X ) -module, so from the Corollaries 4.5.12 and 4.5.19 it follows that K (cid:16) X C b ( X ) (cid:17) = C b (cid:16) β X , (cid:8) K (cid:0) H β x (cid:1)(cid:9) , K (cid:16) X C b ( X ) (cid:17)(cid:17) .If β K (cid:16) X C ( X ) (cid:17) be the Stone- ˇCech extension of K (cid:16) X C ( X ) (cid:17) (cf. Definition 4.5.23)then from (4.5.31) it turns out that β K (cid:16) X C ( X ) (cid:17) ∼ = C b (cid:16) X , {K ( H x ) } , K (cid:16) X C ( X ) (cid:17)(cid:17) (4.12.6)233nd taking into account K (cid:16) X C b ( X ) (cid:17) ⊂ C b (cid:16) X , {K ( H x ) } , K (cid:16) X C ( X ) (cid:17)(cid:17) we con-clude that K (cid:16) X C b ( X ) (cid:17) ⊂ C b (cid:16) β X , (cid:8) K (cid:0) H β x (cid:1)(cid:9) , β K (cid:16) X C ( X ) (cid:17)(cid:17) .From the Corollaries 4.5.12 and 4.5.19 one has K (cid:16) X C b ( X ) (cid:17) = C b (cid:16) β X , (cid:8) K (cid:0) H β x (cid:1)(cid:9) , β K (cid:16) X C ( X ) (cid:17)(cid:17) , β K (cid:16) X C ( X ) (cid:17) = C b (cid:16) β X , (cid:8) K (cid:0) H β x (cid:1)(cid:9) , β K (cid:16) X C ( X ) (cid:17)(cid:17) so from (4.12.6) it follows that K (cid:16) X C b ( X ) (cid:17) = C b (cid:16) X , {K ( H x ) } , K (cid:16) X C ( X ) (cid:17)(cid:17) .So we proved the following lemma. Lemma 4.12.7.
Suppose that X is locally compact Hausdorff space. If X C ( X ) is a HilbertC ( X ) -module, and X C b ( X ) def = C b (cid:16) X , {H x } , X C ( X ) (cid:17) is a Hilbert C b ( X ) -module givenby the Lemma 4.12.3 then there is the natural C b ( X ) -isomorphism K (cid:16) X C b ( X ) (cid:17) ∼ = C b (cid:16) K (cid:16) X C ( X ) (cid:17)(cid:17) . (4.12.7) If (cid:8) L ( N ) x (cid:9) x ∈X is a family of isomorphic to L ( N ) Hilbert spaces thenthe standard Hilbert module ℓ ( C ( X )) (cf. Definition D.4.11) is a continuitystructure for X and the (cid:8) L ( N ) x (cid:9) (cf. Definition D.8.27). From the Corollary4.5.19 it turns out that ℓ ( C ( X )) ∼ = C (cid:0) X , (cid:8) L ( N ) x (cid:9) , ℓ ( C ( X )) (cid:1) (4.12.8)If (cid:8) e j (cid:9) j ∈ N is an orthogonal basis then any ξ ∈ ℓ ( C ( X )) is represented by ξ = ∞ ∑ j = f j e j ; ∀ j ∈ N f j ∈ C ( X ) ; x ∞ ∑ j = (cid:12)(cid:12) f j ( x ) (cid:12)(cid:12) ! ∈ C ( X ) . (4.12.9)If D ∗ ( X ) is the C ∗ -algebra of complex-valued bounded (discontinuous) maps X → C then there is the natural inclusion ℓ ( C ( X )) ⊂ ℓ ( D ∗ ( X )) . (4.12.10) Lemma 4.12.9.
The natural inclusion ℓ ( C ( X )) ⊂ ℓ ( D ∗ ( X )) is exact (cf. Definition1.3.1). roof. If η = ∑ ∞ j = g j e j ∈ ℓ ( D ∗ ( X )) \ ℓ ( C ( X )) is such that g j ∈ D ∗ ( X ) for all j =
1, ..., ∞ then there are x ∈ X and k ∈ N such that g k is not continuous at x .If f ∈ C ( X ) is such that f ( x ) = ξ = f e k ∈ X C ( X ) then h ξ , η i D ∗ ( X ) / ∈ C ( X ) . (4.12.11)The equation (4.12.11) is a specialization of the condition (1.3.2). Lemma 4.12.10.
Let X C ( X ) be a Hilbert C ( X ) -module. Consider the natural isomor-phism X C ( X ) ∼ = C (cid:16) X , {H x } x ∈X , X C ( X ) (cid:17) . If X C b ( X ) def = C b (cid:16) X , {H x } , X C ( X ) (cid:17) thenthere is the natural *-isomorphism of C ∗ -algebras L (cid:16) X C ( X ) (cid:17) ∼ = L (cid:16) X C b ( X ) (cid:17) . Proof.
Any a ∈ L (cid:16) X C b ( X ) (cid:17) corresponds to a family { a x ∈ B ( H x ) } x ∈X such that a ξ ∈ C b (cid:16) X , {H x } , X C ( X ) (cid:17) for any ξ = { ξ x ∈ H x } ∈ C b (cid:16) X , {H x } , X C ( X ) (cid:17) . Forany η ∈ C (cid:16) X , {H x } , X C ( X ) (cid:17) one has norm η ∈ C ( X ) , and taking into accountnorm a ∈ C b ( X ) we conclude that norm a η ≤ norm a norm η ∈ C ( X ) i.e. a η ∈ C (cid:16) X , {H x } , X C ( X ) (cid:17) ∼ = X C ( X ) . So a can be regarded as an element of L (cid:16) X C ( X ) (cid:17) and there is the natural inclusion L (cid:16) X C b ( X ) (cid:17) ⊂ L (cid:16) X C ( X ) (cid:17) . Conversely it is clearthat any a ∈ L (cid:16) X C ( X ) (cid:17) yields a C b ( X ) -linear map C (cid:16) X , {H x } , X C ( X ) (cid:17) → C (cid:16) X , {H x } , X C ( X ) (cid:17) and since k a k < ∞ the operator a gives a linear map C b (cid:16) X , {H x } , X C ( X ) (cid:17) → C b (cid:16) X , {H x } , X C ( X ) (cid:17) ∼ = X C b ( X ) .So a can be regarded as an element of L (cid:16) X C b ( X ) (cid:17) and there is the natural inclusion L (cid:16) X C ( X ) (cid:17) ⊂ L (cid:16) X C b ( X ) (cid:17) . Let F by a continuity structure for X and the { A x } (cf. Definition D.8.27)where A x is a C ∗ -algebra. Let L be a closed left ideal of C b ( X , { A x } , F ) . If p : e X → X then C (cid:16) e X (cid:17) -module n e f lift p ( b ) ∈ lift p [ C ( X , { A x } , F )] (cid:12)(cid:12)(cid:12) b ∈ L ; e f ∈ C b (cid:16) e X (cid:17)o lift p [ C ( X , { A x } , F )] . It turns out that lift p [ L ] is closed left idealof lift p [ C ( X , { A x } , F )] (cf. Definition 4.5.53). Similar reasons can be applied toright, two sided ideals and hereditary subalgebras, i.e. L is a closed left ideal of C b ( X , { A x } , F ) ⇒⇒ lift p [ L ] is a closed left ideal of lift p [ C b ( X , { A x } , F )] ; R is a closed right ideal of C b ( X , { A x } , F ) ⇒⇒ lift p [ R ] is a closed right ideal of lift p [ C b ( X , { A x } , F )] ; I is a closed two sided ideal of C b ( X , { A x } , F ) ⇒⇒ lift p [ I ] is a closed two sided of lift p [ C b ( X , { A x } , F )] ; B is a hereditary subalgebra of C b ( X , { A x } , F ) ⇒⇒ lift p [ B ] is a hereditary subalgebra of lift p [ C b ( X , { A x } , F )] . (4.12.12) Lemma 4.12.12.
Let X C ( X ) be a Hilbert C ( X ) -module. If p : e X → X is a coveringthen C b (cid:16) lift p h X , {H x } , X C ( X ) i(cid:17) is a Hilbert C b (cid:16) e X (cid:17) -module.Proof. From the Definition 4.5.40 and the Lemma 4.5.11 it turns out the norm e ξ = (cid:16)e x (cid:13)(cid:13)(cid:13) e ξ e x (cid:13)(cid:13)(cid:13)(cid:17) ∈ C b (cid:16) e X (cid:17) for all e ξ ∈ C b (cid:16) lift p h X , {H x } , X C ( X ) i(cid:17) . For any e ξ , e η ∈ C b (cid:16) lift p h X , {H x } , X C ( X ) i(cid:17) from (D.4.3) it turns out h ξ x , η x i = ∑ k = i k D ξ x + i k η x , ξ x + i k η x E ⇒ ( x
7→ h ξ x , η x i ) ∈ C (cid:16) e X (cid:17) . Let X C ( X ) be a full Hilbert C ( X ) -module and let K (cid:16) X C ( X ) (cid:17) be a C ∗ -algebra of compact operators. If {H x } x ∈X is the completion of the given by (4.5.26)and (4.5.27) normed space, then there is a family {K ( H x ) } x ∈X is a family ofcompact operators such that K (cid:16) X C ( X ) (cid:17) is a continuity structure for X and the {K ( H x ) } Lemma 4.12.14.
Consider the situation 4.12.13, If X C b ( X ) def = C b (cid:16) lift p h X , {H x } , X C ( X ) i(cid:17) is a Hilbert C b (cid:16) e X (cid:17) -module given by the Lemma 4.12.3 then there is the natural isomor-phism K (cid:16) X C b ( e X ) (cid:17) ∼ = C b (cid:16) lift p h K (cid:16) X C ( X ) (cid:17)i(cid:17) . (4.12.13)236 roof. If a ∈ K (cid:16) X C ( X ) (cid:17) then there is the C ∗ -norm convergent series a = ∞ ∑ j = ξ j ih η j ξ j , η j ∈ X C ( X ) From (4.5.50) it turns out that the series ∞ ∑ j = lift p (cid:0) ξ j (cid:1) ih lift p (cid:0) η j (cid:1) is norm convergent, hence lift p ( a ) ∈ K (cid:16) X C b ( e X ) (cid:17) . So one has the natural in-clusion lift p h K (cid:16) X C ( e X ) (cid:17)i ֒ → K (cid:16) X C b ( e X ) (cid:17) . Let e x ∈ e X , and let e U be an openneighborhood of e x such that the restriction p | e U : e U ≈ −→ U = p (cid:16) e U (cid:17) is a homeo-morphism. According to 4.1.7 there is e f e x ∈ C c (cid:16) e X (cid:17) and open subset e V such that { e x } ⊂ e V ⊂ e U , e f e x (cid:16) e V (cid:17) = e f e x (cid:16) e X (cid:17) = [
0, 1 ] and supp e f e x ⊂ e U . Let x = p ( e x ) and let f x = desc p (cid:16) e f e x (cid:17) . If e a ∈ K (cid:16) X C b ( e X ) (cid:17) and e a f = e f e x e a then e a ( e x ) = e a f ( e x ) forall e x ∈ e V . Otherwise if a = desc p (cid:0)e a f (cid:1) then e a f ( e x ) = lift p ( a ) ( e x ) for all e x ∈ e V , hence e a ( e x ) = lift p ( a ) ( e x ) for each e x ∈ e V . From the Definition D.8.28 it turns out that e a iscontinuous and taking into account k e a k < ∞ one has e a ∈ lift p h K (cid:16) X C ( X ) (cid:17)i , i.e. K (cid:16) X C b ( e X ) (cid:17) ⊂ C b (cid:16) lift p h K (cid:16) X C ( X ) (cid:17)i(cid:17) . Corollary 4.12.15.
Consider the situation 4.12.13. Suppose that p is a finite fold covering.If lift p : K (cid:16) X C ( X ) (cid:17) ֒ → lift p h K (cid:16) X C ( X ) (cid:17)i (4.12.14) is given by the Lemma 4.12.14 then one has lift p (cid:16) K (cid:16) X C ( X ) (cid:17)(cid:17) ⊂ C (cid:16) lift p h K (cid:16) X C ( X ) (cid:17)i(cid:17) (4.12.15) Proof.
Follows from the Lemma 4.5.43.
Lemma 4.12.16. If X is a locally compact space and p : e X → X is a covering then thereis the natural C ( X ) -linear inclusion lift p : ℓ ( C ( X )) ֒ → ℓ (cid:16) C b (cid:16) e X (cid:17)(cid:17) . (4.12.16)237 oreover if p is a finite-fold covering then there is the natural C ( X ) linear inclusion lift p : ℓ ( C ( X )) ֒ → ℓ (cid:16) C (cid:16) e X (cid:17)(cid:17) . (4.12.17) Proof.
From (4.12.9) it turns out that ξ ∈ ℓ ( C ( X )) ⇒ ξ = ∞ ∑ j = f j e j ; ∀ x ∈ X ∞ ∑ j = (cid:12)(cid:12) f j ( x ) (cid:12)(cid:12) < ∞ ∀ j ∈ N f j ∈ C ( X ) .If ξ = ∑ ∞ j = f j e j then we define lift p ( ξ ) = ∞ ∑ j = lift p (cid:0) f j (cid:1) e j . (4.12.18)Taking into account lift p (cid:0) f j (cid:1) ∈ C b (cid:16) e X (cid:17) , ∑ ∞ j = (cid:12)(cid:12)(cid:12) lift p (cid:0) f j (cid:1) ( x ) (cid:12)(cid:12)(cid:12) < ∞ for all x ∈ X oneconcludes that lift p ( ξ ) ∈ ℓ (cid:16) C b (cid:16) e X (cid:17)(cid:17) .Moreover if p is a finite-fold covering then f j ∈ C (cid:16) e X (cid:17) and (cid:16)e x (cid:13)(cid:13)(cid:13) lift p ( ξ ) ( e x ) (cid:13)(cid:13)(cid:13)(cid:17) ∈ C (cid:16) e X (cid:17) . The Hilbert module ℓ ( C ( X )) yields a family of isomorphic Hilbertspaces (cid:8) L ( N ) x (cid:9) x ∈X . From the equation 4.12.8 it turns out that ℓ ( C ( X )) ∼ = C (cid:0) X , (cid:8) L ( N ) x (cid:9) , ℓ ( C ( X )) (cid:1) . (4.12.19)If p : e X → X is a covering then there is a family of isomorphic Hilbert spaces (cid:8) L ( N ) ′ e x (cid:9) e x ∈ e X such that for any e x ∈ e X there is the natural (uniquely defined)isomorphism L ( N ) ′ e x ∼ = L ( N ) p ( e x ) . Otherwise ℓ (cid:16) C (cid:16) e X (cid:17)(cid:17) naturally inducesa family (cid:8) L ( N ) e x (cid:9) e x ∈ e X . there is the natural (uniquely defined) isomorphism L ( N ) ′ e x ∼ = L ( N ) e x . The space lift p (cid:0) ℓ ( X ) (cid:1) is a continuity structure for X andthe (cid:8) L ( N ) ′ e x (cid:9) , and the space ℓ (cid:16) e X (cid:17) is the continuity structure for for X and the (cid:8) L ( N ) e x (cid:9) . From the isomorphisms L ( N ) ′ e x , Lemma 4.12.16 and the Definition4.5.45 it turns out that 238 ift p (cid:2) C (cid:0) X , (cid:8) L ( N ) x (cid:9) , ℓ ( X ) (cid:1)(cid:3) ∼ = C (cid:16) e X , (cid:8) L ( N ) e x (cid:9) , ℓ (cid:16) e X (cid:17)(cid:17) , C c (cid:16) lift p (cid:2) C (cid:0) X , (cid:8) L ( N ) x (cid:9) , ℓ ( X ) (cid:1)(cid:3)(cid:17) ∼ = C c (cid:16) e X , (cid:8) L ( N ) e x (cid:9) , ℓ (cid:16) e X (cid:17)(cid:17) , C (cid:16) lift p (cid:2) C (cid:0) X , (cid:8) L ( N ) x (cid:9) , ℓ ( X ) (cid:1)(cid:3)(cid:17) ∼ = C (cid:16) e X , (cid:8) L ( N ) e x (cid:9) , ℓ (cid:16) e X (cid:17)(cid:17) , C b (cid:16) lift p (cid:2) C (cid:0) X , (cid:8) L ( N ) x (cid:9) , ℓ ( X ) (cid:1)(cid:3)(cid:17) ∼ = C b (cid:16) e X , (cid:8) L ( N ) e x (cid:9) , ℓ (cid:16) e X (cid:17)(cid:17) , (4.12.20)In particular one has ℓ (cid:16) C (cid:16) e X (cid:17)(cid:17) ∼ = C (cid:16) lift p [ C ( X )] (cid:17) , (4.12.21) ℓ (cid:16) C b (cid:16) e X (cid:17)(cid:17) ∼ = C b (cid:16) lift p [ C ( X )] (cid:17) (4.12.22) Let us consider a pointed topological finite covering category S ( X , x ) = n(cid:16) X λ , x λ (cid:17) → ( X , x ) o .Suppose that (cid:0) X , x (cid:1) is its topological inverse limit, and denote by p : (cid:0) X , x (cid:1) → ( X , x ) and p λ : (cid:0) X , x (cid:1) → (cid:0) X λ , x λ (cid:1) the natural coverings. If X C ( X ) is a full count-ably generated Hilbert C ( X ) from the Theorem D.4.12 one has the isomorphismof C ∗ -Hilbert modules. X C ( X ) ⊕ ℓ ( C ( X )) ∼ = ℓ ( C ( X )) .The Lemma 1.3.9 yields inclusions ι L : L (cid:16) X C ( X ) (cid:17) ֒ → L (cid:0) ℓ ( C ( X )) (cid:1) , ι K : K (cid:16) X C ( X ) (cid:17) ֒ → K (cid:0) ℓ ( C ( X )) (cid:1) , ι K (cid:16) K (cid:16) X C ( X ) (cid:17)(cid:17) is a hereditary subalgebraof both K (cid:0) ℓ ( C ( X )) (cid:1) and L (cid:0) ℓ ( C ( X )) (cid:1) . (4.12.23)239therwise from the (4.12.1) and (4.12.21), (4.12.22) there are C ( X ) isomorphisms X C ( X ) = C (cid:16) X , {H x } , X C ( X ) (cid:17) , ℓ ( C ( X λ )) ∼ = C (cid:16) lift p λ (cid:2) ℓ ( C ( X )) (cid:3)(cid:17) , ℓ (cid:0) C (cid:0) X (cid:1)(cid:1) ∼ = C (cid:16) lift p (cid:2) ℓ ( C ( X )) (cid:3)(cid:17) , ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) ∼ = C b (cid:16) lift p (cid:2) ℓ ( C ( X )) (cid:3)(cid:17) .From the above equations and the Lemmas 4.5.42, 4.5.43 it turns out that there arefollowing inclusions ℓ ( p λ ) : ℓ ( C ( X )) ֒ → ℓ ( C ( X λ )) , ℓ (cid:16) p νµ (cid:17) : ℓ (cid:0) C (cid:0) X µ (cid:1)(cid:1) ֒ → ℓ ( C ( X ν )) where ν ≥ µ , ℓ ( p λ ) : ℓ ( C ( X λ )) ֒ → ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) , ℓ ( p ) = ℓ ( p λ ) ◦ ℓ ( p λ ) : ℓ ( C ( X )) ֒ → ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) ,and taking into account (4.5.49) one has ν > λ > µ ⇒ ℓ (cid:16) p νµ (cid:17) = ℓ ( p νλ ) ◦ ℓ (cid:16) p λµ (cid:17) , ν > µ ⇒ ℓ ( p ν ) = ℓ (cid:0) p µν (cid:1) ◦ ℓ ( p µ ) , ℓ ( p ) = ℓ ( p λ ) ◦ ℓ (cid:16) p λ (cid:17) ,If X C ( X λ ) def = C (cid:16) lift p λ h X C ( X ) i(cid:17) , X C ( X ) def = C (cid:16) lift p h X C ( X ) i(cid:17) , X C b ( X ) def = C b (cid:16) lift p h X C ( X ) i(cid:17) .then from (4.5.65) then there are natural inclusions X C ( X λ ) ⊂ ℓ ( C ( X ) λ ) , X C ( X ) ⊂ ℓ (cid:0) C (cid:0) X (cid:1)(cid:1) , X C b ( X ) ⊂ ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) K (cid:16) X C ( X λ ) (cid:17) = C (cid:16) lift p λ h K (cid:16) X C ( X ) (cid:17)i(cid:17) , K (cid:16) X C ( X ) (cid:17) = C (cid:16) lift p h K (cid:16) X C ( X ) (cid:17)i(cid:17) , K (cid:16) X C b ( X ) (cid:17) = C b (cid:16) lift p h K (cid:16) X C ( X ) (cid:17)i(cid:17) (4.12.24)and injective *-homomorphisms K ( p λ ) : K (cid:16) X C ( X ) (cid:17) ֒ → K (cid:16) X C ( X λ ) (cid:17) , K (cid:16) p νµ (cid:17) : K (cid:16) X C ( X µ ) (cid:17) ֒ → K (cid:16) X C ( X ν ) (cid:17) where ν ≥ µ , K ( p λ ) : K (cid:16) X C ( X λ ) (cid:17) ֒ → K (cid:16) X C b ( X ) (cid:17) , K ( p ) = K ( p λ ) ◦ K ( p λ ) : K (cid:16) X C ( X ) (cid:17) ֒ → K (cid:16) X C b ( X ) (cid:17) , (4.12.25)From Lemmas (4.12.23), 4.12.24 and (4.5.64) it turns out that there are the inclu-sions ι K : K (cid:16) X C ( X ) (cid:17) ֒ → K (cid:0) ℓ ( C ( X )) (cid:1) , ι b K : K (cid:16) X C b ( X ) (cid:17) ֒ → K (cid:0) ℓ ( C b ( X )) (cid:1) ,From the 4.12.12 it turns out ι K (cid:16) K (cid:16) X C ( X ) (cid:17)(cid:17) is a hereditary subalgebraof K (cid:0) ℓ ( C ( X )) (cid:1) and L (cid:0) ℓ ( C ( X )) (cid:1) , ι b K (cid:16) K (cid:16) X C b ( X ) (cid:17)(cid:17) is a hereditary subalgebraof K (cid:0) ℓ ( C b ( X )) (cid:1) and L (cid:0) ℓ ( C b ( X )) (cid:1) . (4.12.26) Remark 4.12.18.
Denote by b K def = C ∗ -lim −→ λ ∈ Λ K (cid:16) X C ( X λ ) (cid:17) be the inductive limit of C ∗ -algebras in sense of the Definition 1.2.7 and b π a : b K → B (cid:16) b H a (cid:17) be the atomicrepresentation. If b G def = lim ←− λ ∈ Λ G ( X λ | X ) is the inverse limit of groups then sim-ilarly to the Lemma 3.1.18 one construct the equivariant action b G × b H a → b H a which naturally induces the action b G × B (cid:16) b H a (cid:17) → b B (cid:16) b H a (cid:17) . (4.12.27)24142 hapter 5 Coverings of continuous traceoperator spaces
The notion of the operator space with continuous trace is a generalization of thenotion of the C ∗ -algebra with continuous trace (cf. Section 5.1). Here we considernoncommutative coverings of these spaces. Operator space with continuous trace it generalization of C ∗ -algebra with contin-uous trace. If A be a C ∗ -algebra, such that the spectrum X of A is Hausdorff then A isa CCR -algebra (cf. Remark D.8.24). From the Corollary D.8.11 it follows that A is C ∗ -algebra of type I and its primitive spectrum coincides with its spectrum (cf.Theorem D.8.9). There is the action C ( X ) × A → A ; ( x , a ) xa ; x ∈ C ( X ) , a ∈ A (5.1.1)be the given by the Theorem D.2.26 action. For any x ∈ X there is the irreduciblerepresentation rep x : A → B ( H x ) . Definition 5.1.2.
Let A be a C ∗ -algebra which has continuous trace. We say that asub-unital subspace ( X , Y ) of A (cf. Definition 2.6.1) continuous trace operator space if following conditions hold: 243a) If X is the spectrum of A then one has C ( X ) × X ⊂ X where the action C ( X ) × A → A is given by the Dauns Hofmann theorem (cf. (5.1.1)).(b) For any x ∈ X if both X x and Y x are C ∗ -norm completions of rep x ( X ) and rep x ( Y ) respectively then the C ∗ -algebra rep x ( A ) is the C ∗ -envelope of ( X x , Y x ) , i.e. rep x ( A ) = C ∗ e ( X x , Y x ) (cf. Definition 2.6.4). Lemma 5.1.3.
In the situation of the Definition 5.1.2 A is the C ∗ -envelope of ( X , Y ) (cf.Definition 2.6.4), i.e. A = C ∗ e ( X , Y ) .Proof. Firstly we prove that the minimal unitization A + (cf. Definition D.1.8) of A is the C ∗ -envelope of Y . Denote by k : Y ֒ → A + the natural inclusion. If ( C ∗ e ( Y ) , j ) is the C ∗ -envelope of Y then from the Definition D.7.12 it follows that there is thesurjective unital *-homomorphism π : A + → C ∗ e ( Y ) such that j = π ◦ k . If werepresent A + = C A + ⊕ A then clearly ker π ⊂ ⊕ A ∼ = A . If ker π = { } thenker π is a closed ideal which corresponds to an a nonempty open subset U ⊂ X where X is the spectrum of A (cf. Theorem D.2.17). If x ∈ U is any point thenfrom (b) of the Definition 5.1.2 there is a ∈ X such that rep x ( a ) =
0. Since X is Hausdorff there is f ∈ C ( X ) such that f ( x ) = supp f ⊂ U . Clearly f a =
0. From (a) of the Definition 5.1.2 it follows that f a ∈ Y , otherwise one has k ( f a ) ∈ ker π . Hence π ◦ k is not injective, so there is a contradiction. It turns outthat ker π = { } , i.e. π is injective. Otherwise π is surjective the Definition D.7.12,hence π : A + ∼ = C ∗ e ( Y ) is an isomorphism.Secondly we prove that A = C ∗ e ( X , Y ) . Denote by A ′ def = C ∗ e ( X , Y ) ⊂ C ∗ e ( Y ) .Let us prove that A ′ is a C ( X ) -module. Select f ∈ C ( X ) . For any a ′ ∈ A ′ and ε > b ′ = n ∑ j = x j ,1 · x j ,2 · ... · x j , k j where x , ..., x n , k n ∈ X ∪ X ∗ , (cid:13)(cid:13) b ′ − a ′ (cid:13)(cid:13) < ε k f k .Otherwise X is C ( X ) -module it turns out that f x j ,1 ∈ X for all j =
1, ..., n . So onehas b ′′ = n ∑ j = (cid:0) f x j ,1 (cid:1) · x j ,2 · ... · x j , k j ∈ A ′ , (cid:13)(cid:13) b ′′ − f a ′ (cid:13)(cid:13) < ε ,hence f a ′ ∈ A ′ . From the Definition 5.1.2 it follows that X ∪ X ∗ ⊂ A , hence A ′ ⊂ A . Suppose that that there is x ∈ X rep x ( A ′ ) $ rep x ( A ) . There are a ∈ and ε > k rep x ( a ) − rep x ( a ′ ) k > ε for all a ′ ∈ A ′ . Since A x isgenerated by X x as C ∗ -algebra one has X x $ rep x ( A ′ ) , hence X $ A ′ . There isa contradiction, it follows that rep x ( A ′ ) = rep x ( A ) . From the Corollary 4.5.51 itfollows that A ′ = A . C ∗ -algebras induced by coverings Let A be a C ∗ -algebra with continuous trace which according to the TheoremD.8.34 can be represented by the following way A ∼ = C (cid:0) X , { A x } x ∈X , F (cid:1) (5.2.1)where X = ˆ A is the Hausdorff spectrum of A , and C (cid:0) X , { A x } x ∈X , F (cid:1) is theconverging to zero submodule (cf. Definition 4.5.8). Definition 5.2.1.
Let us consider the specialization of the Definitions 4.6.8 and4.6.9 where A = (cid:0) X , { A x } x ∈X , F (cid:1) is given by (5.2.1). For any finite-fold covering p : e X → X we use the given by (4.6.7) notation. If p is a finite-fold covering thenwe use the following notation A ( p ) : A ֒ → A (cid:16) e X (cid:17) (5.2.2)We say that A (cid:16) e X (cid:17) is the p - lift of A . If p : e X → e X is a morphism of FinTop - X form p : e X → X to p : e X →X , i.e. p = p ◦ p then similarly to to (4.6.6) we write A (cid:16) p (cid:17) def = lift p (cid:12)(cid:12)(cid:12) A ( e X ) : A (cid:16) e X (cid:17) ֒ → A (cid:16) e X (cid:17) . (5.2.3) Let us consider two spectral triples.1. The commutative spectral triple (cid:0) C ∞ ( M ) , L ( M , S ) , D / (cid:1) (cf. Equation (E.4.6)).2. The finite spectral triple (cid:0) M m ( C ) , C k , D fin (cid:1) (cf. E.6).245f Γ fin is the grading operator (cf. Definition E.1.3) of (cid:0) M m ( C ) , C k , D fin (cid:1) then fromE.7 it follows that there is the product of above spectral triples (cid:16) C ∞ ( M ) ⊗ M m ( C ) , L ( M , S ) ⊗ C k , D / ⊗ Γ fin + Id L ( M , S ) ⊗ D fin (cid:17) . (5.2.4)If p : e M → M is a finite-fold covering then there is the given by the Definition4.9.1 geometrical p -lift (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S (cid:17) , e D / (cid:17) of the spectral triple (cid:0) C ∞ ( M ) , L ( M , S ) , D / (cid:1) . Definition 5.2.3.
The given by (cid:16) C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) , L (cid:16) e M , e S (cid:17) ⊗ C k , e D / ⊗ Γ fin + Id L ( e M , e S ) ⊗ D fin (cid:17) product of spectral triples (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S , e µ (cid:17) , e D / (cid:17) and (cid:0) M m ( C ) , C k , D fin (cid:1) issaid to be the geometrical p - lift of (cid:16) C ∞ ( M ) ⊗ M m ( C ) , L ( M , S ) ⊗ C k , D / ⊗ Γ fin + Id L ( M , S ) ⊗ D fin (cid:17) .If p : e M → M is an infinite covering then there is the given by the Definition4.9.2 geometrical p -lift (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S (cid:17) , e D / (cid:17) of the spectral triple (cid:0) C ∞ ( M ) , L ( M , S ) , D / (cid:1) . Definition 5.2.4.
The given by (cid:16) C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) , L (cid:16) e M , e S (cid:17) ⊗ C k , e D / ⊗ Γ fin + Id L ( e M , e S ) ⊗ D fin (cid:17) triple is said to be the geometrical p - lift of (cid:16) C ∞ ( M ) ⊗ M m ( C ) , L ( M , S ) ⊗ C k , D / ⊗ Γ fin + Id L ( M , S ) ⊗ D fin (cid:17) . C ∗ -algebras Lemma 5.3.1. If (cid:16) A , e A , G , π (cid:17) is a noncommutative finite-fold quasi-covering, such thatA is a C ∗ -algebra with continuous trace then the reduced algebra of (cid:16) A , e A , G , π (cid:17) (cf.Definition 2.3.12) is a continuous trace C ∗ -algebra. roof. Let X (resp. e X ) be the spectrum of A (resp. e A ), and let p : e X → X bea continuous map given by the Lemma 2.3.7. Let e x ∈ e X be any point and let x = p ( e x ) ∈ X . From the Proposition D.8.18 it turns out that there is an Abelian a ′ ∈ A such that ˆ a ′ ∈ K ( X ) and tr (cid:0) rep x ( a ′ ) (cid:1) =
1. If e A red is the reduced algebraof (cid:16) A , e A , G , π (cid:17) (cf. Definition 2.3.12) then there is the inclusion φ : A ֒ → e A red .If e a ′ def = φ ( a ) then from the Equation 2.3.20 it follows that the rank of rep e x ( e a ′ ) does not exceed 1 for all e x ∈ e X . Let both ˆ a : X → R and ˆ e a ′ : X → R are givenby x tr ( rep x ( a ′ )) and e x tr ( rep e x ( e a ′ )) respectively. One has f ∈ C ( X ) andtaking into account that the map e X → X is surjective and continuous we concludethat ˆ e a ′ is continuous. It 0 < ε < f ε is given by (3.1.19) then since e a ′ is positivethere is e a = − ε f ε ( e a ) ∈ A + , which satisfies to following conditions(a) dim rep e x ( e a ) ≤ dim rep e x ( e a ′ ) ≤ e x ∈ e X ,(b) ˆ e a = − ε f ε (cid:16) ˆ e a ′ (cid:17) .From (a) it turns out that e aa is an Abelian element of e A red . From (b) it followsthat ˆ e a ∈ K (cid:16) e X (cid:17) . So e a satisfies to (iii) of the Proposition D.8.18. Hence from theProposition D.8.18 it turns out that e A red is a continuous trace C ∗ -algebra. Lemma 5.3.2.
If A is a C ∗ -algebra with the Hausdorff spectrum X then the bijectivemap (D.8.3) yields the one-to-one correspondence between essential ideals and open densesubsets of X .Proof. The given by(D.8.3) map
U ↔ A | U yields the one-to-one correspondencebetween ideals and open subsets of X (cf. Remark D.8.24).Suppose that U ⊂ X is a dense subset of X and a ∈ A be such that a = aA | U = { } . From the Lemma D.8.26 it follows that there is an open subset V ⊂ X such that rep x ( a ) = x ∈ V . Since U is dense, one has V ∩ U 6 = ∅ .There is x ∈ U such that rep x ( a ) =
0. From the Exercise A.1.12 it follows thatthe space X is completely regular (cf. Definition A.1.10). There is a continuousfunction f U : X → [
0, 1 ] such that f U ( x ) = f U ( X \ U ) = { } . Otherwisefrom the Lemma D.8.25 if follows that f U a ∗ ∈ A | U . From (cid:13)(cid:13) rep x ( a ( f U a ∗ )) (cid:13)(cid:13) = (cid:13)(cid:13) rep x ( a ) (cid:13)(cid:13) = aA | U = { } . From the Lemma D.1.4 it turns outthat A | U is an essential ideal.If U is not dense then there is an open subset V ⊂ X such that
U ∩ V = ∅ . There is a ∈ A such that and x ∈ V such that rep x ( a ) =
0. There is acontinuous function f V : X → [
0, 1 ] such that f V ( x ) = f V ( X \ V ) = { } ,247nd rep x ( f V a ) = rep x ( a ) =
0. So f V a =
0. Otherwise for all f ∈ C ( U ) from f V f = ( f V a ) ( f b ) = b ∈ A . On the other hand form theLemma D.8.25 it follows that A | U is the C ∗ -norm closure of C ( U ) A , so one has ( f V a ) A | U = { } , i.e. the ideal A | U is not essential. Lemma 5.3.3.
Let A be a continuous trace C ∗ -algebra, and let be (cid:16) A , e A , G , π (cid:17) be anoncommutative finite-fold quasi-covering. Suppose that there are a subgroup G ′ ⊂ Gand an open subset e U of the spectrum e X of e A such that • For all g ∈ G ′ and e x ∈ e U one has g e x = e x. • For all e a ∈ e A and e x ∈ e U the following condition holds ( rep e x ∑ g ∈ G ′ g e a !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e a ∈ e A ) = rep e x (cid:16) e A red (cid:17) where e A red is the reduced C ∗ -algebra (cf. Definition 2.3.12) of (cid:16) A , e A , G , π (cid:17) .If there is e x ∈ e U such that rep e x (cid:16) e A red (cid:17) $ rep e x (cid:16) e A (cid:17) then the set n g ∈ Aut (cid:16) e A (cid:17)(cid:12)(cid:12)(cid:12) ∀ e a red ∈ e A red g e a red = e a red o is not finite.Proof. From the Theorem D.8.19 it follows that e A contains an essential ideal e I which is a continuous trace C ∗ -algebra. From the Lemma 5.3.2 it follows that thespectrum e X e I of e I is a dense open subset of e X and rep e x (cid:16)e I (cid:17) = rep e x (cid:16) e A (cid:17) for all e x ∈ e X e I . It follows that e I = e A (cid:12)(cid:12)(cid:12) e X e I . The set e X e I is open and dense in e X , hence e U ′ def = e U ∩ e X e I is open and it is not empty. The C ∗ -algebra e A (cid:12)(cid:12)(cid:12) e U ′ is a closed twosided ideal of the continuous trace C ∗ -algebra e I , so from the Proposition D.8.10it follows that e A (cid:12)(cid:12)(cid:12) e U ′ is a continuous trace C ∗ -algebra. For any e x ∈ e A (cid:12)(cid:12)(cid:12) e U ′ thereis an Abelian element e e ∈ e A (cid:12)(cid:12)(cid:12) e U ′ such that rep e x ( e e ) = e V of e x such that e A (cid:12)(cid:12)(cid:12) e V e e | e V is an e A (cid:12)(cid:12)(cid:12) e V − e e | e V A | e V e e | e V ∼ = C (cid:16) e V (cid:17) -imprimitivity bimodule. Wedenote it by X C ( e V ) def = e A (cid:12)(cid:12)(cid:12) e V e e | e V . From the condition (b) of the Definition D.6.1 it248urns out that e A (cid:12)(cid:12)(cid:12) e V ∼ = K (cid:16) X C ( e V ) (cid:17) . From e A red $ e A it turns out that there is aHilbert submodule X k C ( e V ) def = A red | e V e e | e V . From the Equation (2.3.20) it follows that rep e x (cid:16) e A red (cid:17) = rep e x ( A ) so for any e x ∈ e V one has rep e x ∑ g ∈ G ′ e a ! ∈ rep e x (cid:16) e A red (cid:17) .If e a k def = | G ′ | ∑ g ∈ G ′ g e a and e a ⊥ def = e a − e a k then rep e x (cid:16)e a k (cid:17) ∈ rep e x (cid:16) e A red (cid:17) and rep e x (cid:16) ∑ g ∈ G ′ g e a ⊥ (cid:17) = e x ∈ e V . The natural decomposition e a = e a k + e a ⊥ yields the direct sum e A (cid:12)(cid:12)(cid:12) e V = e A red (cid:12)(cid:12)(cid:12) e V ⊕ e A (cid:12)(cid:12)(cid:12) e V⊥ which naturally induces the direct sum X C ( e V ) = X k C ( e V ) ⊕ X ⊥ C ( e V ) of Hilbert modules. Let e f e x : e X → [
0, 1 ] be a continuous map such that supp e f e x ⊂ e U ′′ and there is an open neighborhood e U which satisfies to the condi-tion e f e x (cid:16) e U (cid:17) = { } (cf. Lemma 4.1.5). There is a continuous map ϕ : e X → [
0, 1 ] such that ϕ ( e x ) = supp ϕ ⊂ e U . For any ǫ ∈ (
0, 2 π ) denote by ϕ ǫ ∈ C b (cid:16) e X (cid:17) , e x e i εϕ ( e x ) .There is the automorphism χ ε of the Hilbert C (cid:16) e V (cid:17) -module X C ( e V ) given by e ξ k + e ξ ⊥ e ξ k + ϕ ǫ e ξ ⊥ . This automorphism yields the *-automorphism ψ ǫ ∈ Aut (cid:16) K (cid:16) X C ( e V ) (cid:17)(cid:17) ∼ = Aut (cid:18) e A (cid:12)(cid:12)(cid:12) e V (cid:19) given by ∞ ∑ j = e ξ j ih e η j ∞ ∑ j = χ ε (cid:16) e ξ j (cid:17) ih χ ε (cid:0)e η j (cid:1) .If e U ′′ def = n e x ∈ e V (cid:12)(cid:12)(cid:12) e f e x ( e x ) > o then there the natural inclusion e A (cid:12)(cid:12)(cid:12) e U ′′ ⊂ e A (cid:12)(cid:12)(cid:12) e V suchthat for all ǫ the *-automorphism ψ ǫ corresponds to the *-automorphism θ ǫ ∈ Aut (cid:16) e A (cid:12)(cid:12)(cid:12) e U ′′ (cid:17) . Moreover If ǫ = ǫ then θ ǫ = θ ǫ . For any ǫ ∈ (
0, 2 π ) there is abijective C -linear map g ǫ : e A ≈ −→ e A given by e a (cid:16) − e f e x (cid:17) e a + θ ǫ (cid:16) e f e x e a (cid:17) .249ince for any e x ∈ e X the map rep e x (cid:16) e A (cid:17) → rep e x (cid:16) e A (cid:17) , rep e x ( e a ) → rep e x ( g ǫ ( e a )) ∀ e a ∈ e A is *-automorphism, the map g ǫ is ∗ -automorphism. For any e x ∈ e X consider theirreducible representation rep e x : e A → B ( H e x ) and ξ e x ∈ B ( H e x ) . From the TheoremD.2.6 it follows that H e x = rep e x : (cid:16) e A (cid:17) ξ e x . Denote by H k e x def = rep e x : (cid:16) e A red (cid:17) ξ e x andby H ⊥ e x the orthogonal complement of H k e x , i.e. H e x = H k e x ⊕ H ⊥ e x . There is a familyof Banach spaces n rep e x : (cid:16) e A ⊂ B (cid:16) H k e x ⊕ H ⊥ e x (cid:17)(cid:17)o e x ∈ e X . The space e X is Hausdorffso from the Theorem D.8.26 it follows that e A is a continuity structure for e X andthe family n rep e x : (cid:16) e A (cid:17)o e x ∈ e X (cf. Definition D.8.27). Any e a ∈ e A corresponds to afamily ne a e x ∈ B (cid:16) H k e x ⊕ H ⊥ e x (cid:17)o = (cid:26)(cid:18) α ( e a e x ) β ( e a e x ) γ ( e a e x ) δ ( e a e x ) (cid:19) ∈ B (cid:16) H k e x ⊕ H ⊥ e x (cid:17)(cid:27) where α ( e a e x ) ∈ B (cid:16) H k e x (cid:17) , β ( e a e x ) ∈ B (cid:16) H ⊥ e x , H k e x (cid:17) , γ ( e a e x ) ∈ B (cid:16) H k e x , H ⊥ e x (cid:17) , δ ( e a e x ) ∈ B (cid:0) H ⊥ e x (cid:1) . If e a red def = | G red | ∑ g ∈ G red g e a then e a red ∈ e A red and from the Equation (2.3.16)it follows that e a red corresponds to a family (cid:26)(cid:18) α ( e a e x ) δ ( e a e x ) (cid:19) ∈ B (cid:16) H k e x ⊕ H ⊥ e x (cid:17)(cid:27) e x ∈ e X . (5.3.1)The automorphism g ε is such that e a corresponds to a family ( α ( e a e x ) ϕ ǫ ( e x ) β ( e a e x ) ϕ ǫ ( e x ) γ ( e a e x ) δ ( e a e x ) !) e x ∈ e X , (5.3.2)so from the Equation (5.3.1) it follows that g ǫ ( e a red ) = e a red , i.e. g ǫ ∈ G red . Other-wise from ∀ ǫ , ǫ ∈ (
0, 2 π ) ǫ , = ǫ ⇒ ϕ ǫ ( e x ) = ϕ ǫ ( e x ) and taking into account (5.3.2) one has infinitely many different automorphisms g ε . Thus the group G red is not finite. 250 orollary 5.3.4. Let A be a continuous trace C ∗ -algebra, and let (cid:16) A , e A , G , π (cid:17) by anoncommutative finite-fold pre-covering. If e X is the spectrum of e A and e A red is the reducedalgebra of (cid:16) A , e A , G , π (cid:17) (cf. Definition 2.3.12) then the set e X k def = n e x ∈ e X (cid:12)(cid:12)(cid:12) rep b x (cid:16) e A red (cid:17) = rep b x (cid:16) e A (cid:17)o (5.3.3) is dense in e X Proof. If e X k is not dense in e X then there is nonempty open subset e U ⊂ e X \ e X k .For any e x ∈ e U denote by G e x def = { g ∈ G | g e x = x } . If n def = min e x ∈ e U | G e x | then fromthe Corollary 2.3.9 it turns out that the set e U n def = n e x ∈ e U (cid:12)(cid:12)(cid:12) | G e x | ≤ n o is open. If e x ∈ e U n and G ′ def = G \ G e x then for any g ∈ G ′ there is an openneighborhood e U g such that e U g ∩ g U g = ∅ . The set e U e x ⊂ ∩ g ∈ G ′ e U g is an openneighborhood of e x such that g e x = e x for all e x ∈ e U e x and g ∈ G ′ . It follows for each e x ∈ e U e x one has • G e x = G e x . • rep b x (cid:16) e A red (cid:17) $ rep b x (cid:16) e A (cid:17) It is easy check that rep e x ∑ g ∈ G e x g e a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e a ∈ e A = rep e x (cid:16) e A red (cid:17) for all e x ∈ e U . From the Lemma 5.3.3 it follows that n g ∈ Aut (cid:16) e A (cid:17)(cid:12)(cid:12)(cid:12) ∀ e a red ∈ e A red g e a red = e a red o is not finite, it contradicts with the Condition (a) of the Definition 2.1.5. So the set f U n is not open. The contradiction proves this lemma. Corollary 5.3.5.
Let A be a continuous trace C ∗ -algebra, and let (cid:16) A , e A , G , π (cid:17) by a non-commutative finite-fold pre-covering. Then for any Abelian element a ∈ A the generatedby π ( a ) subalgera e A a of e A is commutative. roof. If e X the spectrum of e A and e X k is given by the Equation 5.3.3 then e X k isa dense subset of e X (cf. Corollary 5.3.4). Otherwise from the Equation 2.3.20 itfollows that dim rep e x (cid:16) e A a (cid:17) ≤ e x ∈ e X k . From the Proposition D.2.28 itfollows that the set e X = n e x ∈ e X (cid:12)(cid:12)(cid:12) dim rep e x (cid:16) e A a (cid:17) ≤ o contains the closure of e X k which coincides with e X , i.e. e X = e X hence e A a iscommutative. Corollary 5.3.6.
Let A be a continuous trace C ∗ -algebra, and let be (cid:16) A , e A , G , π (cid:17) be anoncommutative finite-fold precovering then e A ia a continuous-trace C ∗ -algebraProof. Let e X be the spectrum of e A and let e x ∈ e X be a point. Let p : e X → X be thegiven by the Lemma 2.3.7 map from the spectrum of e A to the spectrum of A . If a ∈ A D.8.4 is an Abelian element such that rep p ( e x ) ( a ) = π ( a ) is Abelian such that rep e x ( π ( a )) =
0. From the DefinitionD.8.4 it follows that e A is a continuous-trace C ∗ -algebra. Lemma 5.3.7.
Let A be a continuous trace C ∗ -algebra, and let be (cid:16) A , e A , G , π (cid:17) be anoncommutative finite-fold covering with unitization (cf. Definition 2.1.13). Let A ֒ → B, e A ֒ → e B required by the Condition (a) the Definition 2.1.13. Let e x ∈ e X is a non-isolatedpoint then the ideal e B = ne b ∈ e B | rep e x (cid:16)e b (cid:17) = o is not finitely generated e B-module.Proof.
Assume there is a finite number of generators e b , ..., e b s of e B . Let e a ∈ e A + bean Abelian element of e A such that there is an open neighborhood e V of x whichsatisfies to the following conditiontr ( rep e x ( e a )) = k rep e x ( e a ) k = ∀ e x ∈ e V .There is e f ∈ C (cid:16) e X (cid:17) such that e f ( x ) = e f (cid:16) e V (cid:17) = { } . From e f e a ∈ e B itfollows that there are e c , ..., e c s ∈ e B such that e b e c + ... + e b s e c s = e f e a ,hence tr (cid:16) rep x (cid:16)e a e b e c + ... + e a e b s e c s (cid:17)(cid:17) = e f ( x ) ∀ x ∈ V .252t turns out that ∃ e x ∈ e V ∃ j ∈ {
1, ..., s } rep e x (cid:16)e a e b j (cid:17) = e a e b j ∈ A for any j =
1, ..., s it turns out that the map e f j : e X → R + given by e f j ( e x ) def = (cid:13)(cid:13)(cid:13) rep e x (cid:16)e a e b j (cid:17)(cid:13)(cid:13)(cid:13) lies in C (cid:16) e X (cid:17) + and from ab j ∈ B it follows that e f j ( e x ) =
0. If e f = q e f + ... + q e f s then from e f ( x ) = e a ′ def = e f e a ∈ e B , hence one has e a ′ = e b e a ′ + ... + e b s e a ′ s .The element e a ′ is Abelian andtr (cid:0) rep e x (cid:0)e a ′ (cid:1)(cid:1) = tr (cid:0) rep e x (cid:0)e a ′ e a (cid:1)(cid:1) = tr (cid:0) rep e x (cid:0)e a e a ′ e a (cid:1)(cid:1) = e f ( e x ) ; ∀ e x ∈ e V , ∃ j ∈ {
1, ..., s } ∃ e x ∈ e V rep e x (cid:16)e a ′ j e a (cid:17) = Λ def = n j ∈ {
1, ..., s }| e a ′ j e a = o Let us put m j = k e a ′ j e a k , j ∈ Λ .There is an open neighborhood e U ⊂ e X of e x such that the following inequalities q f j ( e x ) ≤ · m j hold for any e x ∈ e U , j ∈ Λ . In result one has the following contradiction e f ( e x ) = (cid:13)(cid:13) rep e x (cid:0)e a ′ (cid:1)(cid:13)(cid:13) ≤ n ∑ j = (cid:13)(cid:13)(cid:13) rep e x (cid:16)e a e b j (cid:17)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) rep e x (cid:16)e a ′ j e a (cid:17)(cid:13)(cid:13)(cid:13) ≤ ∑ j ∈ Λ e f j ( e x ) m j ≤≤ s ∑ j = q e f j ( e x ) · q e f j ( x ) m j ≤ e f ( e x ) ; ∀ e x ∈ e U . Remark 5.3.8.
The Lemma 5.3.7 is a generalization of the Lemma 3.2 of the article[56].
Lemma 5.3.9.
Let both A and e A are separable continuous trace C ∗ -algebras and there isthe inclusion A ⊂ e A such that. If X is the spectrum spectra of A then the inclusion A ⊂ e A induces a surjectivemap e X → X . • Both X and e X are compact. • The inclusion A ⊂ e A induces the inclusion M ( A ) ⊂ M (cid:16) e A (cid:17) .Let X ′ ⊂ X be a directed set { x α } together with a unique limit point x. Let e X ′ ⊂ e X beequal to the union of directed sets { e x α } and { e x α } with a common limit point e x, andp ( e x α ) = p ( e x α ) = x α , p ( e x ) = x . Then M (cid:16) e A (cid:17) is not a finitely generated module over M ( A ) .Proof. The surjective map e X ′ → X ′ yields an injective *-homomorphism C ( X ′ ) ֒ → C (cid:16) e X ′ (cid:17) . Consider two C ∗ -subalgebras C ( e X ′ ) and C ( e X ′ ) of the C ∗ -algebra C ( e X ′ ) ,where C ( e X ′ ) consists of those continuous functions, which are constant on { e x α } ∪ e x , and C ( e X ′ ) consists of those continuous functions, which are zero on { e x α } ∪ e x .Then any continuous function f on e X ′ can be represented in the unique way asthe sum f = f + f of the function f ∈ C ( e X ′ ) , which is equal to f on { e x α } ∪{ e x } , and the function f = f − f ∈ C ( e X ′ ) . Hence, C ( e X ′ ) = C ( e X ′ ) ⊕ C ( e X ′ ) .Clearly, C ( e X ′ ) is isomorphic to C ( X ′ ) , and C ( e X ′ ) is isomorphic to C ( X ′ ) as C ( X ′ ) -modules, where C ( X ′ ) consists of continuous functions vanishing at x .If both A | X ′ and e A (cid:12)(cid:12)(cid:12) e X ′ are given by the Equation D.2.8 then there are surjective*-homomorphisms A → A | X ′ , e A → e A (cid:12)(cid:12)(cid:12) e X ′ .From the Theorem D.1.34 it follows that there are are surjective *-homomorphisms M ( A ) → M (cid:16) A | X ′ (cid:17) , M (cid:16) e A (cid:17) → M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) ,Both X ′ and e X ′ are compact so from the Lemma 4.6.5 it follows that these spacesare spectra of both A | X ′ and e A (cid:12)(cid:12)(cid:12) e X ′ respectively. From the Theorems D.2.26 and254.8.9 it turns out that there are inclusions C ( X ′ ) ⊂ M (cid:16) A | X ′ (cid:17) , C (cid:16) e X ′ (cid:17) ⊂ M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) .The direct sum C ( e X ′ ) = C ( e X ′ ) ⊕ C ( e X ′ ) yields the direct sum M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) = M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) ⊕ M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) where M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) = C ( e X ′ ) M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) and M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) = C ( e X ′ ) M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) . Clearly there is the natural isomorphism M (cid:16) A | X ′ (cid:17) ∼ = M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) of M (cid:16) A | X ′ (cid:17) modules. Otherwise M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) ∼ = (cid:26) e a ∈ M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) rep e x ( e a ) = (cid:27) Thus, if M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) is finitely generated, then M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) is finitely generatedtoo. A contradiction with Lemma 5.3.7. Since M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) is not finitely generated M (cid:16) A | X ′ (cid:17) module, M (cid:16) e A (cid:17) is not a finitely generated module over M ( A ) . Remark 5.3.10.
The Lemma 5.3.9 is a generalization of the Lemma 3.3 of the article[56].
Lemma 5.3.11.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold covering with uniti-zation, such that A is a separable C ∗ -algebra with continuous trace. Suppose that bothspectra X , e X of both A, e A are compact and p : e X → X is the given by the Lemma 2.3.7continuous map. Suppose X ′ ⊂ X is a closed subset, e X ′ = p − ( X ′ ) , and M (cid:16) e A (cid:17) is afinitely generated M ( A ) -module. Then(i) M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) is a finitely generated M (cid:16) A | X ′ (cid:17) -module.(ii) M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′′ (cid:19) is a finitely generated M (cid:16) A | X ′ (cid:17) -module for any closed subset e X ′′ ⊂ e X ′ .Proof. Let e a , , ..., e a n be such that M (cid:16) e A (cid:17) = e a M ( A ) + ... + e a n M (cid:16) e A (cid:17) .255i) From the Equation D.2.8 it follows that there are surjective *-homomorphisms A → A | X ′ and e A → e A (cid:12)(cid:12)(cid:12) e X ′ . From the Theorem D.1.34 it follows that there are sur-jective *-homomorphisms p ′ : M ( A ) → M (cid:16) A | X ′ (cid:17) and e p ′ : M (cid:16) e A (cid:17) → M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) .For all j =
1, ..., n denote by e a ′ j def = e p ′ (cid:0)e a j (cid:1) . For any e a ′ ∈ M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) there is e a ∈ e A such that e a ′ = e p ′ ( e a ) . On the other hand there are a , ..., a n ∈ M ( A ) such that e a = e a a + ... + e a n a n . It turns out that e a ′ = e a ′ p ′ ( a ) + ... + e a ′ n p ′ ( a n ) ,i.e. M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) is M (cid:16) A | X ′ (cid:17) module generated by e a ′ , , ..., e a ′ n (ii) From the Equation D.2.8 it follows that there are surjective *-homomorphisms A | X ′ → A | X ′′ From the Theorem D.1.34 it follows that there is the surjective *-homomorphisms e p ′′ : M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) → M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′′ (cid:19) . For any e a ′′ ∈ M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′′ (cid:19) thereis e a ∈ e A such that e a ′′ = e p ′′ ◦ e p ′ ( e a ) . On the other hand there are a , ..., a n ∈ M ( A ) such that e a = e a a + ... + e a n a n . It turns out that e a ′ = e a ′ e p ′′ ◦ p ′ ( a ) + ... + e a ′ n e p ′′ ◦ p ′ ( a n ) ,i.e. M (cid:18) e A (cid:12)(cid:12)(cid:12) e X ′ (cid:19) is M (cid:16) A | X ′ (cid:17) module generated by e a ′ , , ..., e a ′ n Remark 5.3.12.
The Lemma 5.3.9 is a generalization of the Lemma 3.4 of the article[56].
Lemma 5.3.13.
Let (cid:16) A , e A , G , π (cid:17) be noncommutative finite-fold covering with unitiza-tion (cf. Definition 2.1.13). Suppose that A is a separable continuous trace C ∗ -algebra,with compact spectrum X . If p : e X → X is the given by the Lemma 2.3.7 continuousmap from the spectrum of e A to the spectrum of A then p is a finite-fold covering.Proof.
Assume k x = (cid:12)(cid:12) p − ( x ) (cid:12)(cid:12) denotes the cardinality of the pre-image of a point x ∈ X . From the Lemma 2.3.7 it follows that k x ≤ | G | for all x ∈ X . Supposethat k is a minimal value of k x ’s over x ∈ X . Let us select a point x with minimal k x . Firstly, we claim that the set X k = { x ∈ X : k x = k } is open. Indeed, inthe opposite case there is a net { x α } in X \ X k converging to a certain point x of256 k . By Lemma A.1.35 one can found a regular neighborhood U of x satisfying thecondition (A.1.2) with m = k , i.e. if p − ( x ) = { e x , ..., e x k } p − ( U ) = e U ⊔ · · · ⊔ e U k ; e x j ∈ e U j ∀ j =
1, ..., k .There is a closed neighborhood V of x such that V ⊂ U , so the following conditionholds p − ( V ) = e V ⊔ · · · ⊔ e V k ; e x j ∈ e V j = p − ( V ) ∩ e U j ∀ j =
1, ..., k . (5.3.4)Moreover, one can assume (passing to a sub-net of { x α } if it is necessary) thatthe net { x α } belongs to U and there is a number t such that the neighborhood e U t has at least two points e x ′ α and e x ′′ α from the pre-image of x α for any α . Put X ′ def = { x α } ∪ { x } and e X ′ def = { e x ′ α } ∪ { e x ′′ α } ∪ { e x } where f def = p − ( x ) ∩ e U t . Then C (cid:16) e X ′ (cid:17) is a finitely generated module over C ( X ′ ) by Lemma 5.3.11. But this contradictsto Lemma 5.3.9.Secondly, let us show that X k is closed. Indeed from the Corollary 2.3.9 itfollows that for any j > k the set X ′ k def = { x ∈ X | k x ≥ k + } is open, hence X k = X \ X ′ k is closed.So we have proved that the set X \ X k is both open and closed and, conse-quently, it has to be empty, because X is supposed to be connected. Thus, allpoints of X have the same number of pre-images, i.e. k x = k ; ∀ x ∈ X (5.3.5)Now for an arbitrary point x ∈ X let us choose its regular neighborhood U satisfying the condition (A.1.2) with m = k , i.e. p − ( U ) = e U ⊔ · · · ⊔ e U k ,The continuous surjection of compact Hausdorff spaces p : e X → X is in particular,a closed map. It turns out that X ′ = p (cid:16) e X \ S kj = e U j (cid:17) is closed. The set V = X \ X ′ is open. Moreover p − ( V ) = e V ⊔ · · · ⊔ e V k ; where e V j = e U j \ p − ( V ) and p (cid:16) e V j (cid:17) = V for every j =
1, ..., k , hence taking into account (5.3.5) the re-striction p e V j : e V j → V is a bijective map. Since p is closed p e V j : e V j → V is ahomeomorphism. So p is a finite-fold covering.257 emark 5.3.14. The Lemma 5.3.13 is an analog of the Theorem 4.4 of [56].
Corollary 5.3.15.
Let (cid:16) A , e A , G , π (cid:17) be a noncommutative finite-fold covering with uni-tization (cf. Definition 2.1.13) Suppose A is a separable C ∗ -algebra with continuous traceand the spectrum of A is compact. Let X and e X be the spectra of A and e A respectively.Then following conditions hold:(i) The given by the Lemma 2.3.7 map p : e X → X is a transitive finite-fold covering,and G ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) .(ii) There is the natural isomorphism e A ∼ = A (cid:16) e X (cid:17) (cf. Equation 4.6.5).Proof. (i) From the Lemma 5.3.13 it follows that p is a covering. From the Lemma2.3.7 it follows that there is a homeomorphism e X / G ∼ = X , hence G transitivelyacts on p − ( x ) for any x ∈ X . Thus from the Corollary 4.3.4 it turns out that thecovering p is transitive. Taking into account the e X / G ∼ = X one concludes that G ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) .(ii) There is the family n e A e x def = rep e x (cid:16) e A (cid:17)o e x ∈ e X of C ∗ -algebras. If there is e x ∈ X such that rep e x ( A ) $ rep e x (cid:16) e A (cid:17) then there is a nontrivial g ∈ G and e a ∈ e A suchthat g e x = e x and rep e x ( e a ) = rep e x ( g e a ) . But the condition g e x = e x contradictswith (i) of this Corollary. It follows that rep e x (cid:16) e A (cid:17) ∼ = rep x ( A ) for all e x ∈ X . Fromthe Lemma 4.5.39 the C ∗ -algebra A yields for continuity structure for e X and the n e A e x o e x ∈ e X . If A (cid:16) e X (cid:17) def = C (cid:16) lift p [ C ( X , { rep x ( A ) } ) , A ] (cid:17) (cf. Definition 4.5.40)then there is the inclusion A ֒ → A (cid:16) e X (cid:17) . Any e a ∈ e A corresponds to a family n rep e x (cid:16) e A (cid:17)o e x ∈ e X and from the inclusion A → e A it follows that for all a ∈ A thefamily given by lift p ( a ) corresponds to the element of e A . From the TheoremsD.2.26 D.8.9 it follows that for any e f ∈ C (cid:16) e X (cid:17) and a ∈ A the element e f lift p ( a ) given by the family n e f ( e x ) rep e x ( a ) o corresponds to an element of e A . From theRemark 4.5.44 it follows that A (cid:16) e X (cid:17) def = C (cid:16) lift p [ C ( X , { rep x ( A ) } ) , A ] (cid:17) ⊂ e A .If e a ∈ e A \ A (cid:16) e X (cid:17) and T (cid:16) e X , n e A x o , A (cid:17) is the total space for (cid:16) e X , n e A x o , A (cid:17) (cf. Definition 4.5.29) then from the Lemma 4.5.31 it follows that e a correspondsto a dicontinuous map j : e X → T (cid:16) e X , n e A x o , A (cid:17) such that k ◦ j = Id e X where k : T (cid:16) e X , n e A x o , A (cid:17) → e X is the total projection (cf. Definition 4.5.26). Suppose258hat j is not continuous at e x ∈ e X . If e f e x : e X → [
0, 1 ] is the given by the Equation(4.1.7) continuous function then g supp e f e x ∩ supp e f e x = ∅ for all nontrivial g ∈ G and there is an open neighborhood e U of e x such that f e x (cid:16) e U (cid:17) = { } . From theseproperties it turns out that the map j f : e X → T (cid:16) e X , n e A x o , A (cid:17) which correspondsto e f e x e a is not continuous at e x , so one has e f e x e a ∈ e A \ A (cid:16) e X (cid:17) . Clearly one has q e f e x e a ∈ e A \ A (cid:16) e X (cid:17) . However a = ∑ g ∈ G g (cid:18)q e f e x e a (cid:19) ∈ π ( A ) ⊂ e A . From theRemark 4.5.44 it follows that q e f e x a ∈ A (cid:16) e X (cid:17) , otherwise q e f e x a = e a ∈ A (cid:16) e X (cid:17) .It contradicts with e a ∈ e A \ A (cid:16) e X (cid:17) , hence from this contradiction it follows that e A \ A (cid:16) e X (cid:17) = ∅ and e A ∼ = A (cid:16) e X (cid:17) . Corollary 5.3.16.
Let A be a separable C ∗ -algebra with continuous trace and X is aspectrum of A. If X is compact there is 1-1 correspondence between transitive coverings e X → X and noncommutative finite-fold coverings (cid:16) A , e A , G , π (cid:17) with unitization (cf.Definition 2.1.13). The correspondence is given by (cid:16) p : e X → X (cid:17) (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , A ( p ) (cid:17) , (5.3.6) (cid:16) A , e A , G , π (cid:17) (cid:16) the given by the Equation (2.3.8) map p : e X → X (cid:17) . (5.3.7)
Proof.
Firstly we proof that the right part of 5.3.6 is a covering with unitiza-tion. Clearly (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , A ( p ) (cid:17) is a noncommutative finite-foldprecovering. From the Theorems D.2.26 D.8.9 it follows that there is the inclu-sion C ( X ) → M ( A ) and C (cid:16) e X (cid:17) → M (cid:16) A (cid:16) e X (cid:17)(cid:17) . Let both B and e B are C ∗ -subalgebras of M ( A ) and M (cid:16) e A (cid:17) generated by A ∪ C ( X ) and A (cid:16) e X (cid:17) ∪ C (cid:16) e X (cid:17) respectively. The *-homomorphism M ( A ) ֒ → M (cid:16) e A (cid:17) naturally induces the in-jective *-homomorphism e π : B ֒ → e B . The quadruple (cid:16) B , e B , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , e π (cid:17) is aprecovering. The map p : e X → X is a transitive finite-fold covering and bothspaces X and e X are compact, hence from the Remark 4.8.7 it follows that there isa finite sum 1 C ( e X ) = ∑ e α ∈ f A e e e α e e e α ∈ C (cid:16) e X (cid:17) (5.3.8)259uch that for any nontrivial g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) one has e e e α ( g e e e α ) =
0. (5.3.9)If e a ∈ e B then from (5.3.8) it follows that e a = ∑ e α ∈ f A e e e α ( e e e α e a ) (5.3.10)From (5.3.9) it follows that if g ∈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is not trivial then e e e α ( g ( e e e α e a )) = e e e α ( g e e e α ) ( g e α ) = · ( g e α ) =
0. (5.3.11)From the equations 5.3.10 and 5.3.10 it follows that e a = ∑ e α ∈ f A e e e α a α where a α def = ∑ g ∈ G ( e X | X ) g ( e e e α e a ) ∈ B ,i.e. e B is a finitely generated B -module. So (cid:16) B , e B , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , e π (cid:17) is an unital non-commutative finite fold covering (cf. Definition 2.1.9). From B ∩ A (cid:16) e X (cid:17) = A and G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × A (cid:16) e X (cid:17) = A (cid:16) e X (cid:17) it follows that (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , A ( p ) (cid:17) is a noncommutative finite-fold covering with unitization.From Corollary 5.3.15 it follows that the image of the map (5.3.7) containsfinite-fold transitive coverings only. Corollary 5.3.17.
Let A be a separable continuous trace C ∗ -algebra such that the spectrum X of A is compact. If U ⊂ X is a dense open subspace and A | U ⊂ A is the given by (D.2.7) two sided ideal. For any finite-fold transitive covering p : e X → X the quadruple (cid:18) A | U , A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) p − ( U ) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , A ( p ) | A | U (cid:19) (5.3.12) is a noncommutative finite-fold covering with unitization (cf. Definition 2.1.13).Proof. From the Corollary (5.3.16) if follows that there is a noncommutative finite-fold covering with unitization (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , A ( p ) (cid:17) , which in par-ticular is a noncommutative finite-fold precovering (cf. Definition 2.1.5). Fromthe Lemma 5.3.2 it follows that both A | U and A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) p − ( U ) are essential ideals of260oth A , hence A (cid:16) e X (cid:17) the quadruple (5.3.12) is also a noncommutative finite-foldprecovering. From the Definition 2.1.13 it follows that there is an unital noncom-mutative finite-fold covering (cid:16) B , e B , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , e π (cid:17) such that both A and A (cid:16) e X (cid:17) are essential ideals of both B and e B . Let us prove that (cid:16) B , e B , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , e π (cid:17) satis-fies to the conditions (a), (b) of the Definition 2.1.13 with respect to the quadruple(5.3.12), i.e.(a) Both A | U and A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) p − ( U ) are essential ideals of both A and A (cid:16) e X (cid:17) , itturns out that both A | U and A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) p − ( U ) are essential ideals of both B and e B .(b) There quadruple (cid:16) B , e B , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , e π (cid:17) unital noncommutative finite-fold cov-ering. such that A ( p ) | A | U = e π | A | U and the action G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) p − ( U ) → A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) p − ( U ) is induced by the action G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) × e B → e B . Theorem 5.3.18.
Let A be a separable continuous trace C ∗ -algebra and X is a spectrumof A. Suppose that X be a locally compact, connected, locally connected, second-countable,Hausdorff space. There is 1-1 correspondence between finite-fold transitive coverings e X →X with and finite-fold noncommutative coverings (cid:16) A , e A , G , π (cid:17) . The correspondence isgiven by (cid:16) p : e X → X (cid:17) (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , A ( p ) (cid:17) , (5.3.13) (cid:16) A , e A , G , π (cid:17) (cid:16) the given by the Equation (2.3.8) map p : e X → X (cid:17) . (5.3.14)
Proof.
For our proof we need the finite or countable sequence U $ ... $ U n $ ...of connected open subsets of X given by the Lemma 4.3.33 which satisfies to thefollowing conditions • For any n ∈ N the closure V n of U n is compact. • ∪ U n = X . • The space p − ( U n ) is connected for any n ∈ N .261irstly proof that the image of the map (5.3.13) contains finite-fold noncommu-tative coverings only. Clearly the quadruple (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , A ( p ) (cid:17) is anoncommutative finite-fold precovering (cf. Definition 2.1.5). If A | V n is given by(D.2.8) then from the the Proposition D.8.10 it follows that A | V n is a continuoustrace C ∗ -algebra. Moreover from the Lemma 4.6.5 it turns out that the spectrumof A | V n equals to V n . If e V n def = p − ( V n ) then p | e V n : e V n → V n is a finite-fold transi-tive covering. From the Corollary 5.3.16 it follows that there is a noncommutativefinite-fold covering with unitization (cid:18) A | V n , A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) p − ( V n ) , G (cid:16) e V n (cid:12)(cid:12)(cid:12) V n (cid:17)(cid:19) From the Corollary 5.3.17 it follows that A | V n (cid:12)(cid:12)(cid:12) U n , A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) p − ( V n ) (cid:12)(cid:12)(cid:12)(cid:12) p − ( U n ) , G (cid:16) e U n (cid:12)(cid:12)(cid:12) U n (cid:17)! == (cid:18) A | U n , A (cid:16) e X (cid:17) (cid:12)(cid:12)(cid:12) p − ( U n ) , G (cid:16) e U n (cid:12)(cid:12)(cid:12) U n (cid:17)(cid:19) is a noncommutative finite-fold covering with unitization. From the Remark 4.8.2it follows that there is the increasing sequence (cid:8) u n ∈ C c ( X ) + (cid:9) n ∈ N such that1 C b ( X ) = β - lim n → ∞ u n supp u n = V n (5.3.15)with respect to the strict topology on the multiplier algebra M ( C ( X )) (cf. Defini-tion D.1.12). The limit can be regarded as a limit with respect to strict topology of M ( A ) . From the Lemma D.8.25 it follows that both C ∗ -algebras A | U n and e A (cid:12)(cid:12)(cid:12) e U n are C ∗ -norm completions of u n Au n and u n e Au n respectively. So right part of (5.3.13)satisfies to the Definition 2.1.17, i.e. it is a finite-fold noncommutative covering.Let us prove that the right part of the Equation (5.3.14) contains finite foldcoverings only. Consider an increasing net { u λ } λ ∈ Λ ⊂ M ( A ) + and coverings withunitizations (cid:16) A λ , e A λ , G , π | A λ (cid:17) required by the Definition 2.1.17. For all λ ∈ Λ thesubalgebra A λ ⊂ A is hereditary (cf. Remark 2.1.20), so from the PropositionD.8.10 is a C ∗ -algebra with continuous trace. If both X λ and e X λ are spectra of A λ and e A λ which are open subsets of X and e X (cf. Proposition D.2.20). From β - lim λ ∈ Λ u λ = M ( A ) and the Lemma 2.1.2 it follows that { u λ } is an approximateunit for M (cid:16) e A (cid:17) , so the union ∪ A λ is dense in A . For any x ∈ X there is a ∈ A (cid:13)(cid:13) rep x ( a ) (cid:13)(cid:13) >
1. Otherwise since ∪ A λ is dense in A there is λ ∈ Λ and a ∈ A λ such that k a − a k < (cid:13)(cid:13) rep x ( a ) (cid:13)(cid:13) > x ∈ X λ . It turns out that X = ∪X λ . Similarly one can prove that e X = ∪ e X λ . Forany n ∈ N the set V n is compact, hence from X = ∪X λ there is λ n ∈ Λ such that V n ⊂ X λ n . If A λ | V n the V n -restriction (cf. 4.6.4 and the Definition 4.5.32) then fromthe Lemma 4.6.5 it follows that A λ | V n is a C ∗ -algebra with continuous trace andthe spectrum of e A λ (cid:12)(cid:12)(cid:12) e V n equals to e V n . (cf. Lemma 4.6.5) .From the Corollary 5.3.16there is a noncommutative finite-fold covering with unitization (cid:18) A λ | V n , e A λ (cid:12)(cid:12)(cid:12) e V n , G (cid:16) e V n (cid:12)(cid:12)(cid:12) V n (cid:17)(cid:19) From the Corollary 5.3.16 it turns out that the given by the Lemma 2.3.7 map p | e V n : e V n → V n is a finite-fold transitive covering. From X = ∪V n and e X = ∪ e V n itfollows that p : e X → X is a transitive covering.
Corollary 5.3.19.
Let A be a separable, continuous trace C ∗ -algebra such that the spec-trum X of A is connected, locally connected, second-countable, Hausdorff space. Denoteby FinCov - X a category of transitive finite-fold coverings of X given by the Definition4.3.25. Denote by FinCov -A a category of finite-fold coverings of A, i. e. • Objects of
FinCov -A are noncommutative finite-fold coverings of A. • If π : A → e A and π : A → e A then the morphism from π to π is a transitivefinite-fold covering covering ρ : e A → A such that the following diagram commu-tativeA e A A ρπ π There is the natural equivalence of the categories A : FinCov - X ∼ = FinCov -A.Proof.
From the Theorem 5.3.18 it follows that there is a one-to-one correspondencebetween objects of
FinCov - X and objects of FinCov - A given by (cid:16) p : e X → X (cid:17) A ( p ) : A ( X ) ֒ → A (cid:16) e X (cid:17) A ( p ) is given by the Definition 5.2.1. If p : e X → X and p : e X → X are objects of FinCov - X and p : e X → e X is a morphism of FinCov - X then fromthe Theorem 5.3.18 it turns out that p is a transitive finite-fold covering. From theequation 5.2.2 it follows that p corresponds to the *-homomorphism A ( p ) : A (cid:16) e X (cid:17) → A (cid:16) e X (cid:17) . (5.3.16)So one has a functor from FinCov - X to FinCov - A . Let us define the inverse functor.From the Theorem 5.3.18 it follows that any object π : A ( X ) ֒ → A (cid:16) e X (cid:17) of FinCov - A corresponds to a transitive finite-fold covering p : e X → X such that π = A ( p ) . If A ( p ) : A ( X ) → A (cid:16) e X (cid:17) and A ( p ) : A ( X ) → A (cid:16) e X (cid:17) areobjects of FinCov - A and π : A (cid:16) e X (cid:17) → A (cid:16) e X (cid:17) is a *-homomorphism such that A ( p ) = π ◦ A ( p ) then π corresponds to a continuous map p : e X → e X suchthan p = p ◦ p . So one has the inverse functor from FinCov - A to FinCov - X . Let ( X , Y ) be a sub-unital subspace having continuous trace (cf. Definition5.1.2) and let A def = C ∗ e ( X , Y ) be the C ∗ -envelope of ( X , Y ) . From the Lemma 5.1.3it follows that A has continuous trace, hence the spectrum X of A is the locallycompact Hausdorff space. Let p : e X → X be a transitive covering, denote by e X def = C (cid:16) lift p [ X ] (cid:17) , e A def = C (cid:16) lift p [ A ] (cid:17) = A (cid:16) e X (cid:17) (cf. Equation 5.2.2), e Y def = e X + C · e A + . (5.3.17)(cf. Equations 4.5.9, 4.5.46). If the covering p is finite-fold then there are thenatural injective *-homomorphism ρ : A → e A and the complete unital isometry ( π X , π Y ) : ( X , Y ) ֒ → (cid:16) e X , e Y (cid:17) from ( X , Y ) to (cid:16) e X , e Y (cid:17) (cf. Definition 2.6.3). Theorem 5.3.21.
Let us consider the given by 5.3.20 situation. If X is a connected, locallyconnected and second-countable space then there is the 1-1 map from the set of finite-foldtransitive coverings p : e X → X (2.6.5) and noncommutative finite-fold coverings of thesubunital operator space ( X , Y ) given by (cid:16) p : e X → X (cid:17) (cid:16) ( X , Y ) , (cid:16) e X , e Y (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , ( π X , π Y ) (cid:17) . (5.3.18)264 roof. Firstly we proof that (cid:16) ( X , Y ) , (cid:16) e X , e Y (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , ( π X , π Y ) (cid:17) is a finite-foldcovering for each transitive finite-fold covering p : e X → X . One needs check (a)and (b) of the Definition 2.6.5.(a) From the Theorem 5.3.18 it turns out that (cid:16) C ∗ e ( X , Y ) , e A , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , C ( p ) (cid:17) a finite-fold noncommutative covering. From X x ⊂ A x for any x ∈ X , e X e x ∼ = X p ( e x ) and e A e x ∼ = A p ( e x ) it follows that C (cid:16) e X e x , e Y e x (cid:17) = rep e x (cid:16) e A (cid:17) for all e x ∈ e X (cf. notationof the Definition 5.1.2). Applying the Lemma 5.1.3 one has e A = C ∗ e (cid:16) e X , e Y (cid:17) , clearlythe following condition holds π Y = ρ | Y , π X = ρ | X .(b) If e X ′ ⊂ C ∗ e (cid:16) e X , e Y (cid:17) is a C -linear space such that X = e X ′ ∩ C ∗ e ( X , Y ) and G e X ′ = e X ′ then from e X e x ∼ = X p ( e x ) it follows that e X ′ e x ⊂ e X e x for all e x ∈ e X . From the Lemma4.5.50 it follows that e X ′ ⊂ e X .Secondly we proof that any noncommutative finite-fold covering (cid:16) ( X , Y ) , (cid:16) e X , e Y (cid:17) , G , ( π X , π Y ) (cid:17) of the subunital operator space ( X , Y ) yields the transitive finite-fold covering p : e X → X . From (a) of the Definition 2.6.5 it follows that (cid:16) ( X , Y ) , (cid:16) e X , e Y (cid:17) , G , ( π X , π Y ) (cid:17) corresponds to noncommutative covering (cid:16) C ∗ e ( X , Y ) , C ∗ e (cid:16) e X , e Y (cid:17) , G , ρ (cid:17) of C ∗ -algebras.However C ∗ e ( X , Y ) has continuous trace, so and taking into account the Theorem5.3.18 on can find required covering p : e X → X . Let us consider a compact Riemannian manifold M with a spinor bundle S . Thealgebra C ( M ) ⊗ M n ( C ) is a homogeneous of order n C ∗ -algebra (cf. DefinitionD.8.20). If p : e M → M is a finite-fold transitive covering then from the Theorem5.3.18 it follows that there is a noncommutative finite-fold covering (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) (5.3.19)265f ϕ fin : M m ( C ) → B (cid:0) C k (cid:1) is a faithful representation then there is the action C ( M ) ⊗ M m ( C ) × C (cid:16) M , n S x ⊗ C k o , Γ ( M , S ) ⊗ C k (cid:17) →→ C (cid:16) M , n S x ⊗ C k o , Γ ( M , S ) ⊗ C k (cid:17) (5.3.20)such that from a = a ′ ⊗ T , a ′ ∈ C ( M ) T ∈ M m ( C ) , ξ = ξ ′ ⊗ x , ξ ′ ∈ Γ ( M , S ) x ∈ C k it follows that a ξ = a ′ ξ ′ ⊗ ϕ fin ( T ) x There is the alternative description of the above action. The C ∗ -algebra C ( M ) ⊗ M m ( C ) is a continuity structure for M and the family { M m ( C ) x } x ∈X such that C ( M ) ⊗ M m ( C ) ∼ = C ( M , { M m ( C ) x } , C ( M ) ⊗ M m ( C )) For any x ∈ X the representation ϕ fin induces the action M m ( C ) x × S x ⊗ C k →S x ⊗ C k such that a x ( s x ⊗ y ) = s ⊗ ϕ fin ( a x ) y , ∀ a x ∈ M m ( C ) x , ∀ s x ∈ S x , ∀ y ∈ C k . (5.3.21)If a ∈ C ( M , { M m ( C ) x } , C ( M ) ⊗ M m ( C )) corresponds to a family { a x ∈ M m ( C ) } x ∈X and ξ ∈ C (cid:0) M , (cid:8) S x ⊗ C k (cid:9) , Γ ( M , S ) ⊗ C k (cid:1) is corresponds to a family (cid:8) ξ x ∈ S x ⊗ C k (cid:9) x ∈X then the product a ξ corresponds to the family (cid:8) a x ξ x ∈ S x ⊗ C k (cid:9) x ∈X where the ac-tion (5.3.21) is implied. From the Lemma 4.8.13 it follows that there is the followingisomorphism (cid:16) C (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) ⊗ C ( M ) ⊗ M m ( C ) (cid:16) C (cid:16) M , n S x ⊗ C k o , Γ ( M , S ) ⊗ C k (cid:17)(cid:17) ≈ −→ ≈ −→ lift p h(cid:16) C (cid:16) M , n S x ⊗ C k o , Γ ( M , S ) ⊗ C k (cid:17)(cid:17)i (5.3.22)of left C (cid:16) e M (cid:17) ⊗ M m ( C ) -modules. Note that S x ⊗ C k ∼ = S kx and taking into accountLemmas 4.5.4 and 4.5.59 the given by (5.3.22) isomorphism can be represented bythe following way (cid:16) C (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) Γ (cid:16) M , S k (cid:17) ≈ −→ Γ (cid:16) e M , e S k (cid:17) (5.3.23)266here e S is the inverse image of S . For any x ∈ X the space S x is a Hilbert space,so S kx = S x ⊗ C k has the natural structure of Hilbert space which is the finitedirect product of Hilbert spaces. It follows that S k is a Hermitian vector bundle(cf. Definition A.3.11) and L (cid:0) M , S k (cid:1) ∼ = L ( M , S ) k is a Hilbert space with thegiven by (A.3.3) scalar product, i.e. ( ξ , η ) L ( M , S k ) def = Z M ( ξ x , η x ) x d µ . (5.3.24) Lemma 5.3.22.
If p : e M → M is a finite-fold transitive covering and e S is the inverseimage of S by p (cf. Definition A.3.7) then the following conditions hold:(i) The map (5.3.23) can be extended up to the following homomorphism (cid:16) C (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) L (cid:16) M , S k (cid:17) ≈ −→ L (cid:16) e M , e S k (cid:17) of left C (cid:16) e M (cid:17) ⊗ M m ( C ) -modules.(ii) The image of (cid:16) C (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) L (cid:0) M , S k (cid:1) is dense in L (cid:16) e M , e S k (cid:17) .Proof. There is the natural inclusion of C ∗ -algebras C (cid:16) e M (cid:17) → C (cid:16) e M (cid:17) ⊗ M m ( C ) , e a e a ⊗ M m ( C ) .Using it one can prove this lemma as well as the Lemma 4.10.11 had been proved. Remark 5.3.23.
Indeed the image of C (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) coincides with L (cid:16) e M , e S (cid:17) .However this fact is not used here.Let both µ and e µ be Riemannian measure (cf. [26]) on both M and e M respec-tively which correspond to both the volume element (cf. (E.4.1)) and its p -lift(cf. (4.9.2)). If A def = C ( M ) ⊗ M m ( C ) , e A def = C (cid:16) e M (cid:17) ⊗ M m ( C ) and e a ⊗ ξ , e b ⊗ η ∈ e A ⊗ A Γ (cid:0) M , S k (cid:1) ⊂ L (cid:16) e M , e S k (cid:17) then the given by the Equation (2.3.1) scalar product267 · , · ) ind on e A ⊗ A Γ (cid:0) M , S k (cid:1) satisfies to the following equation (cid:16)e a ⊗ ξ , e b ⊗ η (cid:17) ind = (cid:18) ξ , De a , e b E C ( e M ) η (cid:19) L ( M , S k ) == ∑ e α ∈ f A (cid:18) ξ , De a e α e a , e b E C ( e M ) η (cid:19) L ( M , S k ) = ∑ e α ∈ f A (cid:16) ξ , desc (cid:16)e a e α e a ∗ e b (cid:17) η (cid:17) L ( M , S k ) == ∑ e α ∈ f A Z M (cid:16) ξ x , desc (cid:16)e a e α e a ∗ e b (cid:17) η x (cid:17) x d µ == ∑ e α ∈ f A Z M (cid:16) desc ( e e e α e a ) ξ x , desc (cid:16)e e e α e b (cid:17) η x (cid:17) x d µ == ∑ e α ∈ f A Z e M (cid:16)e a lift e U e α ( e e α ξ ) e x , e b lift e U e α ( e e α η ) e x (cid:17) e x d e µ == ∑ e α ∈ f A Z e M e a e α (cid:16)e a lift e U e α ( ξ ) e x , e b lift e U e α ( η ) e x (cid:17) e x d e µ == Z e M (cid:16)e a lift p ( ξ ) e x , e b lift p ( η ) e x (cid:17) e x d e µ = (cid:16)e a lift p ( ξ ) , e b lift p ( η ) (cid:17) L ( e M , e S k ) == (cid:16) φ ( e a ⊗ ξ ) , φ (cid:16)e b ⊗ η (cid:17)(cid:17) L ( e M , e S k ) (5.3.25)where φ is given by (4.8.19). The equation (5.3.25) means that ( · , · ) ind = ( · , · ) L ( e M , e S k ) ,and taking into account the dense inclusion C (cid:16) e M (cid:17) ⊗ C ( M ) Γ (cid:0) M , S k (cid:1) ⊂ L (cid:16) e M , e S k (cid:17) with respect to the Hilbert norm of L (cid:16) e M , e S k (cid:17) one concludes that the space ofinduced representation coincides with L (cid:16) e M , e S k (cid:17) . It means that induced repre-sentation C (cid:16) e M (cid:17) × L (cid:16) e M , e S k (cid:17) → L (cid:16) e M , e S k (cid:17) is given by (A.3.4). So one has thefollowing lemma. Lemma 5.3.24.
If C ( M ) ⊗ M m ( C ) → B (cid:0) L (cid:0) M , S k (cid:1)(cid:1) is the described above represen-tation and the representation e ρ : C (cid:16) e M (cid:17) ⊗ M m ( C ) → B (cid:16) e H (cid:17) is induced by the pair (cid:16) ρ , (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17)(cid:17) (cf. Definition 2.3.1) then following conditions holds(a) There is the homomorphism of Hilbert spaces e H ∼ = L (cid:16) e M , e S k (cid:17) ,(b) The representation e ρ is described above action of C (cid:16) e M (cid:17) ⊗ M m ( C ) on L (cid:16) e M , e S k (cid:17) . roof. (a) Follows from (5.3.25),(b) From the Lemma 5.3.22 it follows that the map (cid:16) C (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) L (cid:0) M , S k (cid:1) ≈ −→ L (cid:16) e M , e S k (cid:17) is the homomorphismof left C (cid:16) e M (cid:17) ⊗ M m ( C ) modules, so the given by (A.3.4), i.e. C (cid:16) e M (cid:17) ⊗ M m ( C ) -action coincides with the C (cid:16) e M (cid:17) ⊗ M m ( C ) -action the given by (2.3.2). Remark 5.3.25.
If the spectral triple e T def = (cid:16) C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) , L (cid:16) e M , e S (cid:17) ⊗ C k , e D / ⊗ Γ fin + Id L ( e M , e S ) ⊗ D fin (cid:17) is the geometrical p -lift of the spectral triple (cid:16) C ∞ ( M ) ⊗ M m ( C ) , L ( M , S ) ⊗ C k , D / ⊗ Γ fin + Id L ( M , S ) ⊗ D fin (cid:17) .(cf. Definition 5.2.3), then clearly the corresponding to e T representation of C ( M ) ⊗ M m ( C ) equals to the given by the Lemma 5.3.24 representation. Consider the situation of the Section 5.3.3. Here the notation of the Section isused. Clearly one has C ( M ) ⊗ M m ( C ) ∼ = m M r = s = C ( M ) ⊗ e rs (5.3.26)where e rs are elementary matrices (cf. Definition 1.4.3). Lemma 5.3.26.
Let p : e M → M be a finite-fold transitive covering. Consider the givenby the Corollary 4.10.3 unital finite-fold noncommutative coverings (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) , given the Corollary 4.10.3. If the family { e e e α } e α ∈ f A ⊂ C ∞ (cid:16) e M (cid:17) is the given by (4.10.13) and { e e e α ⊗ e rs } e α ∈ f A ; r , s = m ⊂ C ∞ ( M ) ⊗ M m ( C ) (5.3.27) then the following conditions hold 4.10.15(i) The finite family (5.3.27) satisfies to the Lemma 2.7.3. ii) C (cid:16) e M (cid:17) ⊗ M m ( C ) ∩ M m | f A | ( C ∞ ( M ) ⊗ M m ( C )) == C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) . (5.3.28) (iii) The given by the the Theorem 5.3.18 unital noncommutative finite-fold covering (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is smoothly invariant (cf. Definition 2.7.2).Proof. (i) For any e a ∈ C (cid:16) e M (cid:17) from (4.10.13) it follows that C (cid:16) e M (cid:17) = ∑ e α ∈ f A e e e α C ( M ) .and taking into account (5.3.26) one has C ( M ) ⊗ M m ( C ) ∼ = m ∑ r = s = ∑ e α ∈ f A ( e e e α ⊗ e rs ) C ( M ) ⊗ M m ( C ) i.e. the right C ( M ) ⊗ M m ( C ) -module C (cid:16) e M (cid:17) ⊗ M m ( C ) is generated by the finiteset (5.3.27). Let us prove that the family (5.3.27) satisfies to conditions (a) and (b)of the Lemma 2.7.3.(a) From the proof of the Lemma 5.3.26 it turns out that (cid:10)e e e α ′ , e e f α ′′ (cid:11) C ( e M ) = ∑ g ∈ G ( e M | M ) g (cid:16)e e ∗ e α ′ , e e f α ′′ (cid:17) ∈ C ∞ ( M ) for all e α ′ , f α ′′ ∈ f A , and taking into account that elements 1 C ( e M ) ⊗ e rs are G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -invariant one has (cid:10)e e e α ′ ⊗ e rs , e e f α ′′ ⊗ e vw (cid:11) C ( e M ) ) ⊗ M m ( C ) = (cid:10)e e e α ′ , e e f α ′′ (cid:11) C ( e M ) ⊗ e rs e vw ∈∈ C ∞ ( M ) ⊗ M m ( C ) .(b) From the Lemma 4.10.15 it follows that the given by (4.10.13) family { e e e α } e α ∈ f A is G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -invariant, hence the family (5.3.27) is also G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -invariant.270ii) If the following condition holds e a ⊗ b ∈ C (cid:16) e M (cid:17) ⊗ M m ( C ) ∩ M m | f A | ( C ∞ ( M ) ⊗ M m ( C )) then from (i) of the Lemma 2.7.3 it follows that (cid:10)e e e α ′ ⊗ e rs , ( e a ⊗ b ) e e f α ′′ ⊗ e vw (cid:11) C ( e M ) ) ⊗ M m ( C ) ∈ C ∞ ( M ) ⊗ M m ( C ) ; ∀ e α ′ , f α ′′ ∈ f A , ∀ r , s , w , v =
1, ..., m .On the other hand from (5.3.26) it follows that e a ⊗ b = m ∑ r = s = e a rs ⊗ e rs , (5.3.29)so one has (cid:10)e e e α ′ ⊗ e rr , ( e a ⊗ b ) e e f α ′′ ⊗ e ss (cid:11) C ( e M ) ) ⊗ M m ( C ) = (cid:10)e e e α ′ , e a rs e e f α ′′ (cid:11) C ( e M ) ⊗ e rs ∈∈ C ∞ ( M ) ⊗ M m ( C ) . (5.3.30)From (5.3.30) and the Lemma 5.3.26 it follows that e a rs ∈ C ∞ (cid:16) e M (cid:17) and taking intoaccount (5.3.29) following condition holds e a ⊗ b ∈ C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) ,i.e. C (cid:16) e M (cid:17) ⊗ M m ( C ) ∩ M m | f A | ( C ∞ ( M ) ⊗ M m ( C )) ⊂ C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) .Conversely if e a ⊗ b ∈ C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) and e a ⊗ b is given by (5.3.29) then fol-lowing condition holds (cid:10)e e e α ′ ⊗ e rs , ( e a ⊗ b ) e e f α ′′ ⊗ e vw (cid:11) C ( e M ) ) ⊗ M m ( C ) = (cid:10)e e e α ′ , e a sv e e f α ′′ (cid:11) C ( e M ) ⊗ e rs e vw . (5.3.31)On the other hand from the Lemma 5.3.26 it follows that (cid:10)e e e α ′ , e a sv e e f α ′′ (cid:11) C ( e M ) ∈ C ∞ ( M ) , hence (cid:10)e e e α ′ ⊗ e rs , ( e a ⊗ b ) e e f α ′′ ⊗ e vw (cid:11) C ( e M ) ) ⊗ M m ( C ) ∈ C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) , andtaking into account (5.3.29) one has e a ⊗ b ∈ C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) ,271.e. C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) ⊂ C (cid:16) e M (cid:17) ⊗ M m ( C ) ∩ M m | f A | ( C ∞ ( M ) ⊗ M m ( C )) .(iii) From (ii) the Lemma 2.7.3.Consider the given by (5.2.4) product of spectral triples (cid:16) C ∞ ( M ) ⊗ M m ( C ) , L ( M , S ) ⊗ C k , D / ⊗ Γ fin + Id L ( M , S ) ⊗ D fin (cid:17) and let D be an unbounded operator on L ( M , S ) ⊗ C k given by D = D / ⊗ Γ fin + Id L ( M , S ) ⊗ D fin . (5.3.32)From the evident equations h D , a ⊗ M m ( C ) i = [ D / , a ] ⊗ Γ fin ∀ a ∈ C ∞ ( M ) , h D , 1 C ( e M ) ⊗ b i = Id L ( M , S ) ⊗ [ D fin , b ] ∀ b ∈ M m ( C ) .it follows that [ D / , a ] ⊗ Γ fin , Id L ( M , S ) ⊗ [ D fin , b ] ∈ Ω D ; ∀ a ∈ C ∞ ( M ) ∀ b ∈ M m ( C ) . (5.3.33)where Ω D ⊂ B (cid:0) L ( M , S ) ⊗ C k (cid:1) is the module of differential forms associated withthe spectral triple (5.2.4) (cf. Definition E.3.5). Similarly to (4.8.20) one can definea C ∞ (cid:16) e M (cid:17) -module homomorphism φ ∞ : C ∞ (cid:16) e M (cid:17) ⊗ M n ( C ) ⊗ C ∞ ( M ) ⊗ M n ( C ) Γ ∞ ( M , S ) ≈ −→ Γ ∞ (cid:16) e M , e S (cid:17) ; m ∑ j = e a j ⊗ ξ j e a j lift p (cid:0) ξ j (cid:1) . (5.3.34)where both Γ ∞ ( M , S ) and Γ ∞ (cid:16) e M , e S (cid:17) are defined by the Equations (E.4.2) and(4.9.4) respectively. Lemma 5.3.27.
The given by the Equation (4.10.20) homomorphism is an isomorphism.Proof.
The proof of the Lemma 4.8.9 uses the partition of unity. However from theProposition A.1.26 it turns out that there is a smooth partition of unity. Using itone can proof this lemma as well as the Lemma 4.8.9 has been proved.272ow we have the G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -equivariant connection e ∇ , so on can find thespecialization of explained in 2.7.5 construction of the operator e D : e A ⊗ A H ∞ → e A ⊗ A H ∞ . Following table reflects the mapping between general theory and thecommutative specializationGeneral theory Commutative specializationHilbert spaces H and e H L ( M , S ) k and L (cid:16) e M , e S (cid:17) k Pre- C ∗ -algebras A C ∞ ( M ) ⊗ M n ( C ) e A C ∞ (cid:16) e M (cid:17) ⊗ M n ( C ) Dirac operators
D D / ⊗ Γ fin + Id L ( M , S ) k ⊗ D fin e D ? H ∞ def = T ∞ n = Dom D n ⊂ H Γ ∞ ( M , S ) k = T ∞ n = Dom D n Let e / ∇ : C ∞ (cid:16) e M (cid:17) → C ∞ (cid:16) e M (cid:17) ⊗ C ∞ ( M ) Ω D be a given by (4.10.19) G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -equivariant connection (cf. (2.7.6)). Suppose that e a ∈ C ∞ (cid:16) e M (cid:17) and e / ∇ ( e a ) = r ∑ j = e a j ⊗ (cid:2) D / , a j (cid:3) ∈ C ∞ (cid:16) e M (cid:17) ⊗ C ∞ ( M ) Ω D Denote by e ∇ : C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) → C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) ⊗ C ∞ ( M ) ⊗ M m ( C ) Ω D , e a ⊗ b r ∑ j = (cid:0)e a j ⊗ (cid:2) D / , a j (cid:3)(cid:1) ⊗ b Γ fin + e a ⊗ [ D fin , b ] . (5.3.35)From [ D fin , bb ′ ] = b [ D fin , b ′ ] + [ D fin , b ] b ′ , Γ fin b ′ = b ′ Γ fin and e / ∇ ( e aa ) = e / ∇ ( e a ) a + a [ D / , a ] , ∀ a ∈ C ∞ ( M ) it follows that for any b ′ ∈ M m ( C ) one has e ∇ (cid:0) ( e a ⊗ b ) (cid:0) a ⊗ b ′ (cid:1)(cid:1) == r ∑ j = (cid:0)e a j ⊗ (cid:2) D / , a j (cid:3)(cid:1) a + e a ⊗ [ D / , a ] ! ⊗ bb ′ Γ fin + e aa ⊗ (cid:2) D fin , bb ′ (cid:3) == r ∑ j = (cid:0)e a j ⊗ (cid:2) D / , a j (cid:3)(cid:1) a ! ⊗ b Γ fin b ′ + e aa ⊗ [ D fin , b ] b ′ ++ ( e a ⊗ [ D / , a ]) ⊗ b Γ fin b ′ + e aa ⊗ b (cid:2) D fin , b ′ (cid:3) .From (5.3.35) it follows that r ∑ j = (cid:0)e a j ⊗ (cid:2) D / , a j (cid:3)(cid:1) a ! ⊗ b Γ fin b ′ + e aa ⊗ [ D fin , b ] b ′ = e ∇ ( e a ⊗ b ) (cid:0) a ⊗ b ′ (cid:1) and taking into account that ( e a ⊗ [ D / , a ]) ⊗ b Γ fin b ′ + e aa ⊗ b (cid:2) D fin , b ′ (cid:3) = ( e a ⊗ b ) ⊗ (cid:2) D , a ⊗ b ′ (cid:3) we conclude that e ∇ (cid:0) ( e a ⊗ b ) (cid:0) a ⊗ b ′ (cid:1)(cid:1) = e ∇ ( e a ⊗ b ) (cid:0) a ⊗ b ′ (cid:1) + ( e a ⊗ b ) (cid:2) D , a ⊗ b ′ (cid:3) ,i.e. e ∇ is a connection (cf. (E.3.8)). The connection e / ∇ is G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -equivariant, so e ∇ is also G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -equivariant (cf. (2.7.6)). Let us find the specialization of theoperator e D given by the equation 2.7.7. If ξ ⊗ η ∈ L ( M , S ) ⊗ C k then the from2.7.7 and (5.3.35) it follows that e D (( e a ⊗ b ) ⊗ ( ξ ⊗ η )) = r ∑ j = (cid:0)e a j ⊗ (cid:2) D / , a j (cid:3) ξ (cid:1) ⊗ b Γ fin η + ( e a ⊗ ξ ) ⊗ [ D fin , b ] η ++ ( e a ⊗ b ) ⊗ D ( ξ ⊗ η ) .Taking into account (5.3.32) one has ( e a ⊗ b ) ⊗ D ( ξ ⊗ η ) = ( e a ⊗ D / ξ ) ⊗ b Γ fin η + e a ⊗ ξ ) ⊗ bD fin η it turns out that e D (( e a ⊗ b ) ⊗ ( ξ ⊗ η )) == e D (( e a ⊗ ξ ) ⊗ b η ) = r ∑ j = (cid:0)e a j ⊗ (cid:2) D / , a j (cid:3) ξ (cid:1) ⊗ b Γ fin η + ( e a ⊗ ξ ) ⊗ [ D fin , b ] η ++ ( e a ⊗ D / ξ ) ⊗ b Γ fin η + ( e a ⊗ ξ ) ⊗ bD fin η == r ∑ j = e a j ⊗ (cid:2) D / , a j (cid:3) ξ + e a ⊗ D / ξ ! ⊗ b Γ fin η + ( e a ⊗ ξ ) ⊗ ([ D fin , b ] + bD fin ) η == r ∑ j = e a j ⊗ (cid:2) D / , a j (cid:3) ξ + e a ⊗ D / ξ ! ⊗ b Γ fin η + ( e a ⊗ ξ ) ⊗ D fin b η .(5.3.36)However from (4.10.24) it follows that m ∑ j = e a j ⊗ (cid:2) D / , a j (cid:3) ⇒ m ∑ j = e a j ⊗ (cid:2) D / , a j (cid:3) ξ + e a ⊗ D / ξ = p − / D ( e a ⊗ ξ ) (5.3.37)where p − D / is the p -inverse image of D / (cf. Definition 4.7.11 and Equation 4.7.10).Substitution of (5.3.37) into right part of (5.3.36) gives the following e D (( e a ⊗ ξ ) ⊗ b η ) = p − / D ( e a ⊗ ξ ) ⊗ Γ fin b η + ( e a ⊗ ξ ) ⊗ D fin b η ,or equivalently e D = p − D / ⊗ Γ fin + Id L ( e M , e S ) ⊗ D fin . (5.3.38) Theorem 5.3.28.
Consider the situation of the Definition 5.2.3, i.e. • The spectral triple T = (cid:16) C ∞ ( M ) ⊗ M m ( C ) , L ( M , S ) ⊗ C k , D / ⊗ Γ fin + Id L ( M , S ) ⊗ C k ⊗ D fin (cid:17) . • The regular finite-fold covering p : e M → M. • The given by e T = (cid:16) C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) , L (cid:16) e M , e S (cid:17) ⊗ C k , e D / ⊗ Γ fin + Id L ( e M , e S ) ⊗ C k ⊗ D fin (cid:17) geometrical p-lift of T . he geometrical p-lift of T (cf. Definition 5.2.3) coincides with the (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) -lift of T (cf. Definition 2.7.6).Proof. Denote by (cid:16) e A , e H , e D (cid:17) the (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) -lift of T. Consider the givenby (5.3.27) finite set. From (iii) of the Lemma 5.3.26 it turns that (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is smoothly invariant. Accord-ing to the Definition 2.7.6 and taking into account the Equation (5.3.28) one has e A = C (cid:16) e M (cid:17) ⊗ M m ( C ) ∩ M m | f A | ( C ∞ ( M ) ⊗ M m ( C )) == C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) .If ρ : C ( M ) ⊗ M m ( C ) → L ( M , S ) ⊗ C k is the representation of the spectral triple T then the representation e ρ : C (cid:16) e M (cid:17) ⊗ M m ( C ) → B (cid:16) e H (cid:17) of the spectral triple (cid:16) e A , e H , e D (cid:17) is induced by the pair (cid:16) ρ , (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17)(cid:17) . From the Remark 5.3.25it follows that the representation e ρ is equivalent to the representation, which cor-responds to the spectral triple T . From (5.3.38) it follows that e D = p − D / ⊗ Γ fin + Id L ( e M , e S ) k ⊗ D fin and taking into account (4.10.25) one has e D / = p − D / we conclude that the spectraltriple (cid:16) e A , e H , e D (cid:17) is equivalent to the e T one.
Consider the described in the Section 4.10.4 situation, i.e. one has • The Riemannian manifold e M with the spinor bundle e S such that there is thespectral triple (cid:16) C ∞ (cid:16) e M , (cid:17) , L (cid:16) e M , e S (cid:17) , e D / (cid:17) . • The unoriented Riemannian manifold M with the two listed covering p : e M → M and the bundle S → M such that e S is the p -inverse image of S . • The unoriented spectral triple (cid:0) C ∞ ( M , ) , L ( M , S ) , D / (cid:1) given by the Theo-rem 4.10.18 276 heorem 5.3.29. In the described above situation there is the given by (cid:16) C ∞ ( M ) ⊗ M m ( C ) , L ( M , S ) k , / D ⊗ Γ fin + Id L ( M , S ) k ⊗ D fin (cid:17) . (5.3.39) unoriented spectral triple (cf. Definition 2.9.1).Proof. The following table shows the specialization of objects 1-5 required by theDefinition 2.9.1.Definition 2.9.1 This theorem specialization1 A C ∞ ( M ) ⊗ M m ( C ) (cid:16) A , e A , Z (cid:17) (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , Z (cid:17) ρ : A → B ( H ) C ( M ) ⊗ M m ( C ) → B (cid:16) L ( M , S ) k (cid:17) D / D ⊗ Γ fin + Id L ( M , S ) k ⊗ D fin (cid:16) e A , e H , e D (cid:17) (cid:18) C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) , L (cid:16) e M , e S (cid:17) k , e D (cid:19) where e D def = e D / ⊗ Γ fin + Id L ( e M , e S ) k ⊗ D fin Similarly to the proof of the Theorem 4.10.18 one can proof that above objectssatisfy to the conditions (a)-(d) of the Definition 2.9.1. C ∗ -algebras Lemma 5.4.1.
Let A be a separable C ∗ -algebra with continuous trace let X = ˆ A be thespectrum of A. Suppose that X is a connected, locally connected, second-countable space.Let us consider the situation 4.11.23 and suppose thatA def = C (cid:0) X , { A x } x ∈X , A (cid:1) and A is the given by the Definition 4.6.12 functor. Following conditions hold: i) If S X = { p λ : X λ → X } λ ∈ Λ ∈ FinTop is a topological finite covering category(cf. Definition 4.11.2) then there is the natural algebraical finite covering category(cf. Definition 3.1.4). S A = { π λ = A ( p λ ) : A ֒ → A ( X λ ) } λ ∈ Λ ∈ FinAlg . (ii) Conversely any directed algebraical finite covering category S A = { π λ : A ֒ → A ( X λ ) } λ ∈ Λ ∈ FinAlg . (5.4.1) naturally induces directed topological finite covering category S X = { p λ : X λ → X } λ ∈ Λ ∈ FinTop . Proof. (i) One needs check the conditions (a) and (b) of the Definition 3.1.4.(a) The conditions (a)-(d) of the Definition 3.1.1 directly follow from (a)-(d) ofthe Lemma 4.6.19.(b) Follows from the Corollary 5.3.19.(ii) From the Theorem 5.3.18 it turns out that any object π λ : A ֒ → A ( X λ ) of thecategory S A corresponds to the object p λ : X λ → X of the category S X , i.e. p λ isa transitive finite-fold covering. If µ , ν ∈ Λ are such that µ > ν then from (b) of theDefinition 3.1.4 it follows that there is *-homomorphism π : A ( X ν ) → A (cid:0) X µ (cid:1) such that A (cid:0) p µ (cid:1) = π ◦ A ( p ν ) . (5.4.2)From (a) of the Definition 3.1.4 it turns out that the homomorphism π is a non-commutative finite fold covering, so taking into account the Corollary 5.3.19 onehas the topological finite-fold covering p : X µ → X ν such that π = A ( p ) . From(5.4.2) it follows that p µ = p ◦ p ν . Lemma 5.4.2. If S ( X , x ) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt is a pointedtopological finite covering category such that for any µ > ν there is the unique pointedcovering p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) then S pt A == { A ( p λ ) : A ֒ → A ( X λ ) } , (cid:8) A (cid:0) p µν (cid:1) : A ( X ν ) ֒ → A (cid:0) X µ (cid:1)(cid:9) (5.4.3) is a pointed algebraical finite covering category (cf. Definition 3.1.6). roof. According to our construction for any µ > ν the category S A contains theunique *-homomorphism from A ( X ν ) to A (cid:0) X µ (cid:1) . This lemma follows from theRemark 3.1.7. Consider the specialization of Lemma 4.11.27 such that A = (cid:0) X , { A x } x ∈X , A (cid:1) ,and A , and the given by the Lemma 4.11.27 functor. Suppose that H x is the spaceof the given by (D.2.3) representation rep x : A → B ( H x ) and K def = rep x ( A ) where K = M n ( C ) or K = K (cid:0) L ( N ) (cid:1) . From the Lemma 5.4.2 it follows that (5.4.3) isthe pointed algebraical finite covering category. Let b A def = C ∗ -lim −→ λ ∈ Λ A ( X λ ) the C ∗ -inductive limit, and let π a : b A ֒ → B ( H a ) be the atomic representation (cf. Def-inition D.2.33). If A is the disconnected algebraical inverse noncommutative limitof S pt A (cf. Definition 3.1.25) then there is the natural inclusion A ⊂ B ( H a ) . Oth-erwise H a = L b x ∈ b X H b x where b X is the spectrum of b A and H b x is the space of rep-resentation rep b x (cid:16) b A (cid:17) → B ( H b x ) . If b X is the spectrum of b A then from the Lemma3.1.27 it follows that b X = lim ←− λ ∈ Λ X λ . Otherwise if X is the disconnected inverselimit of S ( X , x ) then there is the natural bijective continuous map φ : X → b X if b p : b X → X is the natural map and both rep x A → B ( H x ) , rep x A → B ( H x ) are natural irreducible representations then H x ∼ = H b p ◦ φ ( x ) , H b x ∼ = H b p ( x ) , hence H x ∼ = H φ ( x ) for all x ∈ X . So one has M b x ∈ b X H b x ∼ = M x ∈X H x The left part of the above equation is the space of the atomic representation of b A , right part is the space of the atomic representation of A (cid:0) X (cid:1) . So the atomicrepresentation of A (cid:0) X (cid:1) can be regarded as the natural inclusion ϕ A : A (cid:0) X (cid:1) ֒ → B ( H a ) . (5.4.4) Remark 5.4.4.
Below the given by the Equation (5.4.4) *-homomorphism will bereplaced with the inclusion A (cid:0) X (cid:1) ⊂ B (cid:16) b H a (cid:17) of C ∗ -algebras. Similarly to theRemark 3.1.38 below for all λ ∈ Λ we implicitly assume that A ( X λ ) ⊂ B ( H a ) .Similarly the following natural inclusion C ∗ - lim −→ λ ∈ Λ A ( X λ ) ⊂ B (cid:16) b H a (cid:17) ∑ g ∈ G λ z ∗ ( ga ) z ∈ A ( X λ ) , ∑ g ∈ G λ f ε ( z ∗ ( ga ) z ) ∈ A ( X λ ) , ∑ g ∈ G λ ( z ∗ ( ga ) z ) ∈ A ( X λ ) . (5.4.5)where G λ def = ker (cid:0) G (cid:0) X (cid:12)(cid:12) X (cid:1) → G ( X λ | X ) (cid:1) . Example 5.4.5.
Let X C ( X ) def = C (cid:16) X , {H x } , X C ( X ) (cid:17) be a Hilbert C ( X ) -modulegiven by (4.12.1). If S ( X , x ) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt is apointed topological finite covering category then for each λ ∈ Λ define X C ( X ) def = C (cid:16) lift p λ h X C ( X ) i(cid:17) If S ( X , x ) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt is a pointed topological fi-nite covering category them from the Lemma 5.4.2 it follows that there is a pointedalgebraical finite covering category (cf. Definition 3.1.6) given by S pt K ( X C ( X ) ) def = def = nn K ( p λ ) : K (cid:16) X C ( X ) (cid:17) ֒ → K (cid:16) X C ( X λ ) (cid:17)o , (cid:8) K (cid:0) p µν (cid:1)(cid:9)o (5.4.6)If b K def = C ∗ - lim −→ λ ∈ Λ K (cid:16) X C ( X λ ) (cid:17) and b K ֒ → B (cid:16) b H a (cid:17) is the atomic representationthen from 5.4.3 it follows that b H a = M x ∈X H x (5.4.7)where L means the Hilbert norm completion of the algebraic direct sum. Lemma 5.4.6.
If the spectrum of A is paracompact and locally connected then there is thenatural inclusion A (cid:0) X (cid:1) ⊂ A . (cf. Equation (5.4.4) ). roof. Denote by p : X → X and p λ : X → X λ . If U ⊂ X is an open set whichis homeomorphically mapped onto U = p (cid:0) U (cid:1) ⊂ X then U is homeomorpfilallymapped onto U λ = p λ (cid:0) U (cid:1) ⊂ X λ for any λ ∈ Λ . Let a ∈ A (cid:0) X (cid:1) + is a positiveelement such that supp a ⊂ U . Following proof contains two parts:(i) Element a is special.(ii) A (cid:0) X (cid:1) ⊂ A .(i) One needs check that. a satisfies to the conditions (a), (b) of the Definition3.1.19.(a) For any ε ≥ λ ∈ Λ and z ∈ A + λ there is a ελ ∈ A λ one has supp z ∗ az , supp f ε ( z ∗ az ) , supp ( z ∗ az ) ⊂ U , so from the Lemma 4.6.18 it turnsout that ∑ g ∈ ker ( b G → G ( X λ | X ) ) g ( z ∗ az ) = a λ == desc p λ ( z ∗ az ) ∈ A ( X λ ) , ∑ g ∈ ker ( b G → G ( X λ | X ) ) g f ε ( z ∗ az ) = desc p λ ( f ε ( z ∗ az )) ∈ A ( X λ ) , ∑ g ∈ ker ( b G → G ( X λ | X ) ) g ( z ∗ az ) = b λ desc p λ ( z ∗ az ) ∈ A ( X λ ) where the strong convergence of the series is implied. Above equationscoincide with (5.4.5).(b) From a λ = desc p λ ( z ∗ az ) and b λ = desc p λ ( z ∗ az ) it turns out that ∀ λ ∈ Λ u ⇒ (cid:13)(cid:13) a λ − b λ (cid:13)(cid:13) = a λ and b λ satisfy the inequality (3.1.21).(ii) Consider a compliant compliant to the covering p : X → X partition of unity ∑ ( g , α ) ∈ G ( X | X ) × A g φ α = C b ( X ) (cf. Definition 4.8.4) If a ∈ A (cid:0) X (cid:1) + is any positive element then from the LemmaD.8.1 it follows that a = ∑ ( g , α ) ∈ G ( X | X ) × A g φ α a .281n the other hand it is already proven that g φ α a is a special element for all ( g , α ) ∈ G (cid:0) X (cid:12)(cid:12) X (cid:1) × A . For any ε > V ⊂ X such that rep x ( a ) < ε for all x ∈ X \ V so from the Corollary 4.2.5 it turns out that there is a coveringsum for e U (cf. Definition 4.2.6) i.e. a finite subset A ⊂ G (cid:0) X (cid:12)(cid:12) X (cid:1) × A such that ∑ ( g , α ) ∈ A g φ α ( x ) = ∀ x ∈ V .It follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ ( g , α ) ∈ A g φ α a − a (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε .Taking into account that the set A is finite one has ∑ ( g , α ) ∈ A g φ α a ∈ A ans since ε is arbitrary small it turns out that a ∈ A . Thus the positive cone A (cid:0) X (cid:1) + of A (cid:0) X (cid:1) is a subset of A . However any C ∗ -algebra is generated by its positive cone,it follows that A (cid:0) X (cid:1) ⊂ A . Consider the situation of the Example 5.4.5, i.e. b K def = C ∗ - lim −→ λ ∈ Λ K (cid:16) X C ( X λ ) (cid:17) and the atomic representation b K ֒ → B (cid:16) b H a (cid:17) where b H a = M x ∈X H x (cf. Equation (5.4.7)). The space X C b ( X ) def = C b (cid:16) lift p λ h X C ( X ) i(cid:17) is a Hilbert C b (cid:0) X (cid:1) - module. Any bounded family a def = { a x ∈ K ( H x ) } x ∈X (5.4.8)yields a bounded operator in B (cid:16) b H a (cid:17) . If X D ∗ ( X ) is the discontinuous extension of X C ( X ) (cf. Definition 4.12.1) and ξ def = (cid:8) ξ x (cid:9) ∈ X D ∗ ( X ) then a ξ def = (cid:8) a x ξ x (cid:9) ∈ X D ∗ ( X ) (5.4.9) Definition 5.4.8.
Let b K def = C ∗ -lim −→ λ ∈ Λ K (cid:16) X C ( X λ ) (cid:17) be the inductive limit of C ∗ -algebras in sense of the Definition 1.2.7, and let b π a : b K → B (cid:16) b H a (cid:17) be the atomicrepresentation (cf. Definition D.2.33). Let b G def = lim ←− λ ∈ Λ G ( X λ | X ) be the pro-jective limit of groups and let G λ def = ker (cid:16) b G → G ( X λ | X ) (cid:17) . An element a = { a x ∈ K ( H x ) } x ∈X ∈ B ( H a ) is coherent if for any λ ∈ Λ , ξ λ ∈ X λ and ε > f ε : R → R is a continuous function given by (3.1.19) then there are f λ , f ελ , h λ , ∈ C ( X λ ) + such that ∑ g ∈ G λ D lift p λ ( ξ λ ) ga , lift p λ ( ξ λ ) E D ∗ ( X ) = f λ , ∑ g ∈ G λ f ε (cid:18)D lift p λ ( ξ λ ) ga , lift p λ ( ξ λ ) E D ∗ ( X ) (cid:19) = f ελ , ∑ g ∈ G λ D lift p λ ( ξ λ ) ga , lift p λ ( ξ λ ) E D ∗ ( X ) = h λ . (5.4.10)where sums of the above series mean the point-wise convergence, and theEquations (4.12.2), (5.4.9) are used.(b) There is µ ∈ Λ such that µ ≥ λ and ∀ λ ∈ Λ λ ≥ µ ⇒ (cid:13)(cid:13) f λ − h λ (cid:13)(cid:13) < ε (5.4.11)where a λ , b λ ∈ C ( X λ ) are given by (5.4.10). Remark 5.4.9.
Similarly to the Lemma 3.1.21 one can proof that there is the stronglimit a = lim b π a ( a ′ λ ) in B ( H a ) . It follows that a corresponds to the family { a x ∈ H x } x ∈X . (5.4.12) Lemma 5.4.10.
Let b K def = C ∗ - lim −→ λ ∈ Λ K (cid:16) X C ( X λ ) (cid:17) be the inductive limit of C ∗ -algebrasin sense of the Definition 1.2.7, and let b π a : b K → B (cid:16) b H a (cid:17) be the atomic representation(cf. Definition D.2.33). Let b G def = lim ←− λ ∈ Λ G ( X λ | X ) be the projective limit of groups andlet G λ def = ker (cid:16) b G → G ( X λ | X ) (cid:17) . Let K (cid:16) X C ( X λ ) (cid:17) be given by (1.2.1) . If a ∈ B (cid:16) b H a (cid:17) is such that for any λ ∈ Λ , ε > and z ∈ A ∼ λ it satisfies to the following conditions:(a) If f ε : R → R is a continuous function given by (3.1.19) . then for all λ ≥ λ thereare a λ , b λ , a ελ ∈ K (cid:16) X C ( X λ ) (cid:17) such that ∑ g ∈ G λ π ( z ) ∗ ( ga ) π ( z ) = π ( a λ ) , ∑ g ∈ G λ f ε (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = π ( a ελ ) , ∑ g ∈ G λ (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = π ( b λ ) (5.4.13)283 here sums of the above series mean the strong convergence in B (cid:16) b H a (cid:17) and theaction G λ × B (cid:16) b H a (cid:17) → B (cid:16) b H a (cid:17) is given by (3.1.18) .(b) There is µ ∈ Λ such that µ ≥ λ and ∀ λ ∈ Λ λ ≥ µ ⇒ (cid:13)(cid:13) a λ − b λ (cid:13)(cid:13) < ε (5.4.14) where a λ , b λ ∈ A λ are given by (3.1.20) .then a is coherent (cf. Definition 5.4.8).Proof. Suppose λ ∈ Λ . If ξ λ ∈ X λ is an arbitrary element then z ′ def = ξ λ ih ξ λ ∈K (cid:16) X C ( X λ ) (cid:17) , such that norm z ′ = norm ξ λ , i.e. z ′ is a multiple of norm ξ λ (cf. Defi-nition 4.5.16). From the Lemma 4.5.15 it follows that there is z def = div (cid:16) z ′ , norm ξ λ (cid:17) ∈K (cid:16) X C ( X λ ) (cid:17) (cf. Equation 4.5.24). For all x ∈ X λ the dimension of rep x ( z ′ ) doesnot exceed 1, hence rep x ( z ) also does not exceed 1. Let us check that a satisfies toconditions (a) and (b) of the Definition 5.4.8.(a) From (5.4.13) it follows that there are f λ , f ελ , h λ , ∈ C ( X λ ) + such that ∑ g ∈ G λ tr (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = f λ , ∑ g ∈ G λ f ε (cid:0) tr (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1)(cid:1) = f ε l a , ∑ g ∈ G λ tr (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = h λ (5.4.15)where tr (cid:16) b (cid:17) means the map x rep x (cid:16) b (cid:17) . On the other hand from ourconstruction it follows thattr (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = D lift p λ ( ξ λ ) ga , lift p λ ( ξ λ ) E D ∗ ( X ) , f ε (cid:0) tr (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1)(cid:1) == f ε (cid:18)D lift p λ ( ξ λ ) ga , lift p λ ( ξ λ ) E D ∗ ( X ) (cid:19) ,tr (cid:0) π ( z ) ∗ ( ga ) π ( z ) (cid:1) = D lift p λ ( ξ λ ) ga , lift p λ ( ξ λ ) E D ∗ ( X ) (5.4.16)284rom the Equations (5.4.15) and (5.4.16) it turns out that ∑ g ∈ G λ D lift p λ ( ξ λ ) ga , lift p λ ( ξ λ ) E D ∗ ( X ) = f λ , ∑ g ∈ G λ f ε (cid:18)D lift p λ ( ξ λ ) ga , lift p λ ( ξ λ ) E D ∗ ( X ) (cid:19) = f ελ , ∑ g ∈ G λ D lift p λ ( ξ λ ) ga , lift p λ ( ξ λ ) E D ∗ ( X ) = h λ .(b) From the Equation (5.4.14) it turns out that ∀ λ ∈ Λ λ ≥ µ ⇒ (cid:13)(cid:13) f λ − h λ (cid:13)(cid:13) = (cid:13)(cid:13) a λ − b λ (cid:13)(cid:13) < ε . Any element of ξ ∈ ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) corresponds to a family (cid:8) ξ ( x ) ∈ L ( N ) x (cid:9) x ∈X . (5.4.17)Similarly any b ∈ K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) corresponds to a family n b ( x ) ∈ K (cid:0) L ( N ) x (cid:1)o x ∈X , (5.4.18)hence for any x ∈ X there is a representation ρ x : K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) → B (cid:0) L ( N ) x (cid:1) (5.4.19)and there is a faithful representation ρ X : K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) ֒ → B M x ∈X L ( N ) x . (5.4.20)Similarly for any x λ ∈ X λ there is a representation ρ x λ : K (cid:0) ℓ ( C b ( X λ )) (cid:1) → B (cid:16) L ( N ) x λ (cid:17) (5.4.21)and there is a faithful representation ρ X λ : K (cid:0) ℓ ( C b ( X λ )) (cid:1) ֒ → B M x λ ∈X λ L ( N ) x λ ! . (5.4.22)285rom (4.12.23) it turns out K (cid:16) X C ( X λ ) (cid:17) ⊂ K (cid:0) ℓ ( C ( X λ )) (cid:1) and K (cid:16) X C ( X λ ) (cid:17) is ahereditary subalgebra of K (cid:0) ℓ ( C ( X λ )) (cid:1) Taking into account (4.5.64) and 4.12.12one has K (cid:16) X C ( X ) (cid:17) ⊂ K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) , K (cid:16) X C ( X ) (cid:17) is a hereditary subalgebra of K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) (5.4.23)Denote by b K def = C ∗ -lim −→ λ ∈ Λ K (cid:16) X C ( X λ ) (cid:17) the C ∗ -inductive limit in sense of the Def-inition 1.2.7. If b π a : b K ֒ → B (cid:16) b H a (cid:17) be the atomic representation (cf. DefinitionD.2.33). The space b H a is the Hilbert direct sum b H a = M b x ∈ b X H b x where b X is the spectrum of b K and H b x is the irreducible representation whichcorresponds to b x (cf. Definition D.2.9). Similarly to the Lemma 3.1.27 one canproof that b X is homeomorphic the inverse limit of the spectra, i.e. b X = lim ←− λ ∈ Λ X λ .Otherwise from the given by the Lemma 4.11.17 of the disconnected inverse limitthere is the bijective continuous map X ∼ = b X and this map gives the followingrepresentation of the space b H a = M x ∈X H x . (5.4.24)For all λ ∈ Λ the C ∗ -algebra K (cid:16) X C ( X λ ) (cid:17) will be regarded as a subalebra of b K . Lemma 5.4.12.
Consider the situation of the (cf. Definition 5.4.8) with the a coherentelement a ∈ B (cid:16) b H a (cid:17) . Let λ ∈ Λ and ξ ∈ X C ( X λ ) . Let p λ : X → X λ be the naturalcovering, and ξ def = lift p λ ( ξ ) ∈ X C b ( X ) corresponds to a family (cid:8) ξ x (cid:9) x ∈X . If f : X → R is given by x (cid:0) ξ x , rep x ( a ) , ξ x (cid:1) then f ∈ C (cid:0) X (cid:1) .Proof. If f λ ∈ C ( X λ ) is given by the equation (5.4.10) and f λ def = lift p λ ( f λ ) ∈ C b (cid:0) X (cid:1) then f = lim λ f λ is a point-wise limit. From the Definition 5.4.8 it followsthat f satisfies to the Theorem 4.11.33, hence f ∈ C (cid:0) X (cid:1) .286 orollary 5.4.13. If a ∈ B ( H a ) is coherent (cf. Definition 5.4.8), ξ ∈ ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) andf ∈ D ∗ (cid:0) X (cid:1) , x (cid:0) ξ ( x ) , ρ x ( a ) ξ ( x ) (cid:1) H x then f ∈ C b (cid:0) X (cid:1) .Proof. From the Lemma 1.3.7 it follows that X C b ( X ) is the norm completion of ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) K (cid:16) X C b ( X ) (cid:17) .If ξ ∈ ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) then the Lemma 1.3.8 it turns out that for any ε > η ∈ X C b ( X ) such that0 ≤ a ≤ K ( p λ ) ( a ) λ ⇒ (cid:13)(cid:13)(cid:13)(cid:10) ξ a , ξ (cid:11) C b ( X ) − h η a , η i C b ( X ) (cid:13)(cid:13)(cid:13) < ε . (5.4.25)where K ( p λ ) is given by (4.12.25). However form the Lemma 5.4.12 it followsthat h η a , η i C b ( X ) ∈ C b (cid:0) X (cid:1) and taking into account (5.4.25) one has (cid:10) ξ a , ξ (cid:11) C b ( X ) ∈ C b (cid:0) X (cid:1) . Corollary 5.4.14.
If a ∈ B (cid:0) L x ∈X L ( N ) x (cid:1) is coherent thena ∈ K b (cid:16) X C ( X ) (cid:17) = C b (cid:16) K (cid:16) X C b ( X ) (cid:17)(cid:17) . (5.4.26) Proof. If ξ ∈ ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) then then from the Corollary 5.4.13 it turns out that D lift p λ ( ξ λ ) a , lift p λ ( ξ λ ) E D ∗ ( X ) ∈ C b (cid:0) X (cid:1) . From the Lemma 4.12.9 and the Corol-lary 1.3.4 it turns out that a is C b (cid:0) X (cid:1) -continuous (cf. Definition 1.3.2), so one has a ∈ L (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) . The C ∗ -algebra K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) is a hereditary subalbebra of L (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) . Otherwise from lift p λ ( a λ ) > a and lift p λ ( a λ ) ∈ K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) it turns out a ∈ K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) . Taking into account (5.4.23) K (cid:16) X C b ( X ) (cid:17) is ahereditary subalgebra of K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) and from lift p λ ( a λ ) ∈ K (cid:16) X C b ( X ) (cid:17) itturns out that a ∈ K (cid:16) X C b ( X ) (cid:17) . (5.4.27) Lemma 5.4.15.
Suppose that X is compact. If a ∈ B (cid:0) L x ∈X L ( N ) x (cid:1) is coherent thena ∈ K (cid:16) X C ( X ) (cid:17) = C (cid:16) K (cid:16) X C ( X ) (cid:17)(cid:17) . (5.4.28)287 roof. From the Remark D.4.15 it follows that a is compact in sense of Mischenko,hence for any ε > n ∈ N suchthat k p n a p n − a k < ε . (5.4.29)where p n ∈ K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) is the projector onto ⊕ nj = C b (cid:0) X (cid:1) ⊂ ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1) . If λ ∈ Λ then since X is compact and the covering X λ → X is finite fold, thespace X λ is compact. It turns out that C ( X λ ) ∼ = C ( X λ ) and 1 C ( X λ ) ∈ C ( X λ ) .Denote by a ′ = p n a p n . (5.4.30)One has a ′ ∈ K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) . If e , ..., e n ∈ ℓ ( C ( X λ )) such that e j = ... , 1 C ( X λ ) | {z } j th − place , ..., 0 and e j def = p λ (cid:0) e j (cid:1) than since the trace of the positive operator equals or exceeds itsnorm one has (cid:13)(cid:13) rep e x (cid:0) a ′ (cid:1)(cid:13)(cid:13) ≤ n ∑ j = (cid:10) e j a , e j (cid:11) C b ( X ) ( e x ) ∀ e x ∈ e X . (5.4.31)From the Lemma 5.4.12 it turns out (cid:10) e j a , e j (cid:11) C b ( X ) ∈ C (cid:0) X (cid:1) so f n = n ∑ j = (cid:10) e j a , e j (cid:11) C b ( X ) ∈ C (cid:0) X (cid:1) . (5.4.32)Otherwise from a ′ ∈ K (cid:0) ℓ (cid:0) C b (cid:0) X (cid:1)(cid:1)(cid:1) it turns outnorm a = (cid:0) x (cid:13)(cid:13) rep e x (cid:0) a ′ (cid:1)(cid:13)(cid:13)(cid:1) ∈ C (cid:0) X (cid:1) ,and taking into account f n ∈ C (cid:0) X (cid:1) , norm a ′ ≤ f n one hasnorm a ′ ∈ C (cid:0) X (cid:1) .or equivalently norm p n ap n ∈ C (cid:0) X (cid:1) .288rom (5.4.29) it follows that norm a = lim n → ∞ norm p n ap n , and taking into account4.5.14 it turns out that a ∈ C (cid:16) K (cid:16) X C ( X ) (cid:17)(cid:17) . Lemma 5.4.16.
Suppose that X is a connected, locally connected, locally compact, locallycompact, Hausdorff space. If a ∈ B (cid:0) L x ∈X L ( N ) x (cid:1) is coherent thena ∈ K (cid:16) X C ( X ) (cid:17) = C (cid:16) K (cid:16) X C ( X ) (cid:17)(cid:17) . (5.4.33) Proof.
From the Corollary 5.4.14 it turns out that there is f ∈ C b (cid:0) X (cid:1) , x
7→ k rep x ( a ) k From the construction 4.11.1 it follows that there is the minimal λ min ∈ Λ suchthat X λ min def = X . We set λ = λ min . If a ′ λ ∈ K (cid:16) X C ( X λ ) (cid:17) is given by (3.1.23) thenfrom the Lemma D.8.26 it turns out that for all ε > Y ′ def = (cid:8) x ∈ X (cid:12)(cid:12)(cid:13)(cid:13) rep x (cid:0) a ′ λ (cid:1)(cid:13)(cid:13) ≥ ε (cid:9) (5.4.34)is compact. From the Lemmas 4.2.1 and 4.2.2 it follows that there is a connectedcompact set Y ⊂ X such that Y ′ ⊂ Y and x ∈ Y . Denote by Y λ def = Y , Y λ def = p − λ ( Y ) ⊂ X λ and Y def = p − ( Y ) ⊂ X . From x ∈ Y it turns out that that base-points x λ and x of X λ and X respectively satisfy to the following equations. Thedirect check yields the proof that there is a pointed topological finite coveringcategory (cf. Definition 4.11.3) S ( Y , x ) = (cid:8)(cid:0) Y λ , x λ (cid:1) → ( Y , x ) (cid:1) } with morphisms p µν (cid:12)(cid:12) Y µ : (cid:0) Y µ , x µ (cid:1) → ( Y ν , x ν ) (5.4.35)Where p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) are morphisms of the pointed topological finitecovering category S ( X , x ) . One can consider the family F = n K (cid:16) X C ( Y λ ) (cid:17)o λ ∈ Λ If b ∈ B (cid:0) L x ∈Y L ( N ) x (cid:1) is given by rep x (cid:16) b (cid:17) def = rep x ( a ) ∀ x ∈ Y . (5.4.36)then since a satisfies to the Definition 5.4.8, the element b also satisfies the toDefinition 5.4.8, i.e. b is coherent with respect to the family F . The set Y is289ompact so from the Lemmas 5.4.15 and D.8.26 it turns out the map Y → R , x (cid:13)(cid:13)(cid:13) rep x (cid:16) b (cid:17)(cid:13)(cid:13)(cid:13) lies in C (cid:0) X (cid:1) . The set Z def = n x ∈ Y (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) rep x (cid:16) b (cid:17)(cid:13)(cid:13)(cid:13) ≥ ε o is compact. However rep x (cid:16) a ′ λ (cid:17) ≥ rep x ( a ) for any x ∈ X and taking into account(5.4.34) one has ∀ x ∈ X \ p − ( Y ) k rep x ( a ) k < ε .From the above equation and (5.4.36) it follows that Z = (cid:8) x ∈ X |k rep x ( a ) k ≥ ε (cid:9) = n x ∈ X (cid:12)(cid:12)(cid:12) f ( x ) ≥ ε o .The set Z is compact, hence from the Definition A.1.21 it follows that f ∈ C (cid:0) X (cid:1) ,and taking into account the Corollary 5.4.14 one has a ∈ C (cid:16) K (cid:16) X C ( X ) (cid:17)(cid:17) . Lemma 5.4.17.
Consider the situation of the Lemma 5.4.1. If a ∈ B (cid:16) b H a (cid:17) is a specialelement (cf. Definition 3.1.19) then a ∈ A (cid:0) X (cid:1) .Proof. If a λ ∈ A ( X λ ) is given by (3.1.22) then from the Lemma 3.1.21 it followsthat a = lim λ ∈ Λ π a ( a λ ) . Let X be the spectrum of A and let (cid:8) V µ ⊂ X (cid:9) µ ∈ M afamily of compact subsets which satisfies to the Proposition D.8.15, i.e. interiorsform a cover (cid:8) U µ (cid:9) of X , such that for each µ ∈ M , there is an A | V µ − C (cid:0) V µ (cid:1) -imprimitivity bimodule X µ . Let {U ν } ν ∈ N be a compliant with p : X → X family(cf. Definition 4.3.13). If (cid:8) U β (cid:9) β ∈ B def = (cid:8) U β ⊂ X (cid:12)(cid:12) ∃ µ ∈ M ∃ ν ∈ N U β = U µ ∩ U ν (cid:9) .From the Definition A.1.24 it follows that there is a locally finite refinement {U α } α ∈ A (cid:8) U β (cid:9) β ∈ B . From the our construction it follows that {U α } is a refinement of both (cid:8) U µ (cid:9) and {U ν } . So the family {U α } satisfies to the following properties:(a) {U α } is a compliant with p : X → X family.(b) The closure V α of U α is compact for all α ∈ A .290c) For each α ∈ A , there is an A | V α − C ( V α ) -imprimitivity bimodule X α .Let ∑ A φ α = C b ( X ) be a partition of unity dominated by {U α } α ∈ A . For any λ ∈ Λ consider the givenby (4.8.5) partition of unity, i.e. ∑ ( g , α ) ∈ G ( X | X ) × A g φ λα = C b ( X λ ) .If V λα def = supp φ λα and U λα def = (cid:8) x λ ∈ X λ | φ λα ( x λ ) > (cid:9) for any λ ∈ Λ and α ∈ A thenfor any g ∈ G ( X λ | X ) there is an A λ | g V λα − C (cid:0) g V λα (cid:1) -imprimitivity bimodule X λα .Let us select any λ ∈ Λ , ε >
0. If z = A ( X λ ) ∼ ∈ A ( X λ ) ∼ then from (3.1.41) itturns out that following strongly convergent in B (cid:16) b H a (cid:17) series a ′ λ def = ∑ g ∈ G λ g ( z ∗ az ) = ∑ g ∈ G λ ga ∈ A ( X λ ) .From norm a ∈ C ( X λ ) it follows that for all ε > V λ ⊂ X λ such (cid:13)(cid:13)(cid:13) norm a | X λ \V λ (cid:13)(cid:13)(cid:13) < ε . It follows that ∀ x ∈ X \ p − λ ( V λ ) k rep x ( a ) k ≤ (cid:13)(cid:13)(cid:13) rep p ( x ) ( a ) (cid:13)(cid:13)(cid:13) < ε . (5.4.37)The set V λ ⊂ X λ is compact so there is a covering sum for V λ ⊂ X λ (cf. Defini-tion 4.2.6), i.e. a finite subset A ⊂ G ( X λ | X ) × A such that ∑ ( g , α ) ∈ A g φ λ α ( x ) = ∀ x ∈ λ .From the inequality (5.4.37) it follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − ∑ ( g , α ) ∈ A g φ λ α a (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε . (5.4.38)Let us fix ( g , α ) ∈ A and denote by Y λ def = g U λ α . Also denote by q λ def = p λλ : X λ →X λ the natural covering and Y λ def = q − λ ( Y λ ) . One has g φ λ α a ′ λ ∈ A ( X λ ) | Y λ .291rom our construction there is an A ( X λ ) | Y λ - C ( Y λ ) -imprimitivity bimodule X C ( Y λ ) , such that A ( X λ ) | Y λ ∼ = K (cid:16) X C ( Y λ ) (cid:17) .If λ ≥ λ and then there is a Hilbert C ( Y λ ) -module X C ( Y λ ) def = C (cid:16) lift q λ (cid:16) X C ( Y λ ) (cid:17)(cid:17) such that A ( X λ ) | Y λ ∼ = K (cid:16) X C ( Y λ ) (cid:17) .Similarly to the Definition 5.4.8 there is a directed set n K (cid:16) X C ( Y λ ) (cid:17)o λ ∈ Λ λ ≥ Λ of C ∗ -algebras. If b K def = C ∗ -lim −→ λ ∈ Λ K (cid:16) X C ( X λ ) (cid:17) the inductive limit then from the Lemma3.1.27 it turns out that the spectrum of b K coincides as set with Y def = p − λ ( Y λ ) .So the space of the atomic representation of b K equals to b H k a def = χ Y b H a where χ Y ∈ B (cid:16) b H a (cid:17) is the characteristic function of Y . There is the direct sum b H a = b H k a ⊕ b H ⊥ a .Since a is special for any z ∈ A ( X λ ) | Y λ there are a λ , a ελ and b λ ∈ A ( X λ ) suchthat ∑ g ′ ∈ G λ (cid:18) g q φ λ α z (cid:19) ∗ (cid:0) g ′ a (cid:1) (cid:18) g q φ λ α z (cid:19) = a λ = ∑ g ′ ∈ G λ z ∗ (cid:16) g (cid:16) g ′ φ λ α a (cid:17)(cid:17) z , ∑ g ′ ∈ G λ f ε (cid:18)(cid:18) g q φ λ α z (cid:19) ∗ (cid:0) g ′ a (cid:1) (cid:18) g q φ λ α z (cid:19)(cid:19) = a ελ == ∑ g ′ ∈ G λ f ε (cid:16) z ∗ (cid:16) g ′ (cid:16) g φ λ α a (cid:17)(cid:17) z (cid:17) , ∑ g ′ ∈ G λ (cid:18)(cid:18) g q φ λ α z (cid:19) ∗ (cid:0) g ′ a (cid:1) (cid:18) g q φ λ α z (cid:19)(cid:19) = b λ == ∑ g ′ ∈ G λ (cid:16) z ∗ (cid:16) g ′ (cid:16) g φ λ α a (cid:17)(cid:17) z (cid:17) (5.4.39)where sums of the above series mean the strong convergence in B (cid:16) b H a (cid:17) . (cf.Equations (3.1.41)). From z ∈ A ( X λ ) | Y λ it follows that z b H ⊥ a = { } , so the series(5.4.39) are strongly convergent in B (cid:16) b H k a (cid:17) . From the Equation 3.1.20 one has ∀ λ ∈ Λ λ ≥ µ ⇒ (cid:13)(cid:13) a λ − b λ (cid:13)(cid:13) < ε (5.4.40)292he space H k a is the space of the atomic representation of b K , so from the equa-tions (5.4.39), (5.4.39) and the Lemma 5.4.10 it follows that g φ λ α a is coherent (cf.Definition 5.4.8). From the Lemma 5.4.16 it follows that g φ λ α a ∈ K (cid:16) X C ( Y ) (cid:17) = C (cid:16) K (cid:16) X C ( Y ) (cid:17)(cid:17) .If U ⊂ X is the open connected set which is homemorphically mapped onto Y λ then Y = F g ∈ G λ g U and K (cid:16) X C ( Y ) (cid:17) is the C ∗ -norm completion of the algebraicdirect sum M g ′ ∈ G λ g ′ K (cid:16) X C ( U ) (cid:17) = M g ′ ∈ G λ g ′ A (cid:0) X (cid:1)(cid:12)(cid:12) U So g φ λ α a can be represented as a sum of the following C ∗ -norm convergent series g φ λ α a = ∑ g ′ ∈ G λ a g ′ ; a g ′ ∈ g ′ A (cid:0) X (cid:1)(cid:12)(cid:12) U .Since any summand of the series lies in A (cid:0) X (cid:1) the sum of the series belongs to A (cid:0) X (cid:1) , i.e. g φ λ α a ∈ A (cid:0) X (cid:1) . The set A is finite, hence one has ∑ ( g , α ) ∈ A g φ λ α a ∈ A (cid:0) X (cid:1) . Otherwise the number ε in the inequality (5.4.38) may be arbitrary small,so one has a ∈ A (cid:0) X (cid:1) . Corollary 5.4.18.
If A is the disconnected algebraical inverse noncommutative limit ofthe given by (5.4.3) pointed algebraical finite covering category then one has A ⊂ A (cid:0) X (cid:1) Proof.
From the Lemma 5.4.17 it turns out that any special a lies in A (cid:0) X (cid:1) . Fromthe Lemma 4.6.15 it turns out that A ( X λ ) ⊂ M (cid:0) A (cid:0) X (cid:1)(cid:1) . It follows that C ∗ -lim −→ λ ∈ Λ A ( X λ ) ⊂ M (cid:0) A (cid:0) X (cid:1)(cid:1) . Hence one has π a ( b x ) b π a ( b y ) ∈ ϕ (cid:0) A (cid:0) X (cid:1)(cid:1) , ∀ b x , b y ∈ C ∗ - lim −→ λ ∈ Λ A ( X λ ) ,i.e. any weakly special element is contained in ϕ A (cid:0) A (cid:0) X (cid:1)(cid:1) . On other hand A isa C ∗ -algebra generated by weakly special elements, so one has A ⊂ A (cid:0) X (cid:1) . From the Lemma 5.4.6 and the Corollary 5.4.18 it follows that disconnectedinverse noncommutative limit of S A is isomorphic to A (cid:0) X (cid:1) . According to theDefinition 3.1.25 one has G (cid:0) A (cid:0) X (cid:1)(cid:12)(cid:12) A (cid:1) = lim ←− λ ∈ Λ G ( A ( X λ ) | A ) G ( A ( X λ ) | A ) ∼ = G ( X λ | X ) and G (cid:0) X (cid:12)(cid:12) X (cid:1) = lim ←− λ ∈ Λ G ( X λ | X ) we conclude that G (cid:0) A (cid:0) X (cid:1)(cid:12)(cid:12) A (cid:1) = G (cid:0) X (cid:12)(cid:12) X (cid:1) . (5.4.41)If e X ⊂ X is a connected component of X then the closed ideal A (cid:16) e X (cid:17) ⊂ A (cid:0) X (cid:1) is a maximal connected C ∗ -subalgebra of A (cid:0) X (cid:1) . If G ⊂ G (cid:0) A (cid:0) X (cid:1)(cid:12)(cid:12) A (cid:1) is the maximal among subgroups among subgroups G ′ ⊂ G (cid:0) A (cid:0) X (cid:1)(cid:12)(cid:12) A (cid:1) such that G ′ A (cid:16) e X (cid:17) = A (cid:16) e X (cid:17) then G is the maximal among subgroups among subgroups G ′′ ⊂ G (cid:0) X (cid:12)(cid:12) X (cid:1) such that G ′′ e X = e X (cf. (4.11.31)), or equivalently G = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) . (5.4.42)If J ⊂ G (cid:0) A (cid:0) X (cid:1)(cid:12)(cid:12) A (cid:1) is a set of representatives of G (cid:0) A (cid:0) X (cid:1)(cid:12)(cid:12) A (cid:1) / G (cid:16) A (cid:16) e X (cid:17)(cid:12)(cid:12)(cid:12) A (cid:17) then from the (4.11.4) it follows that X = G g ∈ J g e X and A (cid:0) X (cid:1) is a C ∗ -norm completion of the direct sum M g ∈ J gA (cid:16) e X (cid:17) . (5.4.43) Theorem 5.4.20.
Let A be a separable C ∗ -algebra with continuous trace and X = ˆ Ais the spectrum of A. Let S ( X , x ) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt be apointed topological finite covering category such that for any µ > ν there is the uniquepointed covering p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) . If S A == (cid:0) { A ( p λ ) : A ֒ → A ( X λ ) } , (cid:8) A (cid:0) p µν (cid:1) : A ( X ν ) ֒ → A (cid:0) X µ (cid:1)(cid:9)(cid:1) is a given by the Lemma 5.4.2 pointed algebraical finite covering category (cf. Definition3.1.6). then following conditions hold: i) S A is good and the triple (cid:16) A , A (cid:16) lim ←− S X (cid:17) , G (cid:16) lim ←− S X | X (cid:17)(cid:17) is the infinitenoncommutative covering of S A (cf. Definition 3.1.34).(ii) There are isomorphisms: • lim ←− S A ≈ A (cid:16) lim ←− S X (cid:17) . • G (cid:16) lim ←− S A | A (cid:17) ≈ G (cid:16) lim ←− S X | X (cid:17) .Proof. From 5.4.19 it follows that the disconnected algebraical inverse noncommu-tative limit of S A is isomorphic to A (cid:0) X (cid:1) , and the closed ideal A (cid:16) e X (cid:17) ⊂ A (cid:0) X (cid:1) is a maximal connected C ∗ -subalgebra of A (cid:0) X (cid:1) (cf. (5.4.19)) One needs checkconditions (a)-(c) of the Definition 3.1.33.(a) Since e X → X λ is a covering for all λ ∈ Λ the natural *-homomorphism A ( X λ ) ֒ → M (cid:16) A (cid:16) e X (cid:17)(cid:17) is injective.(b) We already know that the algebraic direct sum L g ∈ J gA (cid:16) e X (cid:17) is a densesubalgebra of A (cid:0) X (cid:1) (cf. 5.4.19).(c) For every λ ∈ Λ the homomorphism G (cid:16) A (cid:16) e X (cid:17) (cid:12)(cid:12)(cid:12) A (cid:17) → G ( A ( X λ ) | A ) is equivalent to G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) → G ( X λ | X ) . However both coverings e X → X and X λ → X are transitive, hence from the equation (4.3.8) it follows thatthe homomorphism G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) → G ( X λ | X ) is surjective.(ii) Follows from (i) of this lemma. Lemma 5.4.21.
Let A be a C ∗ -algebra with continuous trace, and let X be the spectrumof A. Suppose that X is a connected, locally connected, second-countable, Hausdorff space.If S ( X , x ) = (cid:8)(cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt and S pt A = = { A ( p λ ) : A ֒ → A ( X λ ) } , (cid:8) A (cid:0) p µν (cid:1) : A ( X ν ) ֒ → A (cid:0) X µ (cid:1)(cid:9) is the given by the Lemma 5.4.2 pointed algebraical finite covering category then S pt A allows inner product (cf. Definition 3.4.1). roof. The proof is similar to the proof of the Lemma 4.11.38. Denote by e p : e X →X , e p λ : e X → X λ , p λ : X λ → X , p µλ : X µ → X λ ( µ > λ ) the natural coveringprojections. If e X = lim ←− S ( X , x ) then from the Theorem 5.4.20 it turns out (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X | X (cid:17)(cid:17) is the infinite noncommutative covering of S pt A (cf. Definition 3.1.34). TheoremD.8.34 represented by the following way A = C (cid:0) X , { A x } x ∈X , F (cid:1) where the notation (4.5.12) is used. If A = (cid:0) X , { A x } x ∈X , F (cid:1) From (4.6.5) and theDefinitions 4.6.9, 5.2.1 it turns that A (cid:16) e X (cid:17) def = C (cid:16) lift p [ X , { A x } , F ] (cid:17) .If e a ∈ C (cid:16) lift p [ X , { A x } , F ] (cid:17) lies in the Pederesen’s ideal of C (cid:16) lift p [ X , { A x } , F ] (cid:17) then the given by (4.5.6) set supp e a is a compact subset of e X . From the Lemma4.11.20 it follows that there is λ ∈ Λ such that for all λ ≥ λ the restriction e p λ | supp a is injective. From (ii) of the Lemma 4.6.16 it follows that c λ = ∑ g ∈ ker ( G ( e X | X ) → G ( X λ | X ) ) g (cid:16)e a ∗ e b (cid:17) = desc p λ (cid:16)e a e b (cid:17) where notation desc p λ means the descent (cf. Definition 4.5.56). From the proper-ties of the descent it turns out that desc p λ (cid:16)e a e b (cid:17) ∈ A ( X λ ) , i.e. c λ ∈ A ( X λ ) . Takinginto account the Remark 3.4.2 we conclude that c λ ∈ A ( X λ ) for all λ ∈ Λ . Here we consider the generalization of the Theorem 5.4.20.
Let ( X , Y ) be a sub-unital subspace having continuous trace (cf. Definition5.1.2) and let A def = C ∗ e ( X , Y ) be the C ∗ -envelope of ( X , Y ) . From the Lemma 5.1.3it follows that A has continuous trace, hence the spectrum X of A is the locallycompact Hausdorff space. Suppose that X is a second-countable, connected andlocally connected set. Let S ( X , x ) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) λ ∈ Λ ∈ FinTop pt bea pointed topological finite covering category such that for any µ > ν there is the296nique pointed covering p µν : (cid:0) X µ , x µ (cid:1) → ( X ν , x ν ) . From the Lemma 5.4.2 it turnsout that is a pointed algebraical finite covering category (cf. Definition 3.1.6). S A == { A ( p λ ) : A ֒ → A ( X λ ) } , (cid:8) A (cid:0) p µν (cid:1) : A ( X ν ) ֒ → A (cid:0) X µ (cid:1)(cid:9) From the Theorem 5.4.20 it follows that S A is good. Moreover if e X is the topolog-ical inverse limit of S ( X , x ) then the triple (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17)(cid:17) the infinite noncommutative covering of S A (cf. Definition 3.1.34). From the The-orem 5.3.21 it follows that there is pointed algebraical finite covering category ofoperator spaces (cf. Definition 3.2.2) S pt ( X , Y ) = { ( π X λ , π Y λ ) : ( X , Y ) → ( X λ , Y λ ) } λ ∈ Λ , n(cid:16) π ν X µ , π ν Y µ (cid:17)o µ , ν ∈ Λ ν > µ ! where X λ = C (cid:16) lift p λ [ X ] (cid:17) , π X λ = C ( p λ ) | X , π ν X µ = C (cid:0) p µ (cid:1)(cid:12)(cid:12) X µ . Theorem 5.4.23.
Consider the situation of 5.4.22. If e X the inverse noncommutativelimit (cf. Definition 3.2.4) of S pt ( X , Y ) then there is the natural complete isomorphism e X ∼ = C (cid:16) lift e p [ X ] (cid:17) where e p : e X → X is the natural transitive covering.Proof.
Let b A def = C ∗ -lim −→ λ ∈ Λ A ( X λ ) , and let π a : b A → B ( H a ) be the atomic repre-sentation, Denote by e p λ = e X → X λ for all λ ∈ Λ the natural coverings. Let e U ⊂ e X be an open subset which is homeomorphically mapped onto U = e p (cid:16) e U (cid:17) . Let e b ∈ C (cid:16) lift e p [ X ] (cid:17) supp e b ⊂ e U . (5.4.44)There is the net n desc e p λ (cid:16)e b (cid:17) ∈ X λ o λ ∈ Λ ⊂ b A . From the Lemma 4.6.18 it turns outthat e b = lim λ ∈ Λ π a (cid:16) desc e p λ (cid:16)e b (cid:17)(cid:17) where the strong limit is implied. So e b is subordinated to S pt ( X , Y ) (cf. Definition3.2.3). According to the Definition 3.2.4 one has e b ∈ e X . Following proof containstwo parts: 297i) C (cid:16) lift e p [ X ] (cid:17) ⊂ e X .(ii) e X ⊂ C (cid:16) lift e p [ X ] (cid:17) (i) The C -linear space of the given by (5.4.44) is dense in C (cid:16) lift e p [ X ] (cid:17) ⊂ e X withrespect to C ∗ -norm topology.(ii) For any x ∈ X denote by S x the linear space of all continuous linear functionals s : rep x ( C ∗ e ( X x , Y x )) → C such that s ( X x ) = { } . Since C ∗ e ( X x , Y x ) ∼ = K and K is a reflexive Banach space, i.e. K ∗∗ ∼ = K , and taking into account that X x ⊂ rep x ( C ∗ e ( X , Y )) is a closed subset one has X x = { t ∈ rep x ( C ∗ e ( X , Y )) | s ( t ) = ∀ s ∈ S x } If H x is the space of the representation rep x then for any s ∈ S there are ξ , η ∈ H x such that s ( a ) = ( ξ , a η ) H x for each a ∈ rep x ( C ∗ e ( X , Y )) . Denote by T x = n ( ξ , η ) ∈ H x × H x | a ( ξ , rep x ( a ) η ) H x ∈ S x o If H e x the space of the representation rep e x : A (cid:16) e X (cid:17) → B ( H e x ) then there if thenatural isomorphism c e x : H e x ∼ = H e p ( e x ) . So for any e x ∈ e X there is the the naturalrepresentation rep e x : A → B ( H e x ) . There is the set T e x = n (cid:16) c − e x ( ξ ) , c − e x ( η ) (cid:17) ∈ H e x × H e x (cid:12)(cid:12)(cid:12) ( ξ , η ) ∈ T x o such that X = (cid:26) a ∈ A (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e ξ , rep e x , ( a ) e η (cid:17) H e x = ∀ (cid:16) e ξ , e η (cid:17) ∈ T e x (cid:27) From the Theorem 5.3.21 it follows that X λ = C (cid:16) lift p [ X ] (cid:17) (cf. Equation (5.3.17))so one has X λ = (cid:26) a λ ∈ A λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e ξ , rep e x ( a λ ) e η (cid:17) H e x = ∀ (cid:16) e ξ , e η (cid:17) ∈ T e x (cid:27) If a λ ∈ X λ then (cid:16) e ξ , rep e x ( a λ ) e η (cid:17) H e x = (cid:16)e ξ , e η (cid:17) ∈ T e x . If e a is a strong limit of { a λ } then e a is a weak limit { a λ } so one has (cid:16) e ξ , rep e x ( e a ) e η (cid:17) H e x = ∀ (cid:16) e ξ , e η (cid:17) ∈ T e x ∀ e x ∈ e X . (5.4.45)298ince the linear span of T e x is dense in c − e x (cid:16) S p ( e x ) (cid:17) the equation (5.4.45) is equivalentto rep e x ( e a ) ∈ c − e x (cid:16) X p ( x ) (cid:17) , rep e x ( e a ) ∈ rep e x (cid:16) C (cid:16) lift e p [ X ] (cid:17)(cid:17) ∀ e x ∈ e X .Hence for every e x ∈ e X one has rep e x (cid:16) e X (cid:17) ⊂ rep e x (cid:16) C (cid:16) lift e p [ X ] (cid:17)(cid:17) . From the Lemma4.5.50 it turns out that e X ⊂ C (cid:16) lift e p [ X ] (cid:17) . Following theorem gives universal coverings of operator spaces with continuoustrace.
Theorem 5.4.24.
Let A be a separable C ∗ -algebra with continuous trace, such that thespectrum X of A is a connected, locally connected, locally compact, second-countable,Hausdorff space. If there is the universal topological covering e p : e X → X (cf. DefinitionA.2.20) such that G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is a residually finite group then the triple (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17)(cid:17) is the universal covering of A (cf. Definition 3.3.1).Proof. Denote by the (topological) finite covering category S X = { p λ : X λ → X } λ ∈ Λ (cf. Definition 4.11.2) such that any finite-fold transitive covering e X ′ → X belongsto S X . From the Corollary 5.3.19 it turns out that there is the (algebraical) fi-nite covering category S A = { π λ : A ֒ → A ( X λ ) } λ ∈ Λ . From the Theorem 5.3.18it follows that any noncommutative finite-fold covering A ֒ → A ′ belongs to S A .From the Lemma 5.4.2 the pointed a topological finite covering category S ( X , e p ( e x )) induces the pointed algebraical finite covering category S pt A given by (5.4.3), i.e. S pt A = { A ( p λ ) : A ֒ → A ( X λ ) } , (cid:8) A (cid:0) p µν (cid:1) : A ( X ν ) ֒ → A (cid:0) X µ (cid:1)(cid:9) .From the Lemma 4.11.22 it turns out that the pointed space (cid:16) e X , e x (cid:17) is the in-verse noncommutative limit of S ( X , x ) . From the Theorem 5.4.20 it turns out that (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17)(cid:17) is infinite noncommutative covering of S A (cf. Def-inition 3.1.34), and taking into account the Definition 3.3.1 one concludes that (cid:16) A , A (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17)(cid:17) is the universal covering of A .299 orollary 5.4.25. Let A be a separable C ∗ -algebra, such that the spectrum X of A is isa connected locally connected locally compact second-countable Hausdorff space. If X is alocally path connected semilocally 1-connected space, then(i) A has the algebraical universal approximately finite covering.(ii) Moreover if π ( X , x ) is residually finite then there is the group isomorphism π ( X , x ) ∼ = π (cid:0) A , (cid:8) A (cid:0) p µν (cid:1)(cid:9)(cid:1) . (5.4.46) Proof. (i) From the Lemmas A.2.21 and A.2.22 it turns out that there is the uni-versal covering e X → X . From the Corollary A.2.19 it turns out that π ( X , x ) ≈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) , hence G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is residually finite. From the Theorem 5.4.24 it fol-lows that A (cid:16) e X (cid:17) is the universal covering of A .(ii) Let e p : e X → X be the universal covering of X and let e x ∈ e X be such that e p ( e x ) = x . If e p λ : e X → X λ are natural coverings and x λ = e p λ ( e x ) then there is apointed topological finite covering category S ( X , e p ( e x )) = (cid:8) p λ : (cid:0) X λ , x λ (cid:1) → ( X , x ) (cid:9) .From the Lemma 5.4.2 there is the associated to S ( X , e p ( e x )) pointed algebraicalfinite covering category (cid:18) { A ( p λ ) : A → A ( X λ ) } , (cid:8) A (cid:0) p µν (cid:1)(cid:9) µ , ν ∈ Λ µ > ν (cid:19) . such that A (cid:16) e X (cid:17) is the inverse limit of it (cf. Definition 3.1.34). From (3.3.1) it follows that π (cid:0) A , (cid:8) A (cid:0) p µν (cid:1)(cid:9)(cid:1) = G (cid:16) A (cid:16) e X (cid:17) (cid:12)(cid:12)(cid:12) A (cid:17) .Otherwise from the Corollary A.2.19 it follows that π ( X , x ) ≈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) and taking into account G (cid:16) A (cid:16) e X (cid:17) (cid:12)(cid:12)(cid:12) A (cid:17) ∼ = G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) one has π ( X , x ) ∼ = π (cid:0) A , (cid:8) A (cid:0) p µν (cid:1)(cid:9)(cid:1) . Let ( X , Y ) be sub-unital operator space with continuous trace (cf. Definition5.1.2) C ∗ -algebra with continuous trace. From the Lemma 5.1.3 it follows that the C ∗ -envelope C e ( X , Y ) has continuous trace. Theorem 5.4.27.
Consider the situation described in 5.4.26, and suppose that the spec-trum X of C e ( X , Y ) a connected locally connected locally compact second-countable Haus-dorff space. If there is the universal topological covering e p : e X → X (cf. DefinitionA.2.20) such that G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is residually finite group then C (cid:16) lift e p [ X ] (cid:17) is the univer-sal covering of ( X , Y ) (cf. Definition 3.3.4). roof. From the Theorem 5.4.24 it follows that (cid:16) C e ( X , Y ) , C e ( X , Y ) (cid:16) e X (cid:17) , G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17)(cid:17) is the universal covering of C e ( X , Y ) . From the Definition 3.3.4 and the Theorem5.4.23 it follows that C (cid:16) lift e p [ X ] (cid:17) is the universal covering of ( X , Y ) . Corollary 5.4.28.
Consider the situation described in 5.4.26 and the Theorem 5.4.27 If X is a locally path connected semilocally 1-connected space, and π ( X , x ) is residuallyfinite then π (cid:0) ( X , Y ) , (cid:8) C ∗ e ( X , Y ) (cid:0) p µν (cid:1)(cid:9)(cid:1) ∼ = π ( X , x ) Proof.
Follows from the Theorem 5.4.24 Corollary 5.4.24 and the Definition (3.3.4).
Let us consider a compact Riemannian manifold M with a spin bundle S . Thealgebra C ( M ) ⊗ M n ( C ) is a homogeneous of order n C ∗ -algebra (cf. DefinitionD.8.20). If p : e M → M is an infinite transitive covering then from the Theorem5.4.20 it follows that there is a noncommutative infinite covering (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) . (5.4.47)If ϕ fin : M m ( C ) → B (cid:0) C k (cid:1) is a faithful representation then there is the action C ( M ) ⊗ M m ( C ) × C (cid:16) M , n S x ⊗ C k o , Γ ( M , S ) ⊗ C k (cid:17) →→ C (cid:16) M , n S x ⊗ C k o , Γ ( M , S ) ⊗ C k (cid:17) (5.4.48)such that from a = a ′ ⊗ T , a ′ ∈ C ( M ) T ∈ M m ( C ) , ξ = ξ ′ ⊗ x , ξ ′ ∈ Γ ( M , S ) x ∈ C k it follows that a ξ = a ′ ξ ′ ⊗ ϕ fin ( T ) x There is the alternative description of the above action. The C ∗ -algebra C ( M ) ⊗ M m ( C ) is a continuity structure for M and the family { M m ( C ) x } x ∈X such that C ( M ) ⊗ M m ( C ) ∼ = C ( M , { M m ( C ) x } , C ( M ) ⊗ M m ( C )) x ∈ X the representation ϕ fin induces the action M m ( C ) x × S x ⊗ C k →S x ⊗ C k such that a x ( s x ⊗ y ) = s ⊗ ϕ fin ( a x ) y , ∀ a x ∈ M m ( C ) x , ∀ s x ∈ S x , ∀ y ∈ C k . (5.4.49)If a ∈ C ( M , { M m ( C ) x } , C ( M ) ⊗ M m ( C )) corresponds to a family { a x ∈ M m ( C ) } x ∈X and ξ ∈ C (cid:0) M , (cid:8) S x ⊗ C k (cid:9) , Γ ( M , S ) ⊗ C k (cid:1) corresponds to afamily (cid:8) ξ x ∈ S x ⊗ C k (cid:9) x ∈X then the product a ξ corresponds to the family (cid:8) a x ξ x ∈ S x ⊗ C k (cid:9) x ∈X where the action (5.3.21) is implied. From the Lemma 4.8.16it follows that there is the isomorphism (cid:16) C c (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) ⊗ C ( M ) ⊗ M m ( C ) (cid:16) C (cid:16) M , n S x ⊗ C k o , Γ ( M , S ) ⊗ C k (cid:17)(cid:17) ≈ −→ ≈ −→ C c (cid:16) lift p h(cid:16) C (cid:16) M , n S x ⊗ C k o , Γ ( M , S ) ⊗ C k (cid:17)(cid:17)i(cid:17) (5.4.50)of left C (cid:16) e M (cid:17) ⊗ M m ( C ) -modules. Note that S x ⊗ C k ∼ = S kx and taking into ac-count Lemmas 4.5.4 and 4.5.59 the given by (5.3.22) isomorphism can be repre-sented by the following way (cid:16) C c (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) Γ (cid:16) M , S k (cid:17) ≈ −→ Γ c (cid:16) e M , e S k (cid:17) (5.4.51)where e S is the inverse image of S . For any x ∈ X the space S x is a Hilbert space,so S kx = S x ⊗ C k has the natural structure of Hilbert space which is the finitedirect product of Hilbert spaces. It follows that S k is a Hermitian vector bundle(cf. Definition A.3.11) and L (cid:0) M , S k (cid:1) ∼ = L ( M , S ) k is a Hilbert space with thegiven by (A.3.3) scalar product, i.e. ( ξ , η ) L ( M , S k ) def = Z M ( ξ x , η x ) x d µ . (5.4.52) Lemma 5.4.29.
If p : e M → M is a finite-fold transitive covering and e S is the inverseimage of S by p (cf. Definition A.3.7) then the following conditions hold:(i) The map (5.3.23) can be extended up to the following homomorphism (cid:16) C c (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) L (cid:16) M , S k (cid:17) ≈ −→ L (cid:16) e M , e S k (cid:17) of left C (cid:16) e M (cid:17) ⊗ M m ( C ) -modules. ii) The image of (cid:16) C c (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) L (cid:0) M , S k (cid:1) is dense in L (cid:16) e M , e S k (cid:17) .Proof. There is the natural inclusion of C ∗ -algebras C (cid:16) e M (cid:17) → C (cid:16) e M (cid:17) ⊗ M m ( C ) , e a e a ⊗ M m ( C ) .Using it one can prove this lemma as well as the Lemma 4.11.41 had been proved.Let both µ and e µ be Riemannian measure (cf. [26]) on both M and e M respec-tively which correspond to both the volume element (cf. (E.4.1)) and its p -lift (cf.(4.9.2)). If A def = C ( M ) ⊗ M m ( C ) , e A def = C (cid:16) e M (cid:17) ⊗ M m ( C ) and K (cid:16) e A (cid:17) is the Ped-ersen’s ideal of e A then one has K (cid:16) e A (cid:17) ∼ = C c (cid:16) e M (cid:17) ⊗ M m ( C ) . If e a ⊗ ξ , e b ⊗ η ∈ K (cid:16) e A (cid:17) ⊗ A Γ (cid:0) M , S k (cid:1) ⊂ L (cid:16) e M , e S k (cid:17) then e a ∈ C c (cid:16) e M (cid:17) , so there exists covering sumfor supp e a (cf. Definition 4.2.6), i.e. a finite subset f A supp e a ⊂ f A such that ∑ α ∈ A supp e a e a α ( e x ) = ∀ e x ∈ supp e a .The given by the Equation (3.4.4) scalar product ( · , · ) ind on C c (cid:16) e M (cid:17) ⊗ M m ( C ) ⊗ C ( M ) ⊗ M m ( C ) Γ ( M , S ) k satisfies to the following equation satis-303es to the following equation (cid:16)e a ⊗ ξ , e b ⊗ η (cid:17) ind = (cid:18) ξ , De a , e b E C ( e M ) η (cid:19) L ( M , S ) == ∑ e α ∈ f A supp e a (cid:18) ξ , De a e α e a , e b E C ( e M ) η (cid:19) L ( M , S ) == ∑ e α ∈ f A supp e a (cid:16) ξ , desc (cid:16)e a e α e a ∗ e b (cid:17) η (cid:17) L ( M , S ) == ∑ e α ∈ f A supp e a Z M (cid:16) ξ x , desc (cid:16)e a e α e a ∗ e b (cid:17) η x (cid:17) x d µ == ∑ e α ∈ f A supp e a Z M (cid:16) desc ( e e e α e a ) ξ x , desc (cid:16)e e e α e b (cid:17) η x (cid:17) x d µ == ∑ e α ∈ f A supp e a Z e M (cid:16)e a lift e U e α ( e e α ξ ) e x , e b lift e U e α ( e e α η ) e x (cid:17) e x d e µ == ∑ e α ∈ f A supp e a Z e M e a e α (cid:16)e a lift e U e α ( ξ ) e x , e b lift e U e α ( η ) e x (cid:17) e x d e µ == Z e M (cid:16)e a lift p ( ξ ) e x , e b lift p ( η ) e x (cid:17) e x d e µ = (cid:16)e a lift p ( ξ ) , e b lift p ( η ) (cid:17) L ( e M , e S ) == (cid:16) φ ( e a ⊗ ξ ) , φ (cid:16)e b ⊗ η (cid:17)(cid:17) L ( e M , e S ) (5.4.53)where φ is given by (4.8.21). The equation (5.4.53) means that ( · , · ) ind = ( · , · ) L ( e M , e S k ) ,and taking into account the dense inclusion C c (cid:16) e M (cid:17) ⊗ C ( M ) Γ (cid:0) M , S k (cid:1) ⊂ L (cid:16) e M , e S k (cid:17) with respect to the Hilbert norm of L (cid:16) e M , e S k (cid:17) one concludes that the space ofinduced representation coincides with L (cid:16) e M , e S k (cid:17) . It means that induced repre-sentation C (cid:16) e M (cid:17) × L (cid:16) e M , e S k (cid:17) → L (cid:16) e M , e S k (cid:17) is given by (A.3.4). So one has thefollowing lemma. Lemma 5.4.30.
If C ( M ) ⊗ M m ( C ) → B (cid:0) L (cid:0) M , S k (cid:1)(cid:1) is the described above repre-sentation and the representation e ρ : C (cid:16) e M (cid:17) ⊗ M m ( C ) → B (cid:16) e H (cid:17) is induced by thepair (cid:16) ρ , (cid:16) C ( M ) ⊗ M m ( C ) , C (cid:16) e M (cid:17) ⊗ M m ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17)(cid:17) (cf. Definition 3.4.5) then following conditions hold: a) There is the homomorphism of Hilbert spaces e H ∼ = L (cid:16) e M , e S k (cid:17) ,(b) The representation e ρ is described above action of C (cid:16) e M (cid:17) ⊗ M m ( C ) on L (cid:16) e M , e S k (cid:17) .Proof. (a) Follows from (5.3.25),(b) From the Lemma 5.3.22 it follows that the map (cid:16) C (cid:16) e M (cid:17) ⊗ M m ( C ) (cid:17) ⊗ C ( M ) ⊗ M m ( C ) L (cid:0) M , S k (cid:1) ≈ −→ L (cid:16) e M , e S k (cid:17) is the homomorphismof left C (cid:16) e M (cid:17) ⊗ M m ( C ) modules, so the given by (A.3.4), i.e. C (cid:16) e M (cid:17) ⊗ M m ( C ) -action coincides with the C (cid:16) e M (cid:17) ⊗ M m ( C ) -action the given by (2.3.2). Remark 5.4.31.
If the spectral triple e T def = (cid:16) C ∞ (cid:16) e M (cid:17) ⊗ M m ( C ) , L (cid:16) e M , e S (cid:17) ⊗ C k , e D / ⊗ Γ fin + Id L ( e M , e S ) ⊗ D fin (cid:17) is the geometrical p -lift of the spectral triple (cid:16) C ∞ ( M ) ⊗ M m ( C ) , L ( M , S ) ⊗ C k , D / ⊗ Γ fin + Id L ( M , S ) ⊗ D fin (cid:17) .(cf. Definition 5.2.4), then clearly the corresponding to e T representation of C ( M ) ⊗ M m ( C ) equals to the given by the Lemma 5.4.30 representation. Let /T def = (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) be a commutative spectral triple (cf. Equa-tion (E.4.6)), and let e p : e M → M be an infinite regular covering such that thecovering group G (cid:16) e M | M (cid:17) (cf. Definition A.2.3) is residually finite (cf. Defini-tion B.3.2). From the Lemma 4.11.22 it it turns out that there is a (topological)finite covering category S = { p λ : M λ → M } (cf. 4.11.2) such that the (topolog-ical) inverse limit of S is naturally homeomorphic to e M . From the PropositionE.5.1 it follows that e M and M λ ( ∀ λ ∈ Λ ) have natural structure of the Rieman-nian manifolds. According to the Example 4.7.5 denote by e S def = C (cid:16) lift p ( S ) (cid:17) and S λ def = C b (cid:16) lift p λ ( S ) (cid:17) (for all λ ∈ Λ ) the lifts of the spin bundle (cf. A.3). Similarlyto the Section 4.9 one can prove that for any λ ∈ Λ that there is the p λ -inverseimage p − λ / D of / D (cf. Definition 4.7.11) and the operator p − λ / D can be regardedas an unbounded operator on L ( M λ , S λ ) . Moreover there is the e p -inverse image e p − / D of / D (cf. Definition 4.7.11) and the operator e p − / D is an unbounded operator305n L (cid:16) e M , e S (cid:17) . From the Lemma 4.11.24 and the Theorem 4.11.37 it turns out that S pt C ( M ) = ( C ( M ) ֒ → C ( M λ ) } λ ∈ Λ , (cid:8) C (cid:0) p µν (cid:1) : C ( M ν ) ֒ → C (cid:0) M µ (cid:1)(cid:1) is a good pointedalgebraical finite covering category, and the triple (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is an infinite noncommutative covering of S C ( M ) . Otherwise from the Theorem4.11.49 it follows that S ( C ∞ ( M ) , L ( M , S ) , / D ) == n /T λ def = (cid:16) C ∞ ( M λ ) , L ( M λ , S λ ) , p − λ / D (cid:17)o λ ∈ Λ ∈ CohTriple (5.4.54)From the Theorem 5.4.20 one has a good pointed algebraical finite covering cate-gory S pt C ( M ) ⊗ M m ( C ) == n C ( p λ ) ⊗ Id M m ( C ) C ( M ) ⊗ M m ( C ) ֒ → C ( M λ ) ⊗ M m ( C ) o λ ∈ Λ , n C (cid:0) p µν (cid:1) ⊗ Id M m ( C ) : C ( M ν ) ⊗ M m ( C ) ֒ → C (cid:0) M µ (cid:1) ⊗ M m ( C ) o µ , ν ∈ Λ µ ≥ ν ! (5.4.55)and (cid:16) C ( M ) ⊗ M m , C (cid:16) e M (cid:17) ⊗ M m , G = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is the infinite noncommutative covering of S pt C ( M ) ⊗ M m ( C ) (cf. Definition 3.1.34). IfT fin def = (cid:0) M m ( C ) , C k , D fin (cid:1) is a finite spectral triple (cf. Definition E.6) then one candefine products of spectral triples (cf. Definition E.7)T def = /T × T fin = (cid:16) C ∞ ( M ) ⊗ M m ( C ) , L ( M , S ) ⊗ C k , D (cid:17) ,where D def = D / ⊗ Γ fin + Id L ( M , S ) ⊗ D fin ;T λ def = /T λ × T fin = (cid:16) C ∞ ( M λ ) ⊗ M m ( C ) , L ( M λ , S λ ) ⊗ C k , D λ (cid:17) ,where D λ def = p − λ D / ⊗ Γ fin + Id L ( M λ , S λ ) ⊗ D fin . (5.4.56) Lemma 5.4.32.
The indexed by Λ family of spectral triples S T def = { T λ } λ ∈ Λ (5.4.57) is a coherent set of spectral triples (cf. Definition 3.5.2). roof. Consider the atomic representation π a : C ∗ -lim −→ λ ∈ Λ C ( M λ ) ⊗ M m ( C ) → B (cid:16) b H (cid:17) (cf. Definition D.2.33). Any coherent set of spectral triples is weakly co-herent. So one needs check that S T satisfies to conditions (a)-(d) of the Definition3.5.1.(a) Follows from the Theorem 5.3.28.(b) For all λ ∈ Λ the C ∗ -algebra C ( M λ ) is the C ∗ -norm completion of C ∞ ( M λ ) .(c) Follows from the Theorem 5.4.20.(d) Follows from the Theorem 5.3.28.From the Lemma 5.4.21 it follows that the pointed algebraical finite coveringcategory S pt C ( M ) ⊗ M m ( C ) allows inner product, hence from the Definition 3.5.2 itturns out that S T is a coherent set of spectral triples. Lemma 5.4.33. If e W ∞ is the space of S T -smooth elements (cf. Definition 3.5.3) then e W ∞ = C ∞ c (cid:16) e M (cid:17) ⊗ M m ( C ) . Proof.
The proof of this lemma is similar to the proofs of the Lemmas 4.11.45 and4.11.46Similarly to (4.8.22) one can define a C ∞ c (cid:16) e M (cid:17) -module homomorphism e φ ∞ : (cid:16) C ∞ c (cid:16) e M ⊗ M n ( C ) (cid:17) ⊗ C ∞ ( M ) ⊗ M n ( C ) Γ ∞ ( M , S ) k (cid:17) → Γ ∞ c (cid:16) e M , e S (cid:17) k , n ∑ j = e a j ⊗ ξ j n ∑ j = e a j lift p (cid:0) ξ j (cid:1) (5.4.58)where both Γ ∞ ( M , S ) and Γ ∞ c (cid:16) e M , e S (cid:17) are defined by the Equations (E.4.2) and(4.9.6) respectively. Lemma 5.4.34.
The given by the Equation (5.4.58) homomorphism is an isomorphism.Proof.
The proof of the Lemma 4.8.11 uses the partition of unity. However fromthe Proposition A.1.26 it turns out that there is a smooth partition of unity. Usingit one can proof this lemma as well as the Lemma 4.8.11 has been proved.
Let us explicitly the specialization of the explained in 3.5.6 construction.Following table reflects the mapping between general theory and the specializa-tion. 307eneral theory The specializationHilbert spaces H and e H L ( M , S ) k and L (cid:16) e M , e S (cid:17) k Pre- C ∗ -algebra A C ∞ ( M ) ⊗ M n ( C ) Pedersen’s ideal K (cid:16) e A (cid:17) C c (cid:16) e M (cid:17) The space of e W ∞ C ∞ c (cid:16) e M (cid:17) ⊗ M n ( C ) smooth elementsDirac operators D D / ⊗ Γ fin + Id L ( M , S ) k ⊗ D fin e D ? H ∞ def = T ∞ n = Dom D n ⊂ H Γ ∞ ( M , S ) k = T ∞ n = Dom D n Similarly to the given by 4.11.48 construction we can prove the following explicitequation e D = e p − D / ⊗ Γ fin + Id L ( M , S ) k ⊗ D fin (5.4.59)for the specialization of the given by (3.5.9) operator. Theorem 5.4.36.
In the described in this section situation following conditions hold:(i) The given by (5.4.56) (5.4.57) coherent set S T def = { T λ } λ ∈ Λ of spectral triples is good (cf. Definition 3.5.5).(ii) The (cid:16) C ( M ) ⊗ M n ( C ) , C (cid:16) e M (cid:17) ⊗ M n ( C ) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) -lift of the spectral triple T def = (cid:16) C ∞ ( M ) ⊗ M n ( C ) , L ( M , S , µ ) k , D / ⊗ Γ fin + Id L ( M , S ) k ⊗ D fin (cid:17) (cf. Definition 3.5.7) coincides with the geometrical p-lift of T (cf. Definition 5.2.4).Proof. The proof of this theorem is similar to the proof of the Theorem 4.11.49.308 hapter 6
Coverings of C ∗ -Hilbert modulesover commutative C ∗ -algebras The notion of continuous vector field is discussed in D.8.2, here we consider thespace of continuous vector fields as an operator space. Let X be a locally compactHausdorff topological space and let {H x } x ∈X be a family of infinite dimensionalHilbert spaces. Let F be a continuity structure for X and the {H x } (cf. DefinitionD.8.27). If X C ( X ) def = C ( X , {H x } , F ) (6.1.1)( cf. Definition 4.5.8) then form the Lemma 4.12.3 it turns out that X C ( X ) is a C ∗ -Hilbert C ( X ) -module. It is known that any C ∗ -Hilbert module is an operatorspace. Here we consider a detailed construction of this operator space. Denote by H C ( X ) def = M x ∈X C x , H X def = M x ∈X H x the Hilbert direct sums where C x def = C for all x ∈ X . Any ξ ∈ X C ( X ) correspondsto a family { ξ x ∈ H x } which gives the operator a ( ξ ) ∈ B (cid:16) H X , H C ( X ) (cid:17) ; { η x ∈ H x } 7→ n ( η x , ξ x ) H x ∈ C x o (6.1.2)309enote by o ( ξ ) ∈ B (cid:16) H X M H C ( X ) (cid:17) ; o ( ξ ) def = (cid:18) a ( ξ ) (cid:19) There is the C -linear injective map o : X C ( X ) → B (cid:16) H X L H C ( X ) (cid:17) yields thestructure of the operator space. If o ∗ ( ξ ) is adjoint of o ( ξ ) then o ∗ ( ξ ) correspondsto the operator (cid:18) a ∗ ( ξ ) (cid:19) ∈ B (cid:16) H X M H C ( X ) (cid:17) where a ∗ ( ξ ) ∈ B (cid:16) H C ( X ) , H X (cid:17) ; { c x ∈ C x } 7→ { c x η x ∈ C x } (6.1.3)Denote by A ⊂ B (cid:16) H C ( X ) + H X (cid:17) the C ∗ -algebra generated by the C -linear space n a ∈ B (cid:16) H C ( X ) M H X (cid:17)(cid:12)(cid:12)(cid:12) ∃ ξ ∈ X C ( X ) a = o ( ξ ) o Any a ∈ A corresponds to the family { a x ∈ H x L C x } x ∈X . For any x ∈ X and ξ ∈ X C ( X ) one has k ξ x k = k o ( ξ ) x k it turns for any a ∈ A one has ( x
7→ k a x k ) ∈ C ( X ) where the family { a x ∈ H x L C x } x ∈X corresponds to a . It turns out that A is acontinuity structure for X and the {H x L C x } (cf. Definition D.8.27) so A = C (cid:16) X , n H x M C x o , A (cid:17) (6.1.4)(cf. Definition 4.5.8). For all ξ , η ∈ X C ( X ) one has o ( ξ ) o ( η ) = o ∗ ( ξ ) o ∗ ( η ) = o ( ξ ) o ∗ ( η ) = (cid:18) ξ ih η (cid:19) ∈ K (cid:16) X C ( X ) (cid:17)
00 0 ! , o ∗ ( ξ ) o ( η ) = h ξ , η i X C ( X ) ! ∈ (cid:18) C ( X ) (cid:19) ,it follows that A ∼ = K (cid:16) X C ( X ) (cid:17) a ∗ (cid:16) X C ( X ) (cid:17) a (cid:16) X C ( X ) (cid:17) C ( X ) . (6.1.5)310 emma 6.1.1. If A is given by (6.1.4) and ˆ A is the spectrum of A then there is the naturalhomeomorphism ˆ A ∼ = X .Proof. If B def = K (cid:16) X C ( X ) (cid:17)
00 0 ! ⊂ A then for all b ′ b ′′ ∈ B and a ∈ A one has b ′ ab ′′ ∈ B , i.e. B is a hereditary subalgebraof A (cf. Lemma D.1.17). If ˆ B is the spectrum of B then from the Proposition D.2.20it follows that ˆ A \ hull ( B ) ∼ = ˆ B . If hull ( B ) = ∅ and y ∈ hull ( B ) then ρ y ( B ) = ξ ∈ X C ( X ) such that ρ y ( o ( ξ )) = ρ y ( o ( ξ ) o ∗ ( ξ )) = o ( ξ ) o ∗ ( ξ ) ∈ B one has a contradiction it turns out ρ y ( o ( ξ )) = ρ y ( o ∗ ( ξ )) = ξ ∈ X C ( X ) . If C def = (cid:18) C ( X ) (cid:19) ⊂ A then C is generated by elements o ∗ ( ξ ) o ( η ) where ξ , η ∈ X C ( X ) , so ρ y ( C ) = { } . Itfollows that ρ y ( A ) = { } but it is impossible. So one has hull ( B ) = ∅ and ˆ A ∼ = ˆ B .From B ∼ = K (cid:16) X C ( X ) (cid:17) and the Proposition D.8.18 it follows that ˆ A ∼ = ˆ B ∼ = X . Corollary 6.1.2.
If A is given by (6.1.4) the A is a C ∗ - algebra with continuous trace.Proof. For any x ∈ X there is a positive f ∈ C c ( X ) such that f ( x ) =
1. Theelement (cid:18) f (cid:19) ∈ A satisfies to (iii) of the Proposition D.8.18 so A is a C ∗ - algebra with continuoustrace.The map o : X C ( X ) → A is an inclusion so X C ( X ) can be regarded as thesubspace of B (cid:16) H X L H C ( X ) (cid:17) . If Y def = C · H X L H C ( X ) ⊕ X C ( X ) then (cid:16) X C ( X ) , Y (cid:17) is a sub-unital operator space (cf. Definition 2.6.1). Lemma 6.1.3.
The given by (6.1.4) C ∗ -algebra A is the C ∗ -envelope of (cid:16) X C ( X ) , Y (cid:17) (cf.Definition 2.6.4), i.e. A ∼ = C ∗ e (cid:16) X C ( X ) , Y (cid:17) (cf. (2.6.2) ).Proof. Firstly we prove that A + is the C ∗ -envelope C ∗ e ( Y ) of Y . From the natural in-clusion j Y : Y ֒ → A + of operator spaces it follows that ( A + , j Y ) is the C ∗ -extensionof Y (cf. Definition D.7.9). From the Theorem D.7.11 it turns out that there is the311urjective *-homomorphism k : A + → C ∗ e ( Y ) . If A + is not isomorphic to C ∗ e ( Y ) then ker k = { } . Note that ker k is a closed self-adjoint ideal. Consider a positivenonzero element a = (cid:18) c b ∗ b d (cid:19) ∈ ker k where the matrix representation (6.1.5) is used. It follows that c = d = c = c nonempty ideal I ∈ K (cid:16) X C ( X ) (cid:17) andan open subset U ∈ X such that ker k = K (cid:16) X C ( X ) (cid:17)(cid:12)(cid:12)(cid:12) U (cf. D.8.2). There is thenonzero ξ ∈ X C ( X ) such that supp ξ ⊂ U . From o ( ξ ) o ∗ ( ξ ) ∈ ker k it turns outthat ( k ◦ j Y ( ξ )) ( k ◦ j Y ( ξ )) ∗ = k ◦ j Y ( ξ ) =
0, but it is impossible. If d = V ⊂ X such that d ( x ) > x ∈ V . There is anonzero η ∈ X C ( X ) such that supp η ⊂ V . From o ∗ ( η ) o ( η ) ∈ ker k it turns outthat ( k ◦ j Y ( η )) ∗ ( k ◦ j Y ( η )) = k ◦ j Y ( η ) = k is *-isomorphism and A + ∼ = C ∗ e ( Y ) . Otherwise A is the minimal containing o (cid:16) X C ( X ) (cid:17) C ∗ -subalgebra of A + . From the Definition 2.6.4 it follows that A ∼ = C ∗ e ( X , Y ) . Following Theorems follow from from the Corollary 6.1.2, Lemma 6.1.3.
Theorem 6.2.1.
If X C ( X ) is given by (6.1.1) and X is a second-countable space lo-cally connected space then the Theorem 5.3.21 5.4.23 yields the 1-1 correspondence be-tween transitive finite-fold coverings of X and noncommutative finite-fold coverings of (cid:16) X C ( X ) , Y (cid:17) . Theorem 6.2.2.
If X C ( X ) is given by (6.1.1) and X is a second-countable space locallyconnected space then the Theorem 5.4.23 yields the 1-1 correspondence between transitiveinfinite coverings of X with residually finite covering group and noncommutative infinitecoverings of (cid:16) X C ( X ) , Y (cid:17) . Taking into account the Theorem 5.4.27 one has
Theorem 6.2.3.
If X C ( X ) is given by (6.1.1) and X is a second-countable space locallyconnected space. If there is the universal topological covering e p : e X → X (cf. DefinitionA.2.20) such that G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) is residually finite group then C (cid:16) lift e p h X C ( X ) i(cid:17) is theuniversal covering of (cid:16) X C ( X ) , Y (cid:17) (cf. Definition 3.3.4). Corollary 6.2.4.
Consider the situation described in the Theorem 6.2.3 If X is a locallypath connected semilocally 1-connected space, and π ( X , x ) is residually finite then π (cid:16)(cid:16) X C ( X ) , Y (cid:17) , n C ∗ e (cid:16) X C ( X ) , Y (cid:17) (cid:0) p µν (cid:1)o(cid:17) ∼ = π ( X , x ) hapter 7 Foliations and coverings
For any foliated space ( M , F ) denote by M / F the set of all leaves, and for anyleaf L ∈ M / F we select a point x L ∈ L . We write ρ L def = ρ x L : C ∗ r ( M , F ) → B (cid:0) L ( G x L ) (cid:1) (7.1.1)where the representation ρ x L is given by the Equation (G.1.4). From the TheoremG.1.38 it turns out that the representation (7.1.1) is irreducible if and only if theleaf L has no holonomy. If M / F no hol ⊂ M / F the subset of leaves which have noholonomy then the spectrum of C ∗ r ( M , F ) coincides with M / F no hol . The atomicrepresentation π a of C ∗ r ( M , F ) (cf. Definition D.2.33) is given by π a = M L ∈ M / F no hol ρ L : C ∗ r ( M , F ) ֒ → B M L ∈ M / F no hol L ( G x L ) . (7.1.2) Lemma 7.1.1. If ( M , F ) is a foliated manifold without boundary and p : e M → M is acovering then for any x ∈ M there is a foliated chart ( V , φ ) such that x ∈ V and V isevenly covered by p. roof. If x ∈ M there is a point then there is an open neighborhood U whichis evenly covered by p (cf. Definition A.2.1). There is a foliated atlas {U α } α ∈ A ,hence there is α ∈ A such that x ∈ U α . There is a *-homomorphism ϕ : U α ≈ −→ R q × R n − q . The set W = ϕ ( U ∩ U α ) is an open neighborhood of ϕ ( x ) . Sincecubic neighborhoods are basis of all neighborhoods of ϕ ( x ) there is an inclusion ψ : R q × R n − q → W which is a foliated chart. If V = ϕ − ( W ) and φ = ψ − ◦ ϕ then ( V , φ ) is a foliated chart, such that V is evenly covered by p . Lemma 7.1.2.
Let ( M , F ) be a foliated manifold without boundary, and let p : (cid:16) e M , e F (cid:17) → ( M , F ) be a regular covering (cf. Definition G.2.10). If e U is a foliated chart such thatthe restriction p | e U : e U → X is injective, and U = p (cid:16) e U (cid:17) then there are the naturalinvolutive isomorphisms ψ e U : C ∗ r ( U , F | U ) ≈ −→ C ∗ r (cid:16) e U , e F (cid:12)(cid:12)(cid:12) e U (cid:17) , ψ e U : Γ c ( U , F | U ) ≈ −→ Γ c (cid:16) e U , e F (cid:12)(cid:12)(cid:12) e U (cid:17) . (7.1.3) Proof.
Follows from the isomorphism ( U , F | U ) ∼ = (cid:16) e U , e F (cid:12)(cid:12)(cid:12) e U (cid:17) of foliated manifolds. Definition 7.1.3.
In the situation the Lemma 7.1.2 we say that if e a ∈ C ∗ r (cid:16) e U , e F (cid:12)(cid:12)(cid:12) e U (cid:17) then ψ − e U ( e a ) is the p - descent of e a . We write desc p ( e a ) def = ψ − e U ( e a ) .Conversely if a ∈ C ∗ r ( U , F | U ) then e a = ψ e U ( a ) ∈ C ∗ r (cid:16) e U , e F (cid:12)(cid:12)(cid:12) e U (cid:17) is said to be the p - e U - lift of a . We write lift p e U ( e a ) def = ψ e U ( a ) .Sometimes lift p e U is replaced with lift p e U . Remark 7.1.4.
From the Definition 7.1.3 it follows that lift p e U ◦ desc p ( e a ) = e a , desc p ◦ lift p e U ( a ) = a . (7.1.4)316 emark 7.1.5. If ( M , F ) is a foliated manifold without boundary and both p ′ : M ′ → M , p ′′ : M ′′ → M ′ are coverings then for any open subset U ′′ of M ′′ suchthat the restriction p ′ ◦ p ′′ | U ′′ and any a ′′ ∈ C ∗ r ( U ′′ , F ′′ | U ′′ ) ⊂ C ∗ r ( M ′′ , F ′′ ) one has desc p ′ ◦ p ′′ (cid:0) a ′′ (cid:1) = desc p ′ ◦ desc p ′′ (cid:0) a ′′ (cid:1) . (7.1.5) Let ( M , F ) be a foliated manifold without boundary, and let p : (cid:16) e M , e F (cid:17) → ( M , F ) be a regular covering of foliations (cf. Definition G.2.10). For any x ∈ M we select a chart ( U , ϕ ) such that x ∈ U and U is evenly covered by p . In resultone has a foliated atlas {U α } α ∈ A associated to F (cf. Definition G.1.6) such that U α is evenly covered by p . From the Lemma G.1.13 one can suppose that theatlas is regular. For any α ∈ A we select a connected open subset e U α which ishomemorphically mapped onto U α . If f A = A × G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) and e U ( α , g ) = g e U α then n e U α o e α ∈ f A is a foliated atlas of (cid:16) e M , e F (cid:17) which is regular. Definition 7.1.7.
Let ( M , F ) be a foliated manifold, and let p : e M → M be acovering. A regular foliated atlas A = {U ι } ι ∈I of M is said to be subordinated to p if U ι is evenly covered by p for all ι ∈ I . Definition 7.1.8.
Let ( M , F ) be a foliated manifold, and let p : e M → M be acovering. Let A = {U ι } ι ∈I of M be subordinated to p regular foliated atlas. If e A = n e U e ι o e ι ∈ e I is the regular foliated atlas of e M such that for any e ι ∈ e I there is ι ∈ I such that p (cid:16) e U e ι (cid:17) = U ι then we say that e A is the p - lift of A . Definition 7.1.9.
Let A = {U ι } ι ∈I be a regular foliated atlas of M . We say that thefamily of given by (G.1.3) V α = G ( U ι ) ... G ( U ι k ) ∈ G ( M , F ) , α = ( ι , ..., ι k , ) (7.1.6)open Hausdorff sets is induced by A . Lemma 7.1.10.
Let ( M , F ) be a foliation, and let p : (cid:16) e M , e F (cid:17) → ( M , F ) be a reg-ular covering (cf. Definition G.2.10). Let A = {U ι } ι ∈I of M be a subordinated to p,regular foliated atlas (cf. Definition G.1.12). Let e A = n e U e ι o e ι ∈ e I be the p-lift of A . Let V α = G ( U ι ) ... G ( U ι k ) = ∅ . If p (cid:16) e U e ι k (cid:17) = U ι k then for there is the unique tuple n e U e ι , ...., e U e ι k − o such that p (cid:16) e U e ι j (cid:17) = U ι j for any j =
1, ..., k − . • G (cid:16) e U e ι (cid:17) ... G (cid:16) e U e ι k (cid:17) = ∅ .Proof. If n e U ι λ o λ ∈ Λ ⊂ e A is such that p (cid:16) e U e ι λ (cid:17) = U ι k − for every λ ∈ Λ then thereis the unique λ ∈ Λ such that e U ι λ ∩ e U e ι k = ∅ . It follows that G (cid:16) e U e ι λ (cid:17) G (cid:16) e U e ι k (cid:17) = ∅ and G (cid:16) e U e ι λ (cid:17) G (cid:16) e U e ι k (cid:17) = ∅ for all λ = λ . Similarly on can find the unique e U e ι k − , ..., e U e ι such that G (cid:16) e U e ι (cid:17) ... G (cid:16) e U e ι k (cid:17) = ∅ . Remark 7.1.11.
In the situation of the Lemma 7.1.10 the chart e U e ι k can be replacedwith e U e ι . From (G.1.5) it turns out that there is the surjective map Γ ⊕ : M α Γ c (cid:16) V α , Ω (cid:17) Γ ⊕ −→ Γ c (cid:16) G , Ω (cid:17) .Otherwise from the Definition G.1.27 it follows that Γ c (cid:0) G , Ω (cid:1) is dense in C ∗ r ( M , F ) with respect the given by (G.1.6) to pseudonorm for all α there is the inclusion Γ c (cid:0) V α , Ω (cid:1) ֒ → Γ c (cid:0) G , Ω (cid:1) which can be regarded as a map φ α : Γ c (cid:16) V α , Ω (cid:17) → C ∗ r ( M , F ) . (7.1.7)The algebraic C -linear span of elements φ α ( a α ) is dense in C ∗ r ( M , F ) Lemma 7.1.13. If ( M , F ) is a foliated space, and p : (cid:16) e M , e F (cid:17) → ( M , F ) is a regularcovering (cf. Definition G.2.10) then one has:(i) There is the natural injective *-homomorphismC ∗ b ( p ) : C ∗ r ( M , F ) ֒ → M (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:17) (7.1.8) (ii) If p is a finite-fold covering then C ∗ b ( p ) ( C ∗ r ( M , F )) ⊂ C ∗ r (cid:16) e M , e F (cid:17) , i.e. is thenatural injective *-homomorphismC ∗ r ( p ) : C ∗ r ( M , F ) ֒ → C ∗ r (cid:16) e M , e F (cid:17) . (7.1.9)318 roof. (i) Let A = {U ι } ι ∈I of M be a subordinated to p regular foliated atlas. Let e A = n e U e ι o e ι ∈ e I be the p - lift of A . For any U ι ∈ A and e U e ι ∈ e A consider given by(G.1.7) maps, i.e. j U ι : Γ c (cid:16) G ( U ι ) , Ω (cid:17) ֒ → Γ c (cid:16) G ( M ) , Ω (cid:17) , j e U e ι : Γ c (cid:16) G (cid:16) e U e ι (cid:17) , e Ω (cid:17) ֒ → Γ c (cid:16) G (cid:16) e M (cid:17) , e Ω (cid:17) Let V α ′ = G (cid:16) U ι ′ (cid:17) ... G (cid:16) U ι ′ k ′ (cid:17) = ∅ and let e V f α ′′ = G (cid:16) e U e ι ′′ (cid:17) ... G (cid:16) e U e ι ′′ k ′′ (cid:17) = ∅ . Let a ′ α ′ = j U ι ′ (cid:16) a ′ ι ′ (cid:17) · ... · j U ι k ′ (cid:16) a ′ ι ′ k ′ (cid:17) ∈ Γ c (cid:16) V α ′ , Ω (cid:17) , a ′ ι ′ j ∈ Γ c (cid:16) G (cid:16) U ι ′ j (cid:17) , Ω (cid:17) ; e a ′′ e α ′′ = j e U e ι ′′ (cid:16)e a ′′ e ι ′ (cid:17) · ... · j U e ι k ′′ (cid:16)e a ′′ e ι ′′ k ′′ (cid:17) ∈ Γ c (cid:16) e V e α ′′ , e Ω (cid:17) , e a ′′ e ι ′ j ∈ Γ c (cid:16) G (cid:16) U e ι ′′ j (cid:17) , e Ω (cid:17) and suppose a = φ α ′ (cid:0) a ′ α ′ (cid:1) ∈ C ∗ r ( M , F ) , (7.1.10) e a = φ e α ′′ (cid:0)e a ′′ e α ′′ (cid:1) ∈ C ∗ r (cid:16) e M , e F (cid:17) , (7.1.11)(cf. (7.1.7)). One has p − (cid:16) U ι ′ k ′ (cid:17) = F λ ∈ Λ e U λ e ι ′ k ′ . There is no more then one λ ∈ Λ such that e U λ e ι ′ k ′ ∩ e U e ι ′′ = ∅ . If e U λ e ι ′ k ′ ∩ e U e ι ′′ = ∅ then we set e aa =
0. Otherwisewe set e U e ι ′ k ′ = e U λ e ι ′ k ′ and according to the Lemma 7.1.10 there is the unique tuple (cid:26) e U ′ e ι , ...., e U ′ e ι ′ k ′− (cid:27) such that • p (cid:16) e U ′ e ι ′ (cid:17) = U ι ′ j for any j =
1, ..., k − • G (cid:16) e U ′ e ι ′ (cid:17) ... G (cid:16) e U ′ e ι ′ k (cid:17) = ∅ .We define a e a = φ α ′ lift e U e ι ′ p (cid:16) a ′ ι ′ (cid:17) · ... · lift e U e ι ′ k ′ p (cid:16) a ′ ι ′ k ′ (cid:17)! e a ∈ C ∗ r (cid:16) e M , e F (cid:17) Similarly one can define e aa ∈ C ∗ r (cid:16) e M , e F (cid:17) . By linearity this product can be ex-tended up to any finite sum of given by (7.1.11) elements. Since the C -linear spanof given by (7.1.11) is dense in C ∗ r (cid:16) e M , e F (cid:17) we have products a e a and e aa for every319 ∈ C ∗ r (cid:16) e M , e F (cid:17) . Using the same reasons the products a e a and e aa can be extendedup to the linear span of given by (7.1.10) elements, hence for any a ∈ C ∗ r ( M , F ) and e a = ∈ C ∗ r (cid:16) e M , e F (cid:17) one has a e a , e aa ∈ C ∗ r (cid:16) e M , e F (cid:17) .(ii) If p is a finite-fold covering then for any ι ∈ I and p − ( U ι ) = F λ ∈ Λ e U λι thenthe set Λ is finite. So for any a ∈ C ∗ r (cid:16) U ι , F | U ι (cid:17) ⊂ C ∗ r ( X , F ) (7.1.12)one has ∑ λ ∈ Λ lift e U λι p ( a ) ∈ C ∗ r (cid:16) e M , e F (cid:17) ,i.e. there is a map C ∗ r (cid:16) U ι , F | U ι (cid:17) ֒ → C ∗ r (cid:16) e M , e F (cid:17) So one has ∑ λ ∈ Λ lift e V λα ′ ( a ) ∈ C ∗ r (cid:16) e M , e F (cid:17) . (7.1.13)Above equation gives a map from the minimal algebra X which contains givenby (7.1.12) elements to C ∗ r (cid:16) e M , e F (cid:17) . Otherwise X is dense in C ∗ r ( M , F ) (cf. theCorollary G.1.28) so the map X → C ∗ r (cid:16) e M , e F (cid:17) can be uniquely extended up to theinjective *-homomorphism C ∗ r ( M , F ) ֒ → C ∗ r (cid:16) e M , e F (cid:17) . Definition 7.1.14.
Denote by
FinFol the category of foliated manifolds and finite-fold regular coverings (cf. Definition G.2.10). The given by the Lemma 7.1.13from
FinFol to the category of C ∗ -algebras and ∗ -homomorphisms is said to bethe C ∗ r - functor . It will be denoted by C ∗ r . Lemma 7.1.15.
Let ( M , F ) be a foliated space, and let p : e M → M be a transitivecovering. Let (cid:16) e M , e F (cid:17) be the foliated space which comes from p. If e U ⊂ e M is a connectedopen subset such that the restriction p | e U is injective, e a ∈ C ∗ r (cid:16) e M , e F (cid:17) is such that e b ∈ C ∗ r (cid:16) e U , e F U (cid:17) (cf. Lemma G.2.3) then the following series ∑ g ∈ G ( e M | M ) g e a (7.1.14) is convergent with respect to the strict topology of M (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:17) . Moreover the sumof the series in sense of the strict topology is equal to desc p ( e a ) . roof. Let A = {U ι } ι ∈I of M be a subordinated to p regular foliated atlas. Let e A = n e U e ι o e ι ∈ e I be the p - lift of A . For all e b ′ ∈ C ∗ r (cid:16) e M , e F (cid:17) and ε > e b ∈ Γ c (cid:16) G (cid:16) e M (cid:17) , e Ω (cid:17) such that e b ′ = n ∑ k = j e U e ι k ,0 (cid:16)e a e ι k ,0 (cid:17) · ... · j e U e ι k , jk (cid:16)e a e ι k , jk (cid:17) , e a e ι k , jl ∈ Γ c (cid:16) G (cid:16) U e ι j , l (cid:17) , e Ω (cid:17) , (cid:13)(cid:13)(cid:13)e b ′ − e b (cid:13)(cid:13)(cid:13) < ε k e a k where e U e ι k , l ∈ e A for all k , l and j e U e ι k , l are given by (G.1.7) maps. For all k =
1, ..., n there is no more than one g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) such that g e U ∩ e U e ι k ,0 = ∅ . It follows thatthe set G ′ = ⊂ n g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:12)(cid:12)(cid:12) ∃ k ∈ {
1, ..., n } g e U ∩ e U e ι k ,0 = ∅ o is finite. Similarly the set G ′′ = n g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:12)(cid:12)(cid:12) ∃ k ∈ {
1, ..., n } g e U ∩ e U e ι k , jk = ∅ o is also finite. According to the given by the Lemma 7.1.13 construction one has desc p ( e a ) · j e U e ι k ,0 (cid:16)e a e ι k ,0 (cid:17) · ... · j e U e ι k , jk (cid:16)e a e ι k , jk (cid:17) == ( g e a ) j e U e ι k ,0 (cid:16)e a e ι k ,0 (cid:17) · ... · j e U e ι k , jk (cid:16)e a e ι k , jk (cid:17) g e U ∩ e U e ι k ,0 = ∅ ∀ g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) g e U ∩ e U e ι k ,0 = ∅ , j e U e ι k ,0 (cid:16)e a e ι k ,0 (cid:17) · ... · j e U e ι k , jk (cid:16)e a e ι k , jk (cid:17) desc p ( e a ) == j e U e ι k ,0 (cid:16)e a e ι k ,0 (cid:17) · ... · j e U e ι k , jk (cid:16)e a e ι k , jk (cid:17) ( g e a ) g e U ∩ e U e ι k , jk = ∅ ∀ g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) g e U ∩ e U e ι k , jk = ∅ . (7.1.15)If follows that both sets G ′ def = n g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:12)(cid:12)(cid:12) ( g e a ) e b = o ⊂ G ′ , G ′′ def = n g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:12)(cid:12)(cid:12) e b ( g e a ) = o ⊂ G ′′ (7.1.16)321re finite. From (7.1.15) it follows that ∑ g ∈ G ( e M | M ) ( g e a ) e b ′ = ∑ g ∈ G ′ ∪ G ′′ ( g e a ) e b ′ , ∑ g ∈ G ( e M | M ) e b ′ ( g e a ) = ∑ g ∈ G ′ ∪ G ′′ e b ′ ( g e a ) ,and taking into account (cid:13)(cid:13)(cid:13)e b ′ − e b (cid:13)(cid:13)(cid:13) < ε k e a k one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G ( e M | M ) ( g e a ) e b − ∑ g ∈ G ′ ∪ G ′′ ( g e a ) e b (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G ( e M | M ) e b ( g e a ) − ∑ g ∈ G ′ ∪ G ′′ e b ( g e a ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε . Let
V ⊂ G ( M , F ) be an open subset. For any x ∈ M denote byker V def = \ a ∈ Γ c ( M , Ω ) supp a ⊂V ker ρ x ( a ) ,im V def = [ a ∈ Γ c ( M , Ω ) supp a ⊂V im ρ x ( a ) . (7.1.17)Let H ′′ x be the completion of im V and let H ′ x be the orthogonal complement of thecompletion of ker V . Denote by p x V , q x V ∈ B (cid:0) L ( G x ) (cid:1) orthogonal projections from L ( G x ) onto H ′ x and H ′′ x respectively. Clearly one has W ⊆ V ⇒ p x W ≤ p x V AND q x W ≤ q x V . (7.1.18)For any open V ⊂ G ( M , F ) and a ∈ C ∗ r ( M , F ) we define the operator a | V = M x ∈ M p x V ρ x ( a ) q x V ∈ B M L ∈ M / F no hol L ( G x L ) . (7.1.19)322cf. Equation 7.1.2) The C -space of given by (7.1.19) operators is an Abelian groupdenoted by G ( V ) . If W ⊆ V then there is the map ρ VW : G ( V ) → G ( W ) , M x ∈ M p x V ρ x ( a ) q x V p x W p x V ρ x ( a ) q x V q x W . (7.1.20)Thus there is a presheaf G of Abelian groups (cf. Definition H.4.1). Definition 7.1.17.
The presheaf G is said to be reduced presheaf of ( M , F ) . Theassociated with G sheaf C ∗ r ( M , F ) is said to be the reduced sheaf of ( M , F ) . Remark 7.1.18.
From (7.1.19) it turns out that for any open
V ⊂ G ( M , F ) thereare projectors P V , Q V ∈ B (cid:0) L L ∈ M / F no hol L ( G x L ) (cid:1) a | V = P V π a ( a ) Q V ∈ B M L ∈ M / F no hol L ( G x L ) . (7.1.21)The Equation (7.1.20) can be replaced with ρ VW : G ( V ) → G ( W ) , P V π a ( a ) Q V P W P V π ( a ) Q V Q W . (7.1.22) Lemma 7.1.19.
If the foliation groupoid G ( M , F ) is Hausdorff and a ∈ C r ( M , F ) + is apositive element then for any ε > there is an open set V ⊂ G ( M , F ) such that • The closure of V is compact. • k ( − P V ) π a ( a ) ( − P V ) k < ε (cf. Equations (7.1.2) and (7.1.21) ) .Proof. There is δ > p k a k δ + δ < ε . From the Definition G.1.27 it fol-lows that there is b ∈ Γ c (cid:0) G ( M , F ) , Ω (cid:1) such that (cid:13)(cid:13) √ a − b (cid:13)(cid:13) < δ . From the aboveinequalities it follows that k a − bb ∗ k < ε . Otherwise bb ∗ ∈ Γ c (cid:0) G ( M , F ) , Ω (cid:1) sothe support of bb ∗ is compact. If V be the maximal open subset of the supportof bb ∗ then P V = Q V because bb ∗ is positive (cf. Equation 7.1.21). There is thefollowing representation π a ( a ) = (cid:18) P V π a ( a ) P V P V π a ( a ) ( − P V )( − P V ) π a ( a ) P V ( − P V ) π a ( a ) ( − P V ) (cid:19) , bb ∗ = (cid:18) d
00 0 (cid:19) ,hence from k a − bb ∗ k < ε it follows that k ( − P V ) π a ( a ) ( − P V ) k < ε .323 orollary 7.1.20. If G ( M , F ) is Hausdorff and a ∈ C r ( M , F ) + is a positive then forany ε > there is an open set V ⊂ G ( M , F ) such that • The closure of V is compact. • k π a ( a ) − a | V k < ε (cf. Equations (7.1.2) and (7.1.21) ) .Proof. There is δ > δ + p δ k a k < ε . From the Lemma 7.1.19 it followsthat there is an open set V ⊂ G ( M , F ) such that • The closure of V is compact. • k ( − P V ) π a ( a ) ( − P V ) k < δ (cf. Equations (7.1.2) and (7.1.21)) .There is the following representation π a ( a ) = (cid:18) P V π a ( a ) P V P V π a ( a ) ( − P V )( − P V ) π a ( a ) P V ( − P V ) π a ( a ) ( − P V ) (cid:19) , a | V = (cid:18) P V π a ( a ) P V
00 0 (cid:19)
From δ + p δ k a k < ε , k ( − P V ) π a ( a ) ( − P V ) k < δ and the Equation (1.4.9) onehas k π a ( a ) − a | V k < ε . Here we research local structure of the reduced sheaf, it enables us to constructthe global one.
We suppose that the groupoid G ( M , F ) is Hausdorff. We would like to findstalks of C ∗ r ( M , F ) at different γ ∈ G ( M , F ) (cf. Definition H.4.3). We considerfollowing three types:(I) γ is the image of a constant path.(II) s ( γ ) = r ( γ ) . 324 ype I Let γ ∈ ( M , F ) is a constant path such that s ( γ ) = r ( γ ) = x ∈ M .Let us consider the foliation chart ϕ U : U ≈ −→ (
0, 1 ) q × (
0, 1 ) n − q where U ⊂ M .Suppose that U is an open neighborhood of x , and ϕ U can extended up to thehomeomorphism ϕ U : U → [
0, 1 ] q × [
0, 1 ] n − q where U is the closure of U . Select ω ∈ Ω such that the restriction of ω x ⊗ ω x on U corresponds to the Lebesguemeasure on (
0, 1 ) n − q . Let Π ( M , F ) be the space of paths on leaves (cf. G.1.18),denote by W def = { γ ∈ Π ( M , F ) | s ( γ ) , r ( γ ) ∈ U } .Let V ⊂ Π ( M , F ) be the connected component of W which contain the constantpath γ . Denote by V def = Φ (cid:0) V (cid:1) ⊂ G ( M , F ) where the map Φ is given by theEquation (G.1.2). If V is the closure of V then there are natural homeomorphisms φ : V ∼ = (
0, 1 ) q × (
0, 1 ) n − q × (
0, 1 ) n − q φ : V ∼ = [
0, 1 ] q × [
0, 1 ] n − q × [
0, 1 ] n − q There are natural homomorphisms of Abelian groups φ ∗ : Γ c (cid:16) M , Ω (cid:17) → Γ c (cid:16) V , Ω (cid:12)(cid:12)(cid:12) V (cid:17) , φ ∗ : Γ c (cid:16) M , Ω (cid:17) → Γ c (cid:16) V , Ω (cid:12)(cid:12)(cid:12) V (cid:17) Since V is dense in V one has the isomorphism of Abelian groups φ ∗ (cid:16) Γ c (cid:16) M , Ω (cid:17)(cid:17) ∼ = φ ∗ (cid:16) Γ c (cid:16) M , Ω (cid:17)(cid:17) .Select ω x ∈ Ω such that ω x ⊗ ω x corresponds to the Lebesgue measure on (
0, 1 ) n − q . Any f ∈ C (cid:16) [
0, 1 ] q × [
0, 1 ] n − q × [
0, 1 ] n − q (cid:17) ∩ C ∞ (cid:16) [
0, 1 ] q × [
0, 1 ] n − q × [
0, 1 ] n − q (cid:17) corresponds to f · (cid:0) ω x ⊗ ω y (cid:1) ∈ Γ c (cid:0) V , Ω (cid:1) . On the other hand from the TheoremG.1.36 it follows that the C ∗ -norm completion of Γ c (cid:0) V , Ω (cid:12)(cid:12) V (cid:1) is isomorphic to C (cid:0) [
0, 1 ) q (cid:3) ⊗ K (cid:16) L (cid:16) (
0, 1 ) n − q , dx (cid:17)(cid:17) ,where dx is the Lebesgue measure on (
0, 1 ) n − q It follows that C ∗ r ( M , F ) ( V ) = ϕ ∗U (cid:16) C (cid:0) [
0, 1 ] q (cid:1) ⊗ K (cid:16) L (cid:16) (
0, 1 ) n − q , dx (cid:17)(cid:17)(cid:17) (7.1.23)325ny a ∈ C (cid:0) [
0, 1 ] q (cid:1) ⊗ K (cid:16) L (cid:16) (
0, 1 ) n − q , dx (cid:17)(cid:17) is the C ∗ -norm limit of the followingoperators (cid:0) k y ξ (cid:1) ( z ) = Z ( ) n − q k (cid:0) y , z , z ′ (cid:1) ξ (cid:0) z ′ (cid:1) dz ′ k ∈ Γ (cid:16) G , Ω (cid:17) (7.1.24)where k ∈ C ∞ (cid:16) [
0, 1 ] q × (
0, 1 ) n − q × (
0, 1 ) n − q (cid:17) For any f ∈ C (cid:16) (
0, 1 ) q × (
0, 1 ) n − q × (
0, 1 ) n − q (cid:17) ∩ C ∞ (cid:16) (
0, 1 ) q × (
0, 1 ) n − q × (
0, 1 ) n − q (cid:17) and k ∈ C ∞ (cid:16) [
0, 1 ] q × (
0, 1 ) n − q × (
0, 1 ) n − q (cid:17) there is an operator f · k given by (cid:0) f · k y ξ (cid:1) ( z ) = Z ( ) q f (cid:0) y , z , z ′ (cid:1) k (cid:0) y , z , z ′ (cid:1) ξ (cid:0) z ′ (cid:1) dz ′ k ∈ Γ (cid:16) G , Ω (cid:17) such that k f · k k ≤ k f k k k k (7.1.25)From the equation (7.1.25) it follows that if there is C ∗ -norm convergent net { k α } of given by (7.1.24) operators then the net { f · k α } is also C ∗ -norm convergent.So for all f ∈ C (cid:16) (
0, 1 ) q × (
0, 1 ) n − q × (
0, 1 ) n − q (cid:17) and for all k ∈ C (cid:0) [
0, 1 ] q (cid:1) ⊗K (cid:16) L (cid:16) (
0, 1 ) n − q , dx (cid:17)(cid:17) there is the uniquely defined operator f · k , such that ϕ U ( f · k ) ∈ C ∗ r ( M , F ) , (cid:0) f ′ + f ′′ (cid:1) · k = f · k + f ′′ · k . (7.1.26)Let Y = ϕ − U (cid:16) ϕ U ( x ) × (
0, 1 ) n − q (cid:17) be a transversal. For any y ∈ Y denote by H y def = (cid:8) ξ ∈ L (cid:0) G y (cid:1) (cid:12)(cid:12) γ / ∈ V ⇒ ξ ( γ ) = (cid:9) ∼ = L (cid:16) (
0, 1 ) n − q , dx (cid:17) ,i.e. there is a family (cid:8) H y (cid:9) of indexed by Y Hilbert spaces. Any smooth halfdensity on ω ∈ Γ (cid:0) V , Ω (cid:1) gives a family (cid:8) ξ y ∈ H y (cid:9) such that the map y (cid:13)(cid:13) ξ y (cid:13)(cid:13) is continuous. It turns out that Γ (cid:0) V , Ω (cid:1) is a continuity structure for Y and the (cid:8) H y (cid:9) (cf. Definition D.8.27). If X C b ( Y ) def = C b (cid:16) Y , (cid:8) H y (cid:9) , Γ (cid:16) U , Ω (cid:17)(cid:17) (7.1.27)is the converging to zero submodule (cf. Definition 4.5.8 and (4.5.9)) then from(4.12.3) it follows that X C b ( Y ) has the natural structure of Hilbert C b ( Y ) -module.326oreover for any y ∈ Y the structure of Hilbert space H y comes from the structureof the Hilbert module X C ( Y ) . From the Equation (7.1.23) it follows that C ∗ r ( M , F ) ( V ) ∼ = K (cid:16) X C b ( Y ) (cid:17) . (7.1.28)From the above construction it follows that ∀ f ∈ C ( V ) ∩ C ∞ ( V ) ∀ a ∈ C ∗ r ( M , F ) ( V ) ∃ f · a ∈ C ∗ r ( M , F ) , (cid:0) f ′ + f ′′ (cid:1) · a = f · a + f ′′ · a . (7.1.29)In result we have the following lemma Lemma 7.1.22.
There is the one-to-one correspondence between C ∗ r ( M , F ) ( V ) and K (cid:16) X C ( Y ) (cid:17) . Type II
Consider a subset V α given by (G.1.3) i.e. V α = G ( U ι ) ... G ( U ι k ) ∈ G ( M , F ) α = ( ι , ..., ι k ) such that s ( γ ) ∈ U ι k and r ( γ ) ∈ U ι . We also suppose that U ι ∩ U ι k = ∅ . (7.1.30)Let us consider a transversal Y ∈ U ι k such the s ( γ ) ∈ Y and for any y ∈ Y thereis γ ∈ V α such that s ( γ ) ∈ Y . Denote by V = { γ ∈ V α | the leaf of γ meets Y } , U s = { x ∈ M | ∃ γ ∈ V s ( γ ) = x } , U r = { x ∈ M | ∃ γ ∈ V r ( γ ) = x } (7.1.31)We also suppose that there are the foliation charts ϕ U s : U s → (
0, 1 ) q × (
0, 1 ) n − q , ϕ U r : U r → (
0, 1 ) q × (
0, 1 ) n − q where U s , U r ⊂ M is an open subsets such that s ( γ ) = r ( γ ) ∈ U . Let Y = ϕ − U (cid:16) s ( γ ) × (
0, 1 ) n − q (cid:17) be a transversal such that forany y ∈ Y there is γ ∈ V a such that s ( γ ) ∈ V . Denote by V = { γ ∈ V α | the leaf of γ meets Y } (7.1.32)For any y ∈ Y denote by H sy = (cid:8) ξ ∈ L (cid:0) G y (cid:1) (cid:12)(cid:12) ( γ / ∈ V OR r ( γ ) / ∈ U s ) ⇒ ξ ( γ ) = (cid:9) , H ry = (cid:8) ξ ∈ L (cid:0) G y (cid:1) (cid:12)(cid:12) ( γ / ∈ V OR r ( γ ) / ∈ U r ) ⇒ ξ ( γ ) = (cid:9) Y Hilbert spaces. Any smooth half den-sity on ω ∈ Γ (cid:0) U s , Ω (cid:1) (resp. ω ∈ Γ (cid:0) U r , Ω (cid:1) gives a family n ξ sy ∈ H sy o ( n ξ sy ∈ H sy o )such that both maps y (cid:13)(cid:13)(cid:13) ξ sy (cid:13)(cid:13)(cid:13) and y (cid:13)(cid:13)(cid:13) ξ ry (cid:13)(cid:13)(cid:13) are continuous. Itturns out that Γ (cid:0) U s , Ω (cid:1) (resp. Γ (cid:0) U r , Ω (cid:1) ) is a continuity structure for Y andthe n H sy o (resp. n H ry o ). Similarly to (7.1.27) define Hilbert C ( Y ) - modules X sC ( Y ) def = C (cid:16) Y , n H sy o , Γ (cid:16) U s , Ω (cid:17)(cid:17) , X rC ( Y ) def = C (cid:16) Y , n H ry o , Γ (cid:16) U r , Ω (cid:17)(cid:17) Note that both H ry and H sy agrees with Hilbert spaces defined in 7.1.16. Hence theprojection of p y : L (cid:0) G y (cid:1) → H y coincides with p y V , q y V defined in 7.1.16. Any a ∈ C ∗ r ( M , F ) defines a family (cid:8) ρ y ( a ) ∈ B (cid:0) L (cid:0) G y (cid:1)(cid:1)(cid:9) y ∈Y . Hence a defines a family (cid:8) p y V ρ y ( a ) q y V ∈ B (cid:0) L (cid:0) G y (cid:1)(cid:1)(cid:9) (7.1.33)The family (7.1.33) defines an element s V ∈ C ∗ r ( M , F ) ( V ) otherwise the family(7.1.33) uniquely corresponds to the compact operator in K (cid:16) X sC ( Y ) , X rC ( Y ) (cid:17) . Inresult we have the following lemma Lemma 7.1.23.
There is the one-to-one correspondence between C ∗ r ( M , F ) ( V ) and K (cid:16) X sC ( Y ) , X rC ( Y ) (cid:17) .If V , U s and U r are given by (7.1.31) then from (7.1.30) it follows that U s ∩ U r = ∅ , V ∩ V − = V − ∩ V = V ∩ V · V − = V − ∩ V · V − = V ∩ V − · V = V − ∩ V − · V = V · V − = V − · V = ∅ . Definition 7.1.24.
The set V b given by V b def = V ∪ V − ∪ V · V − ∪ V − · V = V ⊔ V − ⊔ V · V − ⊔ V − · V (7.1.34)is the balanced envelope of V .From (7.1.34) it follows that C ∗ r ( M , F ) ( V b ) = C ∗ r ( M , F ) ( V ) ⊕ C ∗ r ( M , F ) (cid:16) V − (cid:17) ⊕⊕ C ∗ r ( M , F ) (cid:16) V · V − (cid:17) ⊕ C ∗ r ( M , F ) (cid:16) V − · V (cid:17) (7.1.35)328n the other hand one has V · V − = { γ ∈ G ( M , F ) | γ ( t ) ∈ U s ∀ t ∈ [
0, 1 ] } , V − · V = { γ ∈ G ( M , F ) | γ ( t ) ∈ U r ∀ t ∈ [
0, 1 ] } and taking into account the Lemma 7.1.22 following conditions hold C ∗ r ( M , F ) (cid:16) V · V − (cid:17) = K (cid:16) X sC ( Y ) (cid:17) , C ∗ r ( M , F ) (cid:16) V − · V (cid:17) = K (cid:16) X rC ( Y ) (cid:17) .Otherwise from the Lemma 7.1.22 it turns out that C ∗ r ( M , F ) ( V ) = K (cid:16) X sC ( Y ) , X rC ( Y ) (cid:17) , C ∗ r ( M , F ) (cid:16) V − (cid:17) = K (cid:16) X rC ( Y ) , X sC ( Y ) (cid:17) and taking into account (7.1.35) one has C ∗ r ( M , F ) ( V b ) = K (cid:16) X sC ( Y ) (cid:17) ⊕ K (cid:16) X rC ( Y ) (cid:17) ⊕⊕K (cid:16) X sC ( Y ) , X rC ( Y ) (cid:17) ⊕ K (cid:16) X rC ( Y ) , X sC ( Y ) (cid:17) .From K (cid:16) X sC ( Y ) ⊕ X rC ( Y ) (cid:17) = K (cid:16) X sC ( Y ) (cid:17) ⊕ K (cid:16) X rC ( Y ) (cid:17) ⊕⊕K (cid:16) X sC ( Y ) , X rC ( Y ) (cid:17) ⊕ K (cid:16) X rC ( Y ) , X sC ( Y ) (cid:17) .it turns out C ∗ r ( M , F ) ( V b ) = K (cid:16) X sC ( Y ) ⊕ X rC ( Y ) (cid:17) = K (cid:16) X C ( Y ) (cid:17) . (7.1.36)where X C ( Y ) = X sC ( Y ) ⊕ X rC ( Y ) (7.1.37) Remark 7.1.25. If p x V b and p x V b are given by construction 7.1.16 projections then p x V b = p x V b = (cid:0) L ( G x ) → H sx ⊕ H rx (cid:1) (7.1.38)where L ( G x ) → H sx ⊕ H rx means the projection of the Hilbert space L ( G x ) ontoits subspace H sx ⊕ H rx . Remark 7.1.26.
Similarly to the type I one can proof the analog of the Equation(7.1.29), i.e. ∀ f ∈ C ( V b ) ∀ a ∈ C ∗ r ( M , F ) ( V b ) ∃ f · a ∈ C ∗ r ( M , F ) , (cid:0) f ′ + f ′′ (cid:1) · a = f · a + f ′′ · a . (7.1.39)329 .1.5 Extended algebra of foliation Let ( M , F ) be a foliated manifold (cf. Section G). If Ω G ) is given by (G.1.8) andand C ∞ b ( M ) = C b ( M ) ∩ C ∞ ( M ) is the algebra of smooth bounded functions on M then there is a left and right actions C ∞ b ( M ) × Ω G → Ω G and Ω G × C ∞ b ( M ) → Ω G given by f (cid:0) ω x ⊗ ω y (cid:1) = f ( x ) (cid:0) ω x ⊗ ω y (cid:1) ; (cid:0) ω x ⊗ ω y (cid:1) f = f ( y ) (cid:0) ω x ⊗ ω y (cid:1) ;where f ∈ Γ c ( M ) , x , y ∈ M , ω x ⊗ ω y ∈ Ω x ⊗ Ω y .These actions naturally induces left and right actions of C ∞ b ( M ) on the *-algebradescribed Γ c (cid:16) G , Ω G (cid:17) the section G.2. There is the alternative description of theleft and right actions of C ∞ b ( M ) on Γ c (cid:16) G , Ω G (cid:17) . For all x ∈ M , there is the repre-sentation, ρ x : Γ c (cid:16) G , Ω G (cid:17) → B (cid:0) L ( G x ) (cid:1) given by (G.1.4) and the representation ( ρ x ( a ) ξ ) ( γ ) = Z γ ◦ γ = γ a ( γ ) ξ ( γ ) ∀ ξ ∈ L ( G x ) .Above representations yield the representation ρ def = ∏ x ∈ M ρ x : Γ c (cid:16) G , Ω G (cid:17) → B M x ∈ M L ( G x ) ! (7.1.40)Similarly for all x ∈ M , there is the representation, π x : C ∞ b ( M ) → B (cid:0) L ( G x ) (cid:1) given by π x ( f ) ξ = f | G x ξ ,where f | G x ∈ C b ( G x ) and the the product means the well known product C b ( G x ) × L ( G x ) → L ( G x ) . There is the following representation π def = ∏ x ∈ M π x : C ∞ b ( M ) → B M x ∈ M L ( G x ) ! . (7.1.41)Clearly one has ∀ f ∈ C ∞ b ( M ) k π ( f ) k = k f k C b ( M ) . (7.1.42)Now for all f ∈ C ∞ b ( M ) and a ∈ Γ c (cid:16) G , Ω G (cid:17) the products f a and a f can bedefined by the following way ρ ( f a ) = π ( f ) ρ ( a ) , ρ ( a f ) = ρ ( a ) π ( f ) . (7.1.43)330rom the Equations (7.1.42) and (7.1.43) it follows that k f a k C ∗ r ( M , F ) ≤ k f k C b ( M ) k a k C ∗ r ( M , F ) , k a f k C ∗ r ( M , F ) ≤ k f k C b ( M ) k a k C ∗ r ( M , F ) . (7.1.44)If the net { a α } ⊂ Γ c (cid:16) G , Ω G (cid:17) is convergent with respect to the norm k · k C ∗ r ( M , F ) then from the Equation (7.1.44) it turns out that both nets { f a α } ⊂ Γ c (cid:16) G , Ω G (cid:17) and { a α f } ⊂ Γ c (cid:16) G , Ω G (cid:17) are k a k C ∗ r ( M , F ) -norm convergent. Taking into accountthat the C ∗ -algebra C ∗ r ( M , F ) is the completion of Γ c (cid:16) G , Ω G (cid:17) with respect tothe norm k · k C ∗ r ( M , F ) one has f a , a f ∈ C ∗ r ( M , F ) for any a ∈ C ∗ r ( M , F ) and f ∈ C ∞ b ( M ) . It turns out that every f ∈ C ∞ b ( M ) is a multiplier of C ∗ r ( M , F ) , i.e. C ∞ b ( M ) ⊂ M ( C ∗ r ( M , F )) . Since C ∞ b ( M ) is dense in C b ( M ) with respect to givenby the Equation 7.1.41 C ∗ -norm one has C b ( M ) ⊂ M ( C ∗ r ( M , F )) . (7.1.45) Definition 7.1.27.
Let M ֒ → Y be a compactification of M (cf. Definition A.1.31).Let us consider the induced by (7.1.45) inclusion C ∗ r ( M , F ) ⊕ C ( Y ) ⊂ M ( C ∗ r ( M , F )) .The C ∗ -norm completion generated by C ∗ r ( M , F ) ⊕ C ( Y ) unital subalgebra of M ( C ∗ r ( M , F )) is said to be the Y - extended algebra of the foliation ( M , F ) . The Y -extended algebra will be denoted by E ∗Y ( M , F ) . C ∗ -algebras Lemma 7.2.1. If ( M , F ) is a foliated manifold and both p : ( M , F ) → (cid:16) e M , e F (cid:17) p : ( M , F ) → (cid:16) e M , e F (cid:17) are regular finite-fold coverings of foliations and π : C ∗ r (cid:16) e M , e F (cid:17) → C ∗ r (cid:16) e M , e F (cid:17) is an injective *-homomorphism such that C ∗ r ( p ) = π ◦ C ∗ r ( p ) then thereis the regular covering p : (cid:16) e M , e F (cid:17) → (cid:16) e M , e F (cid:17) such that π = C ∗ r (cid:0) p (cid:1) .Proof. Let A = {U ι } ι ∈I be a regular foliated atlas of M . Suppose that A is simul-taneously subordinated to p and p (cf. Definition 7.1.7). Let both e A and e A arethe p -lift and p -lift of A respectively (cf. Definition 7.1.8). Let us select e x ∈ e M and e U be neighborhood of of e x such that e U ∈ e A Let U = p (cid:16) e U (cid:17) ∈ A and331 U ∈ e A be such that p (cid:16) e U (cid:17) = U . p − ( U ) = G g ∈ G ( e M | M ) g e U ,otherwise one has p − ( U ) = G g ∈ G ( e M | M ) g e U .From the Proposition G.1.30 it turns out that the restrictions C ∗ r ( p ) | C ∗ r ( U , F | U ) , C ∗ r ( p ) | C ∗ r ( U , F | U ) can be regarded as following injective *-homomorphisms C ∗ r ( p ) | C ∗ r ( U , F | U ) : C ∗ r ( U , F | U ) ֒ → M g ∈ G ( e M | M ) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) , C ∗ r ( p ) | C ∗ r ( U , F | U ) : C ∗ r ( U , F | U ) ֒ → M g ∈ G ( e M | M ) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) (7.2.1)such that C ∗ r ( U , F | U ) ∼ = C ∗ r (cid:18) e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) ∼ = C ∗ r (cid:18) e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) for all g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) and g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . Indeed both given by (7.2.1) maps are diagonal, i.e. C ∗ r ( p ) | C ∗ r ( U , F | U ) ( a ) = a , ..., a | {z } | G ( e M | M ) | − times , C ∗ r ( p ) | C ∗ r ( U , F | U ) ( a ) = a , ..., a | {z } | G ( e M | M ) | − times ; (7.2.2)for all a ∈ C ∗ r ( U , F | U ) . Let π a : C ∗ r (cid:16) e M , e F (cid:17) → B (cid:16) e H (cid:17) be the atomic representa-tion and for any g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) denote by e H g e U = π a (cid:18) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19)(cid:19) e H .For any g ′ , g ′′ ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) one has g ′ = g ′′ ⇒ e H g ′ e U ⊥ e H g ′′ e U . For all a ∈ C ∗ r ( U , F | U ) one has π a ( a ) ∈ B M g ∈ G ( e M | M ) e H g e U .332ny element of B (cid:18) L g ∈ G ( e M | M ) e H g e U (cid:19) corresponds to a matrix a g g . . . a g g n ... . . . ... a g n g n . . . a g n g n (7.2.3)where n = (cid:12)(cid:12)(cid:12) G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:12)(cid:12)(cid:12) , { g , ..., g n } = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) and a g j g k ∈ B (cid:18) e H g k e U , e H g j e U (cid:19) where j , k =
1, ..., n . From the Equation (7.2.2) it follows that π a ( a ) corresponds tothe diagonal matrix π a ( a ) = a . . . 0... . . . ...0 . . . a (7.2.4)where elements of matrix correspond to bounded operators of B (cid:18) e H g ′ e U , e H g ′′ e U (cid:19) for any g ′ , g ′′ ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . For any g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) denote by π g e U : M g ′ ∈ G ( e M | M ) C ∗ r (cid:18) g ′ e U , e F (cid:12)(cid:12)(cid:12) g ′ e U (cid:19) → C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) the natural projection. If a ∈ C ∗ r ( U , F | U ) + is positive then both e a = C ∗ r ( p ) | C ∗ r ( U , F | U ) ( a ) , e a k = π g e U ( e a ) If e a ⊥ = e a − e a k then e a ⊥ ∈ M g ′ ∈ G ( e M | M ) g ′ = g C ∗ r (cid:18) g ′ e U , e F (cid:12)(cid:12)(cid:12) g ′ e U (cid:19) and taking into account e a k ∈ C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) one has e a k e a ⊥ = e a ⊥ e a k =
0. Oth-erwise e a ⊥ is a sum of (cid:12)(cid:12)(cid:12) G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:12)(cid:12)(cid:12) − e a ⊥ is positive. Both333 a k = π (cid:16)e a k (cid:17) and e a ⊥ = π (cid:0)e a ⊥ (cid:1) . If both A k = a k g g . . . a k g g n ... . . . ... a k g n g n . . . a k g n g n and A ⊥ = a ⊥ g g . . . a ⊥ g g n ... . . . ... a ⊥ g n g n . . . a ⊥ g n g n are matrices which represent e a k and e a ⊥ respectively then from e a k e a ⊥ = e a ⊥ e a k = A k A ⊥ = A ⊥ A k =
0. Otherwise from (7.2.4) it follows that A k + A ⊥ is a diagonal matrix, so one has j = k ⇒ a k g j g k = − a ⊥ g j g k . If there are j = k such that a k g j g k = ξ ∈ e H g j e U such that η = a k g j g k ξ =
0. From η ∈ e H g k e U it turns out that ξ ⊥ η . If H ∼ = C is the C -linear span of { ξ , η } and p H : L g ∈ G ( e M | M ) e H g e U → H is the natural projection then both pA k p and pA ⊥ p correspond to positive complex matrices 2 × B k = b k b k b k b k ! ∈ M ( C ) and B ⊥ = b ⊥ b ⊥ b ⊥ b ⊥ ! ∈ M ( C ) The matrix B ⊥ is positive because the operator A ⊥ is positive, and from b ⊥ = B ⊥ is invertible. There is ξ ∈ H such that B ⊥ ξ = (cid:18) (cid:19) .From b k = B k (cid:18) (cid:19) = B k B ⊥ ξ = B k B ⊥ =
0. Therefore one has A k A ⊥ =
0, i.e. there is a contradiction. Fromthis contradiction it follows that A k is diagonal, so π (cid:16)e a k (cid:17) ∈ L g ∈ G ( e M | M ) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) . On the other hand the composition π g e U ◦ C ∗ r ( p ) | C ∗ r ( U , F | U ) is *-isomorphism, so that334 (cid:18) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19)(cid:19) ⊂ M g ∈ G ( e M | M ) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) , π M g ∈ G ( e M | M ) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) ⊂ M g ∈ G ( e M | M ) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) (7.2.5)i.e. there are direct sum decompositions of both (cid:16) C ∗ r ( M , F ) , C ∗ r (cid:16) e M , e F (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) , C ∗ r ( p ) (cid:17) and (cid:16) C ∗ r ( M , F ) , C ∗ r (cid:16) e M , e F (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) , C ∗ r ( p ) (cid:17) (cf. Definition 2.2.16). Thesystem of equations (7.2.5) is a special instance of the (2.2.13). If p e : L g ∈ G ( e M | M ) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) → C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) is the natural projec-tion then from the Lemma 2.2.17 it follows that there is the unique g ∈ G such that p e ◦ π (cid:18) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19)(cid:19) = { } . Both homeomorphisms e U ∼ = U and g e U ∼ = U yield a homeomorphism ϕ : e U → g e U ∼ = U . Assume that p (cid:0)e x (cid:1) def = ϕ (cid:0)e x (cid:1) . This definition does not depend on choice of the sets U , e U , e U because if we select U ′ ⊂ U , e U ′ ⊂ e U , e U ′ ⊂ e U such that e x ∈ e U ′ then we obtainthe same p (cid:0)e x (cid:1) . In result one has a continuous map p : e X → e X which is sur-jective because π is injective. In result one has a continuous map p : e X → e X which is surjective because π is injective. From the Lemma 2.2.17 it follows thatthere is the surjecive homomorphism φ : G → G such that π (cid:18) C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19)(cid:19) ⊂ M g ∈ ker φ C ∗ r (cid:18) g e U , e F (cid:12)(cid:12)(cid:12) g e U (cid:19) It turns out that (cid:0) p (cid:1) − (cid:16) g e U (cid:17) = G g ∈ ker φ g e U ,i.e. p is a covering. From the Corollary 4.3.8 it turns that the map p is a transitivecovering. Corollary 7.2.2. If ( M , F ) is a foliated manifold and p : ( M , F ) → (cid:16) e M , e F (cid:17) is a regularfinite-fold covering of foliation then there is the natural isomorphismG (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ∼ = G def = n g ∈ Aut (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) ga = a ∀ a ∈ C ∗ r ( M , F ) o .335 roof. From the Lemma 7.2.1 it follows that any g ∈ G corresponds to a covering e p : e M → e M such that p = p ◦ e p . Otherwise from the Definition A.2.3 it turns outthat e p corresponds to an element g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . Definition 7.2.3. If (cid:16) e M , e F (cid:17) → ( M , F ) is a regular finite-fold covering of foli-ations (cf. Definition G.2.10) is such that e M → M is a covering with compacti-fication (cf. Definition 4.10.4) then (cid:16) e M , e F (cid:17) → ( M , F ) is said to be a finite-foldcovering with compactification . Let (cid:16) e M , e F (cid:17) → ( M , F ) is a finite-fold covering with compactification (cf.Definition 7.2.3). Let both M ⊂ Y a e M ⊂ e Y are compactifications such that thereis a transitive finite fold covering p Y : e Y → Y and the covering p M : e M → M is the restriction of p Y , i.e. p M = p Y | M . If both E ∗Y ( M , F ) and E ∗ e Y (cid:16) e M , e F (cid:17) are Y and e Y -extended algebras of foliations (cf. Definition 7.1.27) respectively thenthe natural inclusions C ( Y ) ֒ → C (cid:16) e Y (cid:17) and C ∗ r ( M , F ) ֒ → C ∗ r (cid:16) e M , e F (cid:17) yield theinclusion E ∗Y ( M , F ) ֒ → E ∗ e Y (cid:16) e M , e F (cid:17) . Lemma 7.2.5.
In the situation 7.2.4 there is the natural isomorphismG (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ∼ = G def = n g ∈ Aut (cid:16) E ∗ e Y (cid:16) e M , e F (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ E ∗Y ( M , F ) o . Proof.
From the equations e a ∈ C (cid:16) e Y (cid:17) + ⊂ E ∗ e Y (cid:16) e M , e F (cid:17) + ⇔ ∑ g ∈ G g e a ∈ C ( Y ) + ⊂ E ∗Y ( M , F ) + , e a ∈ C ∗ r (cid:16) e M , e F (cid:17) + ⊂ E ∗ e Y (cid:16) e M , e F (cid:17) + ⇔ ∑ g ∈ G g e a ∈ C ∗ r ( M , F ) + ⊂ E ∗Y ( M , F ) + it follows that GC (cid:16) e Y (cid:17) + = C (cid:16) e Y (cid:17) + , GC ∗ r (cid:16) e M , e F (cid:17) + = C ∗ r (cid:16) e M , e F (cid:17) + ,hence one has GC (cid:16) e Y (cid:17) = C (cid:16) e Y (cid:17) , GC ∗ r (cid:16) e M , e F (cid:17) = C ∗ r (cid:16) e M , e F (cid:17) . (7.2.6)336rom the equation (7.2.6) any g ∈ G defines a pair (cid:0) g ′ , g ′′ (cid:1) ∈ G (cid:16) C (cid:16) e Y (cid:17) (cid:12)(cid:12)(cid:12) C ( Y ) (cid:17) ×× n g ∈ Aut (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) ga = a ∀ a ∈ C ∗ r ( M , F ) o so from G (cid:16) C (cid:16) e Y (cid:17) (cid:12)(cid:12)(cid:12) C ( Y ) (cid:17) ∼ = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) and the Corollary 7.2.2 one can assumethat ( g ′ , g ′′ ) ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) × G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . Thus there is a homomorphism ι : G → G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) × G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) The C ∗ -algebra E ∗ e Y (cid:16) e M , e F (cid:17) is generated by C (cid:16) e Y (cid:17) ⊕ C ∗ r (cid:16) e M , e F (cid:17) so if g ∈ G is nottrivial if and only if at least one of the following two conditions hold ∃ e a ′ ∈ C (cid:16) e Y (cid:17) g e a ′ = e a ′ , ∃ e a ′′ ∈ C ∗ r (cid:16) e M , e F (cid:17) g e a ′′ = e a ′′ .Hence ι is an injective homomorphism of groups. If ∆ ⊂ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) × G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) is the image of the diagonal homomorphism G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ֒ → G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) × G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ; g ( g , g ) then from ∀ g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ∀ e a ′ ∈ C ∗ r (cid:16) e M , e F (cid:17) ∀ e a ′ ∈ C (cid:16) e Y (cid:17) g (cid:0)e a ′ (cid:1) g (cid:0)e a ′′ (cid:1) = g (cid:0)e a ′ e a ′′ (cid:1) ; g (cid:0)e a ′′ (cid:1) g (cid:0)e a ′ (cid:1) = g (cid:0)e a ′′ e a ′ (cid:1) it follows that ∆ ⊂ ι ( G ) . Suppose that g ∈ G is such that ι ( g ) = ( g ′ , g ′′ ) / ∈ ∆ ,i.e. g ′ = g ′′ . Let e U ⊂ e M is open subset such that e U ∩ g e U = ∅ for any nontrivial g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . There are e a ′ ∈ C (cid:16) e U (cid:17) and e a ′′ ∈ C ∗ r (cid:16) e U , e F (cid:12)(cid:12)(cid:12) e U (cid:17) such that e a ′ e a ′′ = g ′ e U ∩ g ′′ e U = ∅ it follows that g ( e a ′ e a ′′ ) = ( g ′ e a ′ ) ( g ′′ e a ′′ ) =
0, it isimpossible because g is ∗ -automorphism. From this contradiction one has ι ( G ) = ∆ ∼ = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) and since ι is injective it turns out that G ∼ = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) .337 emma 7.2.6. Let (cid:16) e M , e F (cid:17) → ( M , F ) be a regular finite-fold covering of foliated spaces(cf. Definition G.2.10) which corresponds to a covering p : e M → M with compactification(cf. Definition 4.10.4) then (cid:16) C ∗ r ( M , F ) , C ∗ r (cid:16) e M , e F (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) , C ∗ r ( p ) (cid:17) is a noncommutative finite-fold covering with unitization (cf. Definition 2.1.13).Proof. From the Lemma 7.2.5 it turns out that the quadruple (cid:16) C ∗ r ( M , F ) , C ∗ r (cid:16) e M , e F (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) , C ∗ r ( p ) (cid:17) is a finite-fold pre-covering (cf. Definition 2.1.5). Now one needs prove that thequadruple satisfies to conditions (a), (b) of the Definition 2.1.13.(a) Let M ֒ → Y , e M ֒ → e Y be compactifications such that there is the finitefold covering p : e Y → Y such that p = p | e Y . Denote by E ∗Y ( M , F ) (resp. E ∗ e Y (cid:16) e M , e F (cid:17) ) the C ( Y ) (resp. C (cid:16) e Y (cid:17) )-extended algebra of the foliation ( M , F ) (resp. (cid:16) e M , e F (cid:17) ) (cf. Definition 7.1.27). Both C ( Y ) (resp. C (cid:16) e Y (cid:17) are uni-tal, hence both E ∗Y ( M , F ) and E ∗ e Y (cid:16) e M , e F (cid:17) are unital. From E ∗Y ( M , F ) ⊂ M ( C ∗ r ( M , F )) and E ∗ e Y (cid:16) e M , e F (cid:17) ⊂ M (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:17) it turns out that that thereare unitizations C ∗ r ( M , F ) ֒ → E ∗Y ( M , F ) , C ∗ r (cid:16) e M , e F (cid:17) ֒ → E ∗ e Y (cid:16) e M , e F (cid:17) . (7.2.7)From the Lemma 7.2.5 it follows that G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ∼ = n g ∈ Aut (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ C ∗ r ( M , F ) o ∼ = ∼ = (cid:26) g ∈ Aut (cid:18) E ∗ C ( e X ) (cid:16) e M , e F (cid:17)(cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ E ∗ C ( X ) ( M , F ) (cid:27) .(b) One need check that the triple (cid:16) E ∗Y ( M , F ) , E ∗ e Y (cid:16) e M , e F (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) (7.2.8)is an unital noncommutative finite-fold covering. From the Lemma 7.2.5 itfollows that (7.2.8) is a noncommutative finite-fold pre-covering. From the338emark 4.8.7 it follows that there is a finite set { e e α } α ∈ A ⊂ C (cid:16) e Y (cid:17) whichsatisfies to the following equations1 C ( e Y ) = ∑ α ∈ A e e α (7.2.9) e e α ( g e e α ) =
0; for any notrivial g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) (7.2.10)If e a ∈ E ∗ e Y (cid:16) e M , e F (cid:17) then from (7.2.9) it turns out that e a = ∑ α ∈ A e e α ( e e α e a ) .Otherwise from (7.2.10) it follows that e e α ( g ( e e α e a )) = g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) , it turns out e e α ( e e α e a ) = e e α ∑ g ∈ G ( e M | M ) g ( e e α e a ) From ∑ g ∈ G ( e M | M ) g ( e e α e a ) ∈ E ∗Y ( M , F ) it follows that E ∗ e Y (cid:16) e M , e F (cid:17) is a E ∗Y ( M , F ) -module generated by the finite set { e e α } . Let (cid:16) e M , e F (cid:17) → ( M , F ) be a regular finite-fold covering of foliated spaces(cf. Definition G.2.10). Suppose that there is an open connected subset U ⊂ M with compact closure such that the e U def = p − ( U ) is connected. Suppose that u ∈ C ( U ) ⊂ C ( X ) is a strictly positive element of C ( U ) (cf. Definition D.1.28). The C ∗ -norm completions of both uC ∗ r ( M , F ) u and uC ∗ r (cid:16) e M , e F (cid:17) u will be denotedby C ∗ r ( M , F ) | U and C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U respectively. Lemma 7.2.8.
In the situation of 7.2.7 the triple (cid:16) C ∗ r ( M , F ) | U , C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U , G (cid:16) e U (cid:12)(cid:12)(cid:12) U (cid:17) ∼ = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) (7.2.11) is a noncommutative finite-fold covering with unitization (cf. Definition 2.1.13). roof. Firstly one needs prove that the triple (7.2.8) is a noncommutative finite-foldpre-covering, i.e. it satisfies to the Definition 2.1.5. Similarly to the proof of theCorollary 7.2.2 one can prove that n g ∈ Aut (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U (cid:17) (cid:12)(cid:12)(cid:12) ga = a ∀ a ∈ C ∗ r ( M , F ) | U o ∼ = G (cid:16) e U (cid:12)(cid:12)(cid:12) U (cid:17) , (7.2.12) C ∗ r ( M , F ) | U ∼ = C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) G ( e U | U ) e U . (7.2.13)The equations (7.2.12) and (7.2.13) are special instances of the conditions (a) and(b) of the Definition 2.1.5, i.e. the triple (7.2.11) is a a noncommutative finite-foldpre-covering.Secondly we prove that the triple (7.2.11) satisfies to the conditions (a) and (b)of the Definition 2.1.13.(a) If both V and e V are closures of both U and e U then similarly to the Equation(7.1.45) one can obtain the following inclusions C ( V ) ⊂ M (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U (cid:17) , C (cid:16) e V (cid:17) ⊂ M (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U (cid:17) .If both E ∗ r ( M , F ) | U and E ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U are subalgebras of M ( C ∗ r ( M , F ) | U ) and M ( C ∗ r ( M , F ) | U ) generated by both C ∗ r ( M , F ) | U ∪ C ( V ) and C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U ∪ C (cid:16) e V (cid:17) then one has: • The C ∗ -algebras E ∗ r ( M , F ) | U and E ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U are unital. • The inclusions C ∗ r ( M , F ) | U ⊂ E ∗ r ( M , F ) | U and C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U ⊂ E ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U are unitizations.(b) Similarly to the proof of the Lemma 7.2.6 one can proof that the triple (cid:16) E ∗ r ( M , F ) | U , E ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U , G (cid:16) e U (cid:12)(cid:12)(cid:12) U (cid:17) ∼ = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is a unital noncommutative finite-fold covering (cf. Definition 2.1.9). From C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U ∩ E ∗ r ( M , F ) | U = C ∗ r ( M , F ) | U and G (cid:16) e U (cid:12)(cid:12)(cid:12) U (cid:17) C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U = C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U it turns out that that the triple (7.2.11) satisfies to the condition(b) of the Definition 2.1.13. 340 heorem 7.2.9. Let (cid:16) e M , e F (cid:17) → ( M , F ) be a regular finite-fold covering of foliatedspaces (cf. Definition G.2.10). If the manifold M is second-countable and connected thenthe quadruple (cid:16) C ∗ r ( M , F ) , C ∗ r (cid:16) e M , e F (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) , C ∗ r ( p ) (cid:17) is a noncommutative finite-fold covering (cf. Definition 2.1.17).Proof. The space M is second-countable hence from the Theorem A.1.29 it followsthat M is a Lindelöf space. Let us consider a finite or countable sequence U ′ $ ... $ U ′ n $ ... of connected open subsets of M given by the Lemma 4.3.33. In particularfollowing condition hold: • For any n ∈ N the closure V n of U ′ n is compact. • ∪ U ′ n = X . • The space p − ( U ′ n ) is connected for any n ∈ N .From the Remark 4.8.2 it turns out that there there is an increasing net { f n } n ∈ N ⊂ C c ( M ) such that supp f n ⊂ U ′ n and there is the limit 1 C b ( M ) = β - lim n → ∞ f n withrespect to the strict topology on the multiplier algebra M ( C ( X )) ∼ = C b ( X ) (cf.Definition D.1.12). This limit is also the limit1 M ( C ∗ r ( M , F )) = β - lim n → ∞ f n (7.2.14)with respect to the strict topology of the multiplier algebra M ( C ∗ r ( M , F )) . Let uscheck that the net { f n } n ∈ N satisfies to the conditions (a) and (b) of the Definition2.1.17.(a) Follows from the Equation (7.2.14).(b) If U n def = { x ∈ M | f n ( x ) > } , e U n def = p − ( U n ) then the C ∗ -norm completions of both f n C ∗ r ( M , F ) | e U f n and f n C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U f n are the explained in 7.2.7 C ∗ -algebras C ∗ r ( M , F ) | U n and341 ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U n Otherwise from the Lemma 7.2.8 it turns out that for every n ∈ N the triple (cid:18) C ∗ r ( M , F ) | U n , C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U n , G (cid:16) e U n (cid:12)(cid:12)(cid:12) U n (cid:17) ∼ = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:19) is a noncommutative finite-fold covering with unitization (cf. Definition2.1.13). The action G (cid:16) e U n (cid:12)(cid:12)(cid:12) U n (cid:17) × C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U n → C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U n , is therestriction on C ∗ r (cid:16) e M , e F (cid:17)(cid:12)(cid:12)(cid:12) e U n ⊂ , C ∗ r (cid:16) e M , e F (cid:17) of the action G (cid:16) e U n (cid:12)(cid:12)(cid:12) U n (cid:17) × C ∗ r (cid:16) e M , e F (cid:17) → , C ∗ r (cid:16) e M , e F (cid:17) . Lemma 7.2.10.
Let ( M , F ) be a foliated space such that M is a connected, second-countable manifold. If both p : (cid:16) e M , e F (cid:17) → ( M , F ) and p : (cid:16) e M , e F (cid:17) → ( M , F ) areregular coverings then the unordered pair of *-homomorphisms (cid:16) C ∗ r ( p ) : C ∗ r ( M , F ) ֒ → C ∗ r (cid:16) e M , e F (cid:17) , C ∗ r ( p ) : C ∗ r ( M , F ) ֒ → C ∗ r (cid:16) e M , e F (cid:17)(cid:17) is compliant (cf. Definition 3.1.1).Proof. One needs check (a)-(d) of the Definition 3.1.1(a) Suppose that there is a *-homomorphism π : C ∗ r (cid:16) e M , e F (cid:17) → C ∗ r (cid:16) e M , e F (cid:17) such that C ∗ r ( p ) = π ◦ C ∗ r ( p ) . From the Lemma 7.2.1 it turns out that thereis a regular covering p : (cid:16) e M , e F (cid:17) → (cid:16) e M , e F (cid:17) such that π = C ∗ r (cid:0) p (cid:1) .Both e M and e M are compact hence from the Lemma 7.2.6 it follows that C ∗ r (cid:0) p (cid:1) is a noncommutative finite-fold covering.(b) There are natural isomorphisms G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ∼ = G (cid:16) C ∗ r (cid:16) e M , e F (cid:17) (cid:12)(cid:12)(cid:12) C ∗ r ( M , F ) (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ∼ = G (cid:16) C ∗ r (cid:16) e M , e F (cid:17) (cid:12)(cid:12)(cid:12) C ∗ r ( M , F ) (cid:17) ,hence the natural surjective homomorphism h : G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) → G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) induces the surjective homomorphism h ′ : G (cid:16) C ∗ r (cid:16) e M , e F (cid:17) (cid:12)(cid:12)(cid:12) C ∗ r ( M , F ) (cid:17) → G (cid:16) C ∗ r (cid:16) e M , e F (cid:17) (cid:12)(cid:12)(cid:12) C ∗ r ( M , F ) (cid:17) .(7.2.15)342or all g ∈ G (cid:16) C ∗ r (cid:16) e M , e F (cid:17) (cid:12)(cid:12)(cid:12) C ∗ r ( M , F ) (cid:17) and e a ∈ C ∗ r (cid:16) e M , e F (cid:17) one has g e a = h ′ ( g ) e a and taking into account that h ′ ( g ) is a *-automorphism of C ∗ r (cid:16) e M , e F (cid:17) one has g e a ∈ C ∗ r (cid:16) e M , e F (cid:17) .(c) It is already proven the existence of the surjecive homomorphism (7.2.15).(d) If ρ : C ∗ r (cid:16) e M , e F (cid:17) → C ∗ r (cid:16) e M , e F (cid:17) is any *-homomorphism such that C ∗ r ( p ) = ρ ◦ C ∗ r ( p ) then from the the Lemma 7.2.1 it turns out that there is g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) e M (cid:17) such that q = g ◦ p . (7.2.16)If we consider g as element of G (cid:16) C ∗ r (cid:16) e M , e F (cid:17) (cid:12)(cid:12)(cid:12) C ∗ r (cid:16) e M , e F (cid:17)(cid:17) from (7.2.16)it follows that ρ = π ◦ g . If p : (cid:16) e M , e F (cid:17) → ( M , F ) is the regular finite-fold covering then from the Lemma7.2.6 it follows that there is the natural noncommutative finite-fold covering C ∗ r ( M , F ) ֒ → C ∗ r (cid:16) e M , e F (cid:17) . From the Lemma 2.3.7 it turns out that there is the natu-ral surjective map from the spectrum of C ∗ r (cid:16) e M , e F (cid:17) to the spectrum of C ∗ r ( M , F ) . Lemma 7.2.11.
In the above situation the given by the Lemma 2.3.7 map from the spec-trum of C ∗ r (cid:16) e M , e F (cid:17) to the spectrum of C ∗ r ( M , F ) corresponds to natural map e M / e F no hol → M / F no hol , e L p (cid:16)e L (cid:17) . (7.2.17) Proof. If e x ∈ e L then from (7.1.1) it turns out that ρ e L = ρ e x if L = p (cid:16)e L (cid:17) and x = p ( e x ) then ρ L = ρ x . One has ρ e x | C ∗ r ( M , F ) ∼ = ρ x . (7.2.18)The equation (7.2.18) is the special case of (2.3.8).343 emma 7.2.12. Consider the situation of the Lemma 7.1.15. If π a : (cid:16) e M , e F (cid:17) ֒ → B ( H a ) is the atomic representation (cf. Definition D.2.33) then the series (7.1.14) is convergentwith respect to the strong topology of B ( H a ) . Moreover the sum of the series in sense ofthe strong topology of B ( H a ) is equal to desc p ( e a ) (cf. Definition 7.1.3).Proof. From (7.1.2) it follows that H a ∼ = L e L ∈ e M / e F no hol L ( G x L ) . If ξ ∈ H a is given by ξ = ... , ξ e L |{z} e L th − place , ... and ε > F = ne L , ..., e L m o ⊂ e M / e F no hol such that η e L = ( ξ e L e L ∈ F e x / ∈ F AND η = ... , η e L |{z} e L th − place , ... ⇒ (cid:13)(cid:13)(cid:13) e ξ − e η (cid:13)(cid:13)(cid:13) < ε k a k .For any e L ∈ F there is the irreducible representation ρ L def = ρ x L : C ∗ r (cid:16) e M , e F (cid:17) → B (cid:16) L (cid:16) G e x e L (cid:17)(cid:17) (cf. (7.1.1)) and the state τ e L : C ∗ r (cid:16) e M , e F (cid:17) → C , such that L (cid:16) G e x e L (cid:17) ∼ = L (cid:16) C ∗ r (cid:16) e M , e F (cid:17) , τ τ e L (cid:17) where L (cid:16) C ∗ r (cid:16) e M , e F (cid:17) , τ τ e L (cid:17) is given by the GNS-construction(cf. Section D.2.1). Otherwise there is the specialization f τ e L : C ∗ r (cid:16) e M , e F (cid:17) → L (cid:16) C ∗ r (cid:16) e M , e F (cid:17) , τ e L (cid:17) of given by (D.2.1) homomorphism of left C ∗ r (cid:16) e M , e F (cid:17) -modulessuch that f τ e L : C ∗ r (cid:16) e M , e F (cid:17) is dense in L (cid:16) C ∗ r (cid:16) e M , e F (cid:17) , τ e L (cid:17) . Let A = {U ι } ι ∈I of M be a subordinated to p regular foliated atlas. Let e A = n e U e ι o e ι ∈ e I be the p - lift of A . e b ∈ Γ c (cid:16) G (cid:16) e M (cid:17) , e Ω (cid:17) such that e b e L = n ∑ k = j e U e ι k ,0 (cid:16)e a e ι k ,0 (cid:17) · ... · j e U e ι k , jk (cid:16)e a e ι k , jk (cid:17) , e a e ι k , jl ∈ Γ c (cid:16) G (cid:16) U e ι j , l (cid:17) , e Ω (cid:17) , (cid:13)(cid:13)(cid:13) f τ e L (cid:16)e b e L (cid:17) − e η e L (cid:13)(cid:13)(cid:13) < ε | F | k e a k where e U e ι k , l ∈ e A for all k , l and j e U e ι k , l are given by (G.1.7) maps. Similarly to (7.1.16)the set G ′ e L def = n g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:12)(cid:12)(cid:12) ( g e a ) e b e L = o G e L def = n g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:12)(cid:12)(cid:12) ( g e a ) f τ e L (cid:16)e b e L (cid:17) = o ⊂ G e L is also finite. If G def = ∩ e L ∈ F G e L then ∑ g ∈ G g e a ∑ e L ∈ F f τ e L (cid:16)e b e L (cid:17)! = ∑ g ∈ G ( e M | M ) g e a ∑ e L ∈ F f τ e L (cid:16)e b e L (cid:17)! ,so taking into account (cid:13)(cid:13)(cid:13) e ξ − e η (cid:13)(cid:13)(cid:13) < ε k a k and (cid:13)(cid:13)(cid:13) f τ e L (cid:16)e b e L (cid:17) − e η e L (cid:13)(cid:13)(cid:13) < ε | F |k e a k for all e L ∈ F one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G ( g e a ) ξ − ∑ g ∈ G ( e M | M ) ( g e a ) ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε .So the series (7.1.14) is strongly convergent. On the other hand from ( g e a ) e b e L = ⇒ desc p ( e a ) e b e L = ( g e a ) e b e L ⇒ desc p ( e a ) f τ e L (cid:16)e b e L (cid:17) = ( g e a ) f τ e L (cid:16)e b e L (cid:17) it follows that the limit of the series (7.1.14) equals to desc p ( e a ) . Definition 7.3.1.
Let ( M , F ) be a foliated space such that M is second-countableand connected. Let Λ be a countable directed set (cf. Definition A.1.3) with theminimal element λ min ∈ Λ . Let us consider a following category S ( M , F ) .(a) Objects are regular coverings p λ : ( M λ , F λ ) → ( M , F ) indexed by λ ∈ Λ .We do not require that any finite-fold covering p : (cid:16) e M , e F (cid:17) → ( M , F ) is anobject of S ( M , F ) .(b) Morphism from p µ : (cid:0) M µ , F µ (cid:1) → ( M , F ) to p ν : ( M ν , F ν ) → ( M , F ) is aregular covering p : (cid:0) M µ , F µ (cid:1) → ( M ν , F ν ) such that p µ = p ν ◦ p .We claim that µ ≥ ν if and only if there is a morphismfrom p µ : (cid:0) M µ , F µ (cid:1) → ( M , F ) to p ν : ( M ν , F ν ) → ( M , F ) (7.3.1)We set ( M λ min , F λ min ) def = ( M , F ) The category S ( M , F ) is said to be a foliation finitecovering category and we write S ( M , F ) ⊂ FinFol .345 emma 7.3.2. If S ( M , F ) = { p λ : ( M λ , F λ ) → ( M λ , F λ ) } ∈ FinFol is a foliation finitecovering category (cf. Definition 7.3.1) then there is the natural algebraical finite coveringcategory (cf. Definition 3.1.4) S C ∗ r ( M , F ) == { π λ = C ∗ r ( p λ ) : C ∗ r ( M , F ) ֒ → C ∗ r ( M λ , F λ ) } λ ∈ Λ ∈ FinAlg . (7.3.2)
Proof.
One needs check (a), (b) of the Definition 3.1.4.(a) Follows from the Lemma 7.2.10.(b) From (7.3.1) it follows that µ ≥ ν if and only if there isan *-homomorphism C ∗ r ( p ) : C ∗ r ( M ν , F ν ) ֒ → C ∗ r (cid:0) M µ , F µ (cid:1) ;such that C ∗ r (cid:0) p µ (cid:1) = C ∗ r ( p ) ◦ C ∗ r (cid:0) p µ (cid:1) . (7.3.3) Suppose that S ( M , F ) = { p λ : ( M λ , F λ ) → ( M , F ) } ⊂ FinFol be a foliation finite covering category. Let x ∈ M be a base-point suppose thatfor any λ ∈ Λ there is a base point x λ ∈ X λ such that x = p λ (cid:0) x λ (cid:1) . Suppose thatfor any µ > ν ∈ Λ there is the unique regular covering p µν : (cid:0) M µ , F µ (cid:1) → ( M ν , F ν ) such that p µν (cid:0) x µ (cid:1) = x ν . There is a category S ( M , F , x ) == ( { p λ : ( M λ , F λ ) → ( M , F ) } λ ∈ Λ , (cid:8) p µν : (cid:0) M µ , F µ (cid:1) → ( M ν , F ν ) (cid:9) µ , ν ∈ Λ µ ≥ ν ) . (7.3.4)The category S ( M , F , x ) is the pre-order category which is equivalent to Λ (cf. Def-inition H.1.1). Definition 7.3.4.
The given by (7.3.4) category is said to be a pointed finite foliationcategory . Lemma 7.3.5. If S ( M , F , x ) def = ( { p λ : ( M λ , F λ ) → ( M , F ) } λ ∈ Λ , (cid:8) p µν : (cid:0) M µ , F µ (cid:1) → ( M ν , F ν ) (cid:9) µ , ν ∈ Λ µ ≥ ν ) s a pointed finite foliation category then there is a pointed algebraical finite coveringcategory S C ∗ r ( M , F , x ) (cf. Definition (ref. 3.1.6) given by (cid:8) { C ∗ r ( p λ ) : C ∗ r ( M , F ) ֒ → C ∗ r ( M λ , F λ ) } , (cid:8) C ∗ r (cid:0) p µν (cid:1) : C ∗ r ( M ν , F ν ) ֒ → C ∗ r (cid:0) M µ , F µ (cid:1)(cid:9)(cid:9) . Proof.
Follows from the Lemma 7.3.2 and the Remark 3.1.7.
Let S ( M , F , x ) (cid:8) ( M , F , x ) → (cid:0) M λ , F λ , x λ (cid:1)(cid:9) λ ∈ Λ be a pointed foliation finitecovering category. There is the natural pointed topological finite covering category S ( M , x ) = { p λ : M λ → M } λ ∈ Λ (cf. Definition 4.11.3). Let M be the disconnectedinverse limit of S ( M , x ) (cf. Definition 4.11.19), and let p : M → M , p : M → M λ be the natural transitive coverings. If (cid:0) M , F (cid:1) is the p -lift of ( M , F ) (cf. DefinitionG.2.5) then for any x ∈ M there is the natural homeomorphism G p λ ( x ) ∼ = G x . Hencethere is the natural isomorphism L (cid:16) G p λ ( x ) (cid:17) ∼ = L ( G x ) . Denote by L no hol def = { L is a leaf of ( M , F ) | L has no holomomy } , L no hol λ def = { L λ is a leaf of ( M λ , F ) | L λ has no holomomy } , L no hol def = (cid:8) L is a leaf of (cid:0) M , F (cid:1) (cid:12)(cid:12) L has no holomomy (cid:9) ,Denote by M no hol ⊂ M such that for all L ∈ L no hol there is just one point M no hol such that x ∈ L . We suppose that any having holonomy leaf has emptyintersection with M no hol . Similarly one defines M no hol λ ⊂ M λ and M no hol ⊂ M .Moreover we require that ∀ λ ∈ Λ p λ (cid:16) M no hol λ (cid:17) = M no hol , ∀ µ , ν ∈ Λ ν > µ ⇒ p νµ (cid:16) M no hol ν (cid:17) = M no hol µ , ∀ λ ∈ Λ ⇒ p λ (cid:0) M no hol (cid:1) = M no hol λ (7.3.5)The direct calculation shows that the above requirements are compatible. Fromthe Theorem G.1.38 it turns out that if ρ x is given by (G.1.4) then the following347epresentations π a def = M x ∈ M no hol ρ x : C ∗ r ( M , F ) ֒ → B M x ∈ M no hol L ( G x ) ! , π λ a def = M x λ ∈ M no hol λ ρ x λ : C ∗ r ( M λ , F λ ) ֒ → B M x ∈ M no hol λ L ( G x λ ) , π a def = M x ∈ M no hol ρ x : C ∗ r (cid:0) M , F (cid:1) ֒ → B M x ∈ M no hol L ( G x ) ,are atomic. If S C ∗ r ( M , F ) is a given by the Lemma 7.3.5 pointed algebraical finitecovering category then for any µ , ν the following condition holds µ > ν ⇒ π µ a = C ∗ r (cid:0) p µν (cid:1) ◦ π ν a . (7.3.6)If \ C ∗ r ( M , F ) def = C ∗ -lim −→ C ∗ r ( M λ , F λ ) then from the Lemma 3.1.27 it follows thatthe spectrum of \ C ∗ r ( M , F ) as a set is the inverse limit b M no hol def = lim ←− λ ∈ Λ M no hol λ .From (7.3.5) it follows that φ : b M no hol ∼ = M no hol , i.e. the spectrum of \ C ∗ r ( M , F ) coincides with the spectrum of C ∗ r (cid:0) M , F (cid:1) as a set. For any b x ∈ b M no hol there is theirreducible representation π b x : \ C ∗ r ( M , F ) → B (cid:16) b H b x (cid:17) . If b p : b M → M is the naturalcontinuous map then b p (cid:16) b M no hol (cid:17) M no hol and H b x ∼ = L (cid:16) G b p ( b x ) (cid:17) . Otherwise forany x ∈ M there is the natural isomorphism L ( G x ) ∼ = L (cid:16) G p ( x ) (cid:17) .It follows that there is the natural isomorphism of Hilbert spaces φ : M x ∈ M no hol L ( G x ) ∼ = M b x ∈ b M no hol H b x . (7.3.7)The left part of (7.3.7) is the space of the atomic representation π a : C ∗ r (cid:0) M , F (cid:1) ֒ → B (cid:0) H a (cid:1) the right part is the space of the atomic representation b π a : \ C ∗ r ( M , F ) ֒ → B (cid:16) b H a (cid:17) . From (7.3.7) it follows that there is the natural inclusion ϕ : C ∗ r (cid:0) M , F (cid:1) ֒ → B (cid:16) b H a (cid:17) . (7.3.8)348 emark 7.3.7. Below the given by the Equation (7.3.8) *-homomorphism will bereplaced with the inclusion C ∗ r (cid:0) M , F (cid:1) ⊂ B (cid:16) b H a (cid:17) of C ∗ -algebras. Similarly tothe Remark 3.1.38 below for all λ ∈ Λ we implicitly assume that C ∗ r ( M λ , F λ ) ⊂ B ( widehat H a ) . Similarly the following natural inclusion \ C ∗ r ( M , F ) ⊂ B (cid:16) b H a (cid:17) will be implicitly used. These inclusions enable us replace the Equations 3.1.20with the following equivalent system of equations ∑ g ∈ G λ z ∗ ( ga ) z ∈ C ∗ r ( M λ , F λ ) , ∑ g ∈ G λ f ε ( z ∗ ( ga ) z ) ∈ C ∗ r ( M λ , F λ ) , ∑ g ∈ G λ ( z ∗ ( ga ) z ) ∈ C ∗ r ( M λ , F λ ) . (7.3.9) Select λ ∈ Λ and z ∈ C ∗ r ( M λ , F λ ) ∼ . Let A = {U ι } ι ∈I be a regularsubordinated to p : M → M foliated atlas of M (cf. Definition 7.1.7). Let A λ = (cid:8) U λι λ ⊂ M λ (cid:9) , A = (cid:8) U ι ⊂ M (cid:9) be p λ and p -lifts of A respectively (cf. Def-inition 7.1.8). Let q ∈ C ∗ r ( M λ , F λ ) be given by q = n ∑ j = q ι λ j ,0 · ... · q ι λ j , kj ; q ι λ j , kj ∈ C ∗ r U λ ι λ j , kj , F λ | U λ ιλ j , kj U λ ι λ j , kj ∈ A λ , (7.3.10)Suppose that there is U ∈ A such that for any j ∈ {
1, ..., n } there is l ∈ (cid:8)
1, ..., k j (cid:9) such that p λ (cid:0) U (cid:1) = U λ ι λ j ,0 . From the Lemma 7.1.10 it follows that for any j ∈{
1, ..., n } there is the unique tuple n U ι j ,1 , ...., U ι j , k o such that • p λ (cid:16) U ι j , l (cid:17) = U λ ι λ j , l for any l =
1, ..., k j . • G (cid:16) U ι j ,0 (cid:17) ... G (cid:16) U ι j , kj (cid:17) = ∅ .Denote by U q = n [ j = k j [ l = U ι j , l . (7.3.11)349or any j ∈ {
1, ..., n } and l ∈ (cid:8)
1, ..., k j (cid:9) , we select q j , l ∈ C ∗ r (cid:18) U ι j , l , F (cid:12)(cid:12) U ι j , l (cid:19) such that desc p λ (cid:16) q j , l (cid:17) = q ι λ j , l . Let q def = n ∑ j = q ι j ,0 · ... · q ι j , kj ∈ C ∗ r (cid:16) U q , F (cid:12)(cid:12) U q (cid:17) . (7.3.12) Definition 7.3.9.
In the situation 7.3.8 we say that q is p λ - U - extended lift of q andwe write ext - lift U p λ ( q ) def = q . Remark 7.3.10.
The given by (7.3.11) set U q is a finite union of the sets with com-pact closure. So the closure of U q is compact. From the Lemma 4.11.20 there is λ U q ∈ Λ such that for any λ ≥ λ U q ∈ Λ such that for any λ ≥ λ U q the set U q ismapped homeomorphically onto p λ (cid:16) U q (cid:17) ⊂ X λ . Lemma 7.3.11.
Consider the above situation. Let
U ∈ A and let a ∈ C ∗ r (cid:0) U , F (cid:12)(cid:12) U (cid:1) apositive element. If λ ∈ Λ and z ∈ C ∗ r (cid:0) M , F (cid:1) ∼ . For any ε > there is y ∈ C ∗ r (cid:0) M , F (cid:1) ∼ and an open subset U y ⊂ X with compact closure such that k y − z k < ε , y ∗ ay ∈ C ∗ r (cid:16) U y , F (cid:12)(cid:12) U y (cid:17) . Proof.
For any λ ∈ Λ denote by a λ def = desc p λ ( a ) . If in the equation (7.3.10) q ι λ j ,0 = √ a λ for all j ∈ {
1, ..., n } and q is given by the Equation (7.3.12) then q ∗ q = n ∑ j = q ∗ ι j , kj · ... · q ∗ ι j ,1 ! a n ∑ j = q ι j ,1 · ... · q ι j , kj ! ∈ C ∗ r (cid:16) U q , F (cid:12)(cid:12) U q (cid:17) (7.3.13)Any z ∈ C ∗ r (cid:0) M , F (cid:1) ∼ can be presented by the sum Let z = z + c where z ∈ C ∗ r ( M λ , F λ ) and c ∈ C . From (i) of the Lemma 7.1.13 it turns out that z can beregarded as an element of M (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:1) so one has z ∗ az ∈ C ∗ r (cid:0) M , F (cid:1) . For all350 > t ∈ C ∗ r ( M λ , F λ ) such that t = n ∑ j = t ι λ j ,1 · ... · t ι λ j , kj ; t ι λ j , kj ∈ C ∗ r U λ ι λ j , kj , F λ | U λ ιλ j , kj U λ ι λ j , kj ∈ A λ , k t − z k < δ , y def = c + t = c + n ∑ j = t ι λ j ,1 · ... · t ι λ j , kj ; k y − z k < ε . (7.3.14)If I ⊂ {
1, ..., n } is such that j ∈ I ⇔ p λ (cid:0) U (cid:1) ∩ U λι λ kj = ∅ and r = c + ∑ j ∈ I t ι λ j ,1 · ... · t ι λ j , kj then one has r ∗ ar = y ∗ ay , ∀ λ ∈ Λ λ ≥ λ ⇒ r ∗ desc p λ ( a ) r = y ∗ desc p λ ( a ) (7.3.15)On the other hand from j ∈ I ⇔ p λ (cid:0) U (cid:1) ∩ U λι λ kj = ∅ if follows that the sum q = c q desc p λ ( a ) + ∑ j ∈ I q desc p λ ( a ) t ι λ j ,1 · ... · t ι λ j , kj complies with the construction 7.3.8, so there is p λ - U -extended lift ext - lift U p λ ( q ) of q (cf. Definition 7.3.9) given by q = √ a + ∑ j ∈ I √ a t ι j ,0 · ... · t ι j , kj ∈ C ∗ r (cid:16) U q , F (cid:12)(cid:12) U q (cid:17) . (7.3.16)where the closure of U q is compact. From the Equations 7.3.13 and (7.3.15) itfollows that y ∗ ay = q ∗ q ∈ C ∗ r (cid:16) U q , F (cid:12)(cid:12) U q (cid:17) .Define U y def = U q . Corollary 7.3.12.
Consider the above situation. Let
U ∈ A and let a ∈ C ∗ r (cid:0) U , F (cid:12)(cid:12) U (cid:1) apositive element. If λ ∈ Λ and z ∈ C ∗ r (cid:0) M , F (cid:1) ∼ then following conditions hold: i) The series ∑ g ∈ G λ g ( z ∗ az ) is strongly convergent in B (cid:16) b H a (cid:17) .(ii) For any ε > there is y ∈ C ∗ r (cid:0) M , F (cid:1) ∼ such that for all λ ≥ λ one has k y − z k < ε , (7.3.17) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ g ( z ∗ az ) − ∑ g ∈ G λ g ( y ∗ ay ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε , (7.3.18) ∃ µ ≥ λ λ ≥ µ ⇒ ∑ g ∈ G λ g ( y ∗ ay ) = ∑ g ∈ G λ g ( y ∗ ay ) ! . (7.3.19) Proof. (i) For all g ∈ G λ one has ( ga ) zz ∗ ( ga ) ≤ k z k ( ga ) ( ga ) .On the other hand ∑ g ∈ G λ k z k ( ga ) ( ga ) = ∑ g ∈ G λ k z k ( ga ) = k z k π a (cid:16) desc p λ ( a ) (cid:17) ,hence for any finite subset G λ ⊂ G λ one has ∑ g ∈ G λ ( ga ) zz ∗ ( ga ) ≤ k z k π a (cid:16) desc p λ ( a ) (cid:17) ,i.e. an increasing series ∑ g ∈ G λ ( ga ) zz ∗ ( ga ) is bounded. From the Lemma D.1.25it turns ot that the series ∑ g ∈ G λ ( ga ) zz ∗ ( ga ) is strongly convergent in B (cid:16) b H a (cid:17) . Itfollows that the series ∑ g ∈ G λ g ( z ∗ az ) == z ∗ ∑ g ∈ G λ ( ga ) zz ∗ ( ga ) ! z is also strongly convergent in B (cid:16) b H a (cid:17) . 352ii) For all ∆ ∈ C ∗ r ( M λ , F λ ) ∼ such that k ∆ k = δ one has ∑ g ∈ G λ ( ga ) zz ∗ ( ga ) ≤≤ k z k ∑ g ∈ G λ ( ga ) ( ga ) = k z k desc p λ ( a ) . ∑ g ∈ G λ ( ga ) π a ( z + ∆ ) ( z + ∆ ) ∗ ( ga ) ≤ ( k z k + δ ) desc p λ ( a ) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ ( ga ) zz ∗ ( ga ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k z k k a k , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ ( ga ) π a ( z + ∆ ) ( z + ∆ ) ∗ ( ga ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ( k z k + δ ) k a k (7.3.20)The series in the Equations (7.3.20) contain positive summands only and from(7.3.20) it turns out that they are bounded. From the D.1.25 it follows that theseries in the Equations (7.3.20) are strongly convergent in B (cid:16) b H a (cid:17) . ( z + ∆ ) ∗ ∑ g ∈ G λ ( ga ) ( z + ∆ ) ( z + ∆ ) ∗ ( ga ) ! ( z + ∆ ) ∗ == z ∗ ∑ g ∈ G λ ( ga ) ( z + ∆ ) ( z + ∆ ) ∗ ( ga ) ! ( z + ∆ ) ∗ ++ π a ( ∆ ) ∗ ∑ g ∈ G λ ( ga ) ( z + ∆ ) ( z + ∆ ) ∗ ( ga ) ! ( z + ∆ ) ∗ .From the Equation (7.3.20) it follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) π a ( ∆ ) ∗ ∑ g ∈ G λ ( ga ) ( z + ∆ ) ( z + ∆ ) ∗ ( ga ) ! ( z + ∆ ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤≤ δ ( k z k + δ ) k a k Similarly one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) z ∗ ∑ g ∈ G λ ( ga ) ( z + ∆ ) ( z + ∆ ) ∗ ( ga ) ! ∆ ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤≤ δ ( k z k + δ ) k z k k a k (cid:13)(cid:13) ( z + ∆ ) ( z + ∆ ) ∗ − zz ∗ (cid:13)(cid:13) < δ + δ k z k If P def = (cid:12)(cid:12) ( z + ∆ ) ( z + ∆ ) ∗ − zz ∗ (cid:12)(cid:12) ∈ B (cid:16) b H a (cid:17) is given by the Proposition D.1.26 thenfollowing conditions hold • k P k = (cid:13)(cid:13) ( z + ∆ ) ( z + ∆ ) ∗ − zz ∗ (cid:13)(cid:13) < δ + δ k z k • (cid:12)(cid:12) ( ga ) (cid:0) ( z + ∆ ) ( z + ∆ ) ∗ − zz ∗ (cid:1) ( ga ) (cid:12)(cid:12) == ( ga ) P ( ga ) . • (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ ( ga ) (cid:0) ( z + ∆ ) ( z + ∆ ) ∗ − zz ∗ (cid:1) ( ga ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ ( ga ) P ( ga ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k P k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ ( ga ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) == k P k k a k ≤ (cid:0) δ + δ k z k (cid:1) k a k (7.3.21)If y def = z + ∆ then From the equation 7.1.42 it follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) z ∗ ∑ g ∈ G λ ( ga ) ( yy ∗ − zz ∗ ) ( ga ) ! z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤≤ (cid:0) δ + δ k z k (cid:1) k a k k z k .From the above equations it follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ g ( y ∗ ay ) − ∑ g ∈ G λ g ( z ∗ az ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ φ ( δ ) (7.3.22)where δ = k y − z k and φ : R + → R + is a continuous function given by φ ( δ ) = δ ( k z k + δ ) k a k + δ ( k z k + δ ) k z k k a k ++ (cid:0) δ + δ k z k (cid:1) k a k k z k == δ k a k (cid:16) ( k z k + δ ) + ( k z k + δ ) k z k + ( δ + k z k ) k z k (cid:17) . (7.3.23)354ince φ ( ) = ε > δ ε > φ ( δ ε ) < ε . From the Lemma7.3.11 it follows that there is y ∈ C ∗ r (cid:0) M , F (cid:1) ∼ an open subset U y ⊂ X with compactclosure such that k y − z k < min ( ε , δ ε ) , y ∗ ay ∈ C ∗ r (cid:16) U y , F (cid:12)(cid:12) U y (cid:17) .From our above construction it turns out that y satisfies to inequalities 7.3.17,7.3.18. From the Lemma 4.11.20 it follows that there is µ ∈ Λ such that for any λ ≥ λ U the set U y is mapped homeomorphically onto p λ (cid:16) U y (cid:17) . From the Lemma7.2.12 it turns out that for all λ ≥ µ following conditions hold ∑ g ∈ G λ g ( y ∗ ay ) = desc p λ ( y ∗ ay ) , ∑ g ∈ G λ g ( y ∗ ay ) = desc p λ ( y ∗ ay ) ,hence one has ∑ g ∈ G λ g ( y ∗ ay ) = ∑ g ∈ G λ g ( y ∗ ay ) ! i.e. y satisfies to the Equation 7.3.19. Lemma 7.3.13.
Consider the above situation. If
U ∈ A then any positive element a ∈ C ∗ r (cid:0) U , F (cid:12)(cid:12) U (cid:1) ⊂ C ∗ r (cid:0) M , F (cid:1) ⊂ B (cid:16) b H a (cid:17) is special (cf. Definition 3.1.19).Proof. Let λ ∈ Λ , ε > z ∈ C ∗ r ( M λ , F λ ) ∼ . Let us check conditions (a), (b)of the Definition 3.1.19.(a) One needs check that ∑ g ∈ G λ z ∗ ( ga ) z ∈ C ∗ r ( M λ , F λ ) , (7.3.24) ∑ g ∈ G λ f ε ( z ∗ ( ga ) z ) ∈ C ∗ r ( M λ , F λ ) , (7.3.25) ∑ g ∈ G λ ( z ∗ ( ga ) z ) ∈ C ∗ r ( M λ , F λ ) . (7.3.26)(cf. Remark 3.1.38). 355. Proof of (7.3.24). One has ∑ g ∈ G λ z ∗ ( ga ) z = z ∗ (cid:16) ∑ g ∈ G λ ga (cid:17) z . From theLemma 7.2.12 it turns out that there is the strong limit ∑ g ∈ G λ ga = desc p λ ( a ) ∈ C ∗ r ( M λ , F λ ) , so one has ∑ g ∈ G λ z ∗ ( ga ) z ∈ C ∗ r ( M λ , F λ ) .2. Proof of (7.3.25). From f ε ( z ∗ ( ga ) z ) ≤ z ∗ ( ga ) z it follows that ∑ g ∈ G λ f ε ( z ∗ ( ga ) z ) ≤ a λ for any finite subset G λ ⊂ G λ so from the Lemma D.1.25 it turns out thatthe series ∑ g ∈ G λ f ε ( z ∗ ( ga ) z ) is strongly convergent in B (cid:16) b H a (cid:17) . Moreover one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ f ε ( z ∗ ( ga ) z ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k a λ k . (7.3.27)From the Lemma 7.3.11 it follows that for all For all δ > y ∈ C ∗ r ( M λ , F λ ) ∼ such that k y − z k < δ and an open subset U y ⊂ X withcompact closure such that y ∗ ay ∈ C ∗ r (cid:16) U y , F (cid:12)(cid:12) U y (cid:17) .If δ ′ > δ is such that δ ( δ + k z k ) k a k < δ ′ then k y − z k < δ ⇒ k y ∗ ay − z ∗ az k < δ ′ , k y − z k < δ ⇒ (cid:13)(cid:13) y ∗ desc p λ ( a ) y − z ∗ desc p λ ( a ) z (cid:13)(cid:13) < δ ′ . (7.3.28)If b H ⊥ a is given by b H ⊥ a def = n ξ ∈ b H a (cid:12)(cid:12)(cid:12) (cid:16) y ∗ desc p λ ( a ) y (cid:17) ξ = o ⊂ b H a ,and p ⊥ y : b H a → b H ⊥ a is the natural projection and p y def = (cid:16) − p ⊥ y (cid:17) then thereis the following matrix representation z ∗ desc p λ ( a ) z = p y z ∗ desc p λ ( a ) zp y p y z ∗ desc p λ ( a ) zp ⊥ y p ⊥ y z ∗ desc p λ ( a ) zp y p ⊥ y z ∗ desc p λ ( a ) zp ⊥ y ! , y ∗ desc p λ ( a ) y = (cid:18) p y y ∗ desc p λ ( a ) yp y
00 0 (cid:19) , z ∗ az = p y z ∗ azp y p y z ∗ azp ⊥ y p ⊥ y z ∗ azp y p ⊥ y z ∗ azp ⊥ y ! , y ∗ ay = (cid:18) p y y ∗ ayp y
00 0 (cid:19) ,356ence from the inequality (7.3.28) it turns out that (cid:13)(cid:13)(cid:13) p ⊥ y z ∗ azp ⊥ y (cid:13)(cid:13)(cid:13) < δ ′ and (cid:13)(cid:13)(cid:13) p ⊥ y z ∗ desc p λ ( a ) zp ⊥ y (cid:13)(cid:13)(cid:13) < δ ′ . Taking into account that p ⊥ y f ε ( z ∗ az ) p ⊥ y ≤ p ⊥ y z ∗ azp ⊥ y p ⊥ y f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) p ⊥ y ≤ p ⊥ y z ∗ desc p λ ( a ) zp ⊥ y (cid:13)(cid:13)(cid:13) p ⊥ y f ε ( z ∗ az ) p ⊥ y (cid:13)(cid:13)(cid:13) < δ ′ , (cid:13)(cid:13)(cid:13) p ⊥ y f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) p ⊥ y (cid:13)(cid:13)(cid:13) < δ ′ . (7.3.29)From the above inequalities and the Lemma 1.4.7 it follows that (cid:13)(cid:13)(cid:13)(cid:13) f ε ( z ∗ az ) − (cid:18) p y f ε ( z ∗ az ) p y
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) << δ ′ + q δ ′ k z k k a k , (cid:13)(cid:13)(cid:13)(cid:13) f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) − (cid:18) p y f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) p y
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) << δ ′ + q δ ′ k z k k a k . (7.3.30)From the Corollary 4.11.21 it follows that there is µ def = λ U y and a continuousmap f µ : M µ → [
0, 1 ] such that f µ (cid:16) p µ (cid:16) U y (cid:17)(cid:17) = { } and g supp f µ ∩ supp f µ = ∅ for all nontrivial g ∈ G (cid:0) X µ (cid:12)(cid:12) X (cid:1) . From the Equation (7.1.45) it follows that f µ is a multiplier of C ∗ r ( M λ , F λ ) for any λ ≥ µ such that f λ p y = p y f λ = p y .It follows that p y f ε ( z ∗ az ) p y = p y f µ f ε ( z ∗ az ) f µ p y , p y f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) p y = p y f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ p y , (7.3.31)so one has f µ f ε ( z ∗ az ) f µ = p y f ε ( z ∗ az ) p y p y f µ f ε ( z ∗ az ) f µ p ⊥ y p ⊥ y f µ f ε ( z ∗ az ) f µ p y p ⊥ y f µ f ε ( z ∗ az ) f µ p ⊥ y ! , f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ == p y f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) p y p y f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ p ⊥ y p ⊥ y f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ p y p ⊥ y f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ p ⊥ y ! ,357rom f µ ≤ (cid:13)(cid:13)(cid:13) p ⊥ y f µ f ε ( z ∗ az ) f µ p ⊥ y (cid:13)(cid:13)(cid:13) < δ ′ , (cid:13)(cid:13)(cid:13) p ⊥ y f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ p ⊥ y (cid:13)(cid:13)(cid:13) < δ ′ . (7.3.32)On the other hand from k f ε ( z ∗ az ) z k ≤ k z ∗ az k ≤ k z k k a k , (cid:13)(cid:13) f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) z (cid:13)(cid:13) ≤ (cid:13)(cid:13) z ∗ desc p λ ( a ) z (cid:13)(cid:13) ≤ k z k k a k , inequalities (7.3.29) andthe Lemma 1.4.7 it turns out that (cid:13)(cid:13)(cid:13)(cid:13) f µ f ε ( z ∗ az ) f µ − (cid:18) p y f µ f ε ( z ∗ az ) f µ p y
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) < δ ′ + q δ ′ k z k k a k , (cid:13)(cid:13)(cid:13)(cid:13) f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ − (cid:18) p y f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ p y
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) << δ ′ + q δ ′ k z k k a k . (7.3.33)The combination of inequalities (7.3.30) and (7.3.33) yields the following (cid:13)(cid:13) f ε ( z ∗ az ) − f µ f ε ( z ∗ az ) f µ (cid:13)(cid:13) < (cid:18) δ ′ + q δ ′ k z k k a k (cid:19) , (cid:13)(cid:13) f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) − f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ (cid:13)(cid:13) < (cid:18) δ ′ + q δ ′ k z k k a k (cid:19) .For any ε ′ > δ ′ > ε ′ < (cid:18) δ ′ + q δ ′ k z k k a k (cid:19) . Select δ > δ ( δ + k z k ) k a k < δ ′ . From the above description it followsthat k z − y k < δ ⇒ (cid:13)(cid:13) f ε ( z ∗ az ) − f µ f ε ( z ∗ az ) f µ (cid:13)(cid:13) < ε ′ , k z − y k < δ ⇒ (cid:13)(cid:13) f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) − f µ f ε (cid:0) z ∗ desc p λ ( a ) z (cid:1) f µ (cid:13)(cid:13) < ε ′ .(7.3.34)From the inequality (7.3.34) it follows that ∀ λ ≥ λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ g f ε ( z ∗ az ) − ∑ g ∈ G λ g (cid:0) f µ f ε ( z ∗ az ) f µ (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε ′ . (7.3.35)On the other hand from the Lemma 7.2.12 it follows that ∀ λ ≥ µ ∑ g ∈ G λ g (cid:0) f µ f ε ( z ∗ az ) f µ (cid:1) = desc e p λ (cid:0) f µ f ε ( z ∗ az ) f µ (cid:1) ∈ C ∗ r ( M λ , F λ ) ,358o from the Lemma 3.1.20 it turns out that ∀ λ ≥ λ ∑ g ∈ G λ g (cid:0) f µ f ε ( z ∗ az ) f µ (cid:1) ∈ C ∗ r ( M λ , F λ ) (7.3.36)From the Equations (7.3.35) and (7.3.36) it turns out that ∀ ε ′ > ∀ λ ≥ λ ∃ a λ ∈ C ∗ r ( M λ , F λ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ g f ε ( z ∗ az ) − a λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε ′ ,so one has ∀ λ ≥ λ ∑ g ∈ G λ g f ε ( z ∗ az ) ∈ C ∗ r ( M λ , F λ ) .3. Proof of (7.3.26). From the Corollary 7.3.12 it turns out that for any ε > y ∈ C ∗ r ( M λ , F λ ) + which satisfies to the Equations (7.3.18) and(7.3.19). It is already proven that for any λ ≥ λ ∑ g ∈ G λ g ( y ∗ ay ) = y ∗ desc p λ ( a ) y ∈ C ∗ r ( M λ , F λ ) so ∑ g ∈ G λ g ( y ∗ ay ) = y ∗ desc p λ ( a ) y ! ∈ C ∗ r ( M λ , F λ ) .From the Equation (7.3.19) it follows that ∃ µ ≥ λ λ ≥ µ ⇒ ∑ g ∈ G λ g ( y ∗ ay ) ∈ C ∗ r ( M λ , F λ ) and taking into account the Lemma 3.1.20 one has ∀ λ ≥ λ ∑ g ∈ G λ g ( y ∗ ay ) ∈ C ∗ r ( M λ , F λ ) (7.3.37)Using the Equations (7.3.18) and (7.3.37) one concludes that ∀ ε > ∀ λ ≥ λ ∃ b λ ∈ C ∗ r ( M λ , F λ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ g ( f ε ( z ∗ az )) − b λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε .Hence one has ∑ g ∈ G λ g ( f ε ( z ∗ az )) ∈ C ∗ r ( M λ , F λ ) .359b) If δ > δ ( δ + k y − z k ) k a k < δ ′ then from k y − z k < δ it followsthat (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ g ( z ∗ az ) − ∑ g ∈ G λ g ( y ∗ ay ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < δ ′ (7.3.38)If δ ′ is such that δ ′ k a k k z k (cid:16) δ ′ + k a k k z k (cid:17) < ε /2 then from k y − z k < δ itfollows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ g ( z ∗ az ) ! − ∑ g ∈ G λ g ( y ∗ ay ) ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε y ∈ C ∗ r ( M λ , F λ ) such that k y − z k < min ( ε /2, δ ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ g ( z ∗ az ) − ∑ g ∈ G λ g ( y ∗ ay ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε ∃ µ ≥ λ λ ≥ µ ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ G λ g ( z ∗ az ) ! − ∑ g ∈ G λ g ( z ∗ az ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε (7.3.41)The Equation (7.3.41) is version of (3.1.21). Corollary 7.3.14.
Let S ( M , F , x ) def = ( { p λ : ( M λ , F λ ) → ( M , F ) } λ ∈ Λ , (cid:8) p µν : (cid:0) M µ , F µ (cid:1) → ( M ν , F ν ) (cid:9) µ , ν ∈ Λ µ ≥ ν ) be pointed finite foliation category (cf. Definition (ref. 3.1.6), and let M be the disconnectedinverse limit 4.11.19 of a pointed topological finite covering category S ( M , x ) def = def = (n p λ : (cid:16) M λ , x λ (cid:17) → ( M , x ) o λ ∈ Λ , (cid:8) p µν : (cid:0) M µ , x µ (cid:1) → ( X ν , x ν ) (cid:9) µ , ν ∈ Λ µ ≥ ν ) ∈ FinTop pt , (cf. Definition 4.11.3) and p : M → M be the natural covering. Let (cid:0) M , F (cid:1) be theinduced by p foliated space (cf. Definition G.2.5). If (cid:8) { C ∗ r ( p λ ) : C ∗ r ( M , F ) ֒ → C ∗ r ( M λ , F λ ) } , (cid:8) C ∗ r (cid:0) p µν (cid:1) : C ∗ r ( M ν , F ν ) ֒ → C ∗ r (cid:0) M µ , F µ (cid:1)(cid:9)(cid:9) .360 s a pointed algebraical finite covering category S C ∗ r ( M , F , x ) (cf. Definition (ref. 3.1.6)given by the Lemma 7.3.5 and A ∈ B ( H a ) is its disconnected algebraical infinite non-commutative covering (cf. Definition 3.1.25) then there is the natural inclusionC ∗ r (cid:0) M , F (cid:1) ⊂ A . Proof.
The C ∗ -algebra C ∗ r (cid:0) M , F (cid:1) is generated by special elements given by theLemma 7.3.13.Any special element a = b π a ( b x ) b b π a ( b x ) ∈ B (cid:16) b H a ∼ = L x ∈ M no hol L ( G x ) (cid:17) yieldsthe family of operators (cid:8) a x ∈ B (cid:0) L ( G x ) (cid:1)(cid:9) x ∈ M no hol .We would like to prove that { a x } gives an element of C ∗ r (cid:0) M , F (cid:1) (cid:0) G (cid:0) M , F (cid:1)(cid:1) . Todo it one should find the covering G (cid:0) M , F (cid:1) = S V α such that for all α the family { a x } represents element s x ∈ C ∗ r (cid:0) M , F (cid:1) (cid:0) V α (cid:1) . From the construction 7.1.16 oneshould proof that the family p x V α a x q x V α corresponds to the element of C ∗ r (cid:0) M , F (cid:1) (cid:0) V α (cid:1) . Lemma 7.3.15.
Let
V ⊂ G (cid:0) M , F (cid:1) be such that(a) V ∩ g V = ∅ for any nontrivial g ∈ G (cid:0) M (cid:12)(cid:12) M (cid:1) .(b) V is a set of type I (cf. 7.1.4) or V α = V b where V b is given by (7.1.34) .If a ∈ B (cid:16) b H a (cid:17) is a special element (cf. Definition 3.1.19) which corresponds to the familyof operators (cid:8) a x ∈ B (cid:0) L ( G x ) (cid:1)(cid:9) x ∈ M no hol then (cid:8) a x ∈ B (cid:0) L ( G x ) (cid:1)(cid:9) yields the elements ∈ C ∗ r (cid:0) M , F (cid:1) (cid:0) V α (cid:1) .Proof. For each λ ∈ Λ let a λ ∈ C ∗ r ( M λ , F λ ) + be such that ∑ g ∈ ker ( b G → G ( M | M λ )) ga = b π ( a λ ) . (7.3.42)Let V = q (cid:0) V (cid:1) ∈ G ( M , F ) and V λ = q λ (cid:0) V (cid:1) ∈ G ( M λ , F λ ) where q : G (cid:0) M , F (cid:1) →G ( M , F ) and q λ : G (cid:0) M , F (cid:1) → G ( M λ , F λ ) are natural coverings. Denote by V = q − ( V ) = ⊔ g ∈ G ( M | M ) g V and V λ = q − λ ( V ) where q λ : G ( M λ , F ) → G ( M , F λ ) isthe natural covering. Let Y ⊂ M be the transversal to V considered in the section361.1.4, let Y = p (cid:0) Y (cid:1) ⊂ M and Y λ = p λ (cid:0) Y (cid:1) ⊂ M . Denote by Y = p − ( Y ) = ⊔ g ∈ G ( M | M ) g Y and Y λ = p − λ ( Y ) . Any a λ defines an element of C ∗ r ( M λ , F λ ) ( V λ ) .For all λ ∈ Λ denote by X C ( Y λ ) the Hilbert C ( Y λ ) -module which comes from(7.1.27) if V is of type I (resp. from (7.1.37) V = V b (cf. (7.1.34)). From the resultsof the section 7.1.4 for all λ ∈ Λ there is the isomorphism C ∗ r ( M λ , F λ ) ( V λ ) ∼ = K (cid:16) X C b ( Y λ ) (cid:17) it turns out that one has the surjective *-homomorphism φ λ : C ∗ r ( M λ , F λ ) → K (cid:16) X C b ( Y λ ) (cid:17) (7.3.43)Let b K def = C ∗ -lim −→ λ ∈ Λ K (cid:16) X C b ( Y λ ) (cid:17) be the C ∗ -inductive limit, and let π K a : b K → B (cid:0) H K a (cid:1) be the atomic representation. The family n φ λ ( a λ ) ∈ b K o is decreasing,hence from the Lemma D.1.25 it follows that there is the limit a K = lim φ λ ( a λ ) ∈ B (cid:0) H K a (cid:1) with respect to the strong topology of B (cid:0) H K a (cid:1) . The element a is special,so it satisfies to conditions (a) and (b) of the Definition 3.1.19. It turns out thatthe element a K satisfies to conditions (a) and (b) of the Definition 5.4.8. From theLemma 5.4.16 it follows that a K ∈ K (cid:16) X C ( Y ) (cid:17) , (7.3.44)and taking into account the Section 7.1.4 the family { a λ } yields the element s ∈ C ∗ r (cid:0) M , F (cid:1) (cid:0) V α (cid:1) Corollary 7.3.16.
Any special element a ∈ B (cid:16) b H a (cid:17) corresponds to the global sections ∈ C ∗ r (cid:0) M , F (cid:1) (cid:0) G (cid:0) M , F (cid:1)(cid:1) .Proof. There is family (cid:8) V α ⊂ G (cid:0) M , F (cid:1)(cid:9) such that (cid:0) M , F (cid:1) such that V α satisfiesto the conditions (a), (b) of the Lemma 7.3.15. From the Lemma 7.3.15 it followsthat the family { a x } yields the family (cid:8) s α ∈ C ∗ r (cid:0) M , F (cid:1) (cid:0) V α (cid:1)(cid:9) a ⊂ A . Moreover onehas s α ′ | V α ′ ∩V α ′′ = s α ′′ | V α ′ ∩V α ′′ , hence the family { s α } gives the global section s ∈ C ∗ r (cid:0) M , F (cid:1) (cid:0) G (cid:0) M , F (cid:1)(cid:1) . Corollary 7.3.17.
If a ∈ B (cid:16) b H a (cid:17) is a special element (cf. Definition 3.1.19) then a ∈ C ∗ r (cid:0) M , F (cid:1) where we assume the given by the Equation (7.3.8) inclusion C ∗ r (cid:0) M , F (cid:1) ⊂ B (cid:16) b H a (cid:17) . roof. From the Definition 7.3.1 that there is the minimal λ min ∈ Λ such that ( M , F ) = ( M λ min , F λ min ) . Let a = a λ min ∈ C ∗ r ( M , F ) + be given by the Equation3.1.23. If ε > δ > δ + p δ k a k < ε then from the Lemma7.1.19 there is an open set V ⊂ G ( M , F ) such that • The closure of V is compact, • k ( − P V ) π a ( a ) ( − P V ) k < δ where π a : C ∗ r ( M , F ) ֒ → B ( H a ) is the atomic representation (cf. DefinitionD.2.33). If \ C ∗ r ( M , F ) def = C ∗ -lim −→ λ ∈ Λ C ∗ r ( M , F ) and b π a : \ C ∗ r ( M , F ) → B (cid:16) b H a (cid:17) isthe atomic representation then P V can be regarded as the element of B (cid:16) b H a (cid:17) , i.e.there is b P V ∈ B (cid:16) b H a (cid:17) such that there is the natural *-isomorphism P V π a ( C ∗ r ( M , F )) P V ∼ = b P V b π a ( C ∗ r ( M , F )) b P V From a < b π a ( a ) it follows that (cid:16) − b P V (cid:17) a (cid:16) − b P V (cid:17) < (cid:16) − b P V (cid:17) b π a ( a ) (cid:16) − b P V (cid:17) ,so one has (cid:13)(cid:13)(cid:13)(cid:16) − b P V (cid:17) a (cid:16) − b P V (cid:17)(cid:13)(cid:13)(cid:13) < δ .From δ + p δ k a k < ε and the Lemma (1.4.7) it follows that (cid:13)(cid:13)(cid:13) a − b P V a b P V (cid:13)(cid:13)(cid:13) < ε G ( M , F ) can be represented as a union G ( M , F ) ∪ α ∈ A V α where for all α ∈ A the set V α is described in the Section 7.1.4. If q : G (cid:0) M , F (cid:1) → G (cid:0) M , F (cid:1) is the natural covering the using subordinated atlas (cf. Definition 7.1.7) one cansuppose that V α is evenly covered by q (cf. Definition A.2.1). Since the closure of V is compact there is a covering sum for V (cf. Definition 4.2.5), i.e. a finite subset A ⊂ A and a family (cid:8) f α ∈ C c (cid:0) G (cid:0) M , F (cid:1)(cid:1)(cid:9) such that Dom f α = [
0, 1 ] and ∀ x ∈ V f ( x ) = ∑ α ∈ A f α ( x ) = α ∈ A we select a connected set V α which is mapped homeomorphicallyonto V α , and f α def = lift q e U ( f α ) (cf. Definition 4.5.70). From the Corollary 7.3.17 itfollows that the element a corresponds to the section s ∈ C ∗ r (cid:0) M , F (cid:1) (cid:0) G (cid:0) M , F (cid:1)(cid:1) .363rom the Equations (7.1.29), (7.1.39) for any α ∈ A and g ∈ G (cid:0) M (cid:12)(cid:12) M (cid:1) one candefine (cid:16) g f α (cid:17) · s | g V α ∈ C ∗ r (cid:0) M , F (cid:1) . If f ′ def = − ∑ α ∈ A g ∈ G ( M | M ) ∈ C b (cid:0) G (cid:0) M , F (cid:1)(cid:1) then k f ′ k =
1. From the inequalities (7.1.25), (7.3.45) and taking into account a − ∑ α ∈ A g ∈ G ( M | M ) (cid:16) g f α (cid:17) · s | g V α = f ′ · (cid:16) a − b P V a b P V (cid:17) one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − ∑ α ∈ A g ∈ G ( M | M ) (cid:16) g f α (cid:17) · s | g V α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε α ∈ A thefollowing condition holds ∑ g ∈ G ( M | M ) (cid:16) g f α (cid:17) · s | g V α ∈ K (cid:18) X C (cid:16) F g ∈ G ( M | M ) (cid:17) g V α (cid:19) ,hence for all α ∈ A there is G α ⊂ G (cid:0) M (cid:12)(cid:12) M (cid:1) such that ∀ g ∈ G (cid:0) M (cid:12)(cid:12) M (cid:1) \ G α (cid:13)(cid:13)(cid:13)(cid:16) g f α (cid:17) · s | g V α (cid:13)(cid:13)(cid:13) < ε | A | .Otherwise if P g V α and Q g V α are given by (7.1.22) then one has ∀ g ′ , g ′′ ∈ G (cid:0) M (cid:12)(cid:12) M (cid:1) g ′ = g ′′ ⇒⇒ s (cid:0) g ′ V α (cid:1) ∩ s (cid:0) g ′′ V α (cid:1) = ∅ AND r (cid:0) g ′ V α (cid:1) ∩ r (cid:0) g ′′ V α (cid:1) = ∅ ⇒⇒ P g ′ V α b H a ⊥ P g ′′ V α b H a AND Q g ′ V α b H a ⊥ Q g ′′ V α b H a (7.3.47)From the Equation (7.3.47) it follows that ∀ α ∈ A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − ∑ g ∈ G ( M | M ) \ G α (cid:16) g f α (cid:17) · s | g V α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε | A | .364nd taking into account the inequality (7.3.47) one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − ∑ α ∈ A ∑ g ∈ G α (cid:16) g f α (cid:17) · s | g V α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε .One the other hand the sum b def = ∑ α ∈ A ∑ g ∈ G α (cid:16) g f α (cid:17) · s | g V α has a finite set ofsummands and any summand lies in C ∗ r (cid:0) M , F (cid:1) , it follows that b ∈ C ∗ r (cid:0) M , F (cid:1) . Inresult one has ∀ ε > ∃ b ∈ C ∗ r (cid:0) M , F (cid:1) (cid:13)(cid:13)(cid:13) a − b (cid:13)(cid:13)(cid:13) < ε ,hence a ∈ C ∗ r (cid:0) M , F (cid:1) . Corollary 7.3.18.
If A is the disconnected algebraical inverse noncommutative limit ofthe given by the Lemma 7.3.5 pointed algebraical finite covering category (cid:8) { C ∗ r ( p λ ) : C ∗ r ( M , F ) ֒ → C ∗ r ( M λ , F λ ) } , (cid:8) C ∗ r (cid:0) p µν (cid:1) : C ∗ r ( M ν , F ν ) ֒ → C ∗ r (cid:0) M µ , F µ (cid:1)(cid:9)(cid:9) . then one has A ⊂ ϕ (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:1) where ϕ is given by the Equation (7.3.8) .Proof. If π a ( b x ) a π a ( b y ) be weakly special element such that a is special and b x , b y ∈ C ∗ -lim −→ λ ∈ Λ C ∗ r ( M λ , F λ ) then a ∈ ϕ (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:1) . From the Lemma 7.1.13 it turnsout that C ∗ r ( M λ , F λ ) ⊂ M (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:1) , hence one has C ∗ -lim −→ λ ∈ Λ C ∗ r ( M λ , F λ ) ⊂ M (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:1) . It follows that b x , b y ∈ M (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:1) , (7.3.48)hence one has π a ( b x ) a π a ( b y ) ∈ ϕ (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:1) .The C ∗ -algebra A is generated by weakly special elements, so one has A ⊂ ϕ (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:1) From the Corollaries 7.3.14 and 7.3.18 it turns out that A = ϕ (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:1) ,or equivalently, there is the natural *-isomorphism A ∼ = C ∗ r (cid:0) M , F (cid:1) . (7.3.49)According to the Definition 3.1.25 one has G (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:12)(cid:12) C ∗ r ( M , F ) (cid:1) = lim ←− λ ∈ Λ G ( C ∗ r ( M λ , F λ ) | C ∗ r ( M , F )) G ( C ∗ r ( M λ , F λ ) | C ∗ r ( M , F )) ∼ = G ( M λ | M ) and G (cid:0) M (cid:12)(cid:12) M (cid:1) = lim ←− λ ∈ Λ G ( M λ | M ) we conclude that G (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:12)(cid:12) C ∗ r ( M , F ) (cid:1) = G (cid:0) M (cid:12)(cid:12) M (cid:1) . (7.3.50)If e M ⊂ M be a connected component of M then C ∗ r (cid:16) e M , e F (cid:17) def = C ∗ r (cid:16) e M , F (cid:12)(cid:12) e M (cid:17) isthe maximal connected subalgebra of C ∗ r (cid:0) M , F (cid:1) . If G ⊂ G (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:12)(cid:12) C ∗ r ( M , F ) (cid:1) is the maximal among subgroups among subgroups G ′ ⊂ G (cid:0) C ∗ r (cid:0) M , F (cid:1)(cid:12)(cid:12) C ∗ r ( M , F ) (cid:1) such that GC ∗ r (cid:16) e M , e F (cid:17) = C ∗ r (cid:16) e M , e F (cid:17) then G is the maximal among subgroupsamong subgroups G ′′ ⊂ G (cid:0) M (cid:12)(cid:12) M (cid:1) such that G ′′ e M = e M (cf. (7.3.50)), or equiva-lently G = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) (7.3.51)If J ⊂ G (cid:0) M (cid:12)(cid:12) M (cid:1) is a set of representatives of G (cid:0) M (cid:12)(cid:12) M (cid:1) / G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) then fromthe (4.11.4) it follows that M = G g ∈ J g e M and C ∗ r (cid:0) M , F (cid:1) is a C ∗ -norm completion of the direct sum M g ∈ J gC ∗ r (cid:16) e M , e F (cid:17) . (7.3.52) Theorem 7.3.20.
Suppose that (cid:0) M , F (cid:1) is a local fiber bundle. Let S C ∗ r ( M , F , x ) be apointed algebraical finite covering category S C ∗ r ( M , F , x ) (cf. Definition (ref. 3.1.6) de-scribed in the Lemma 7.3.5 and given by (cid:8) { C ∗ r ( p λ ) : C ∗ r ( M , F ) ֒ → C ∗ r ( M λ , F λ ) } , (cid:8) C ∗ r (cid:0) p µν (cid:1) : C ∗ r ( M ν , F ν ) ֒ → C ∗ r (cid:0) M µ , F µ (cid:1)(cid:9)(cid:9) . Let S ( M , x ) = (cid:8) p λ : (cid:0) M λ , x λ (cid:1) → ( M , x ) (cid:9) λ ∈ Λ ∈ FinTop pt be the corresponding pointedtopological finite covering category (cf. Definition 4.11.3). Let e M = lim ←− S ( M , x ) be thetopological inverse limit of S ( M , x ) (cf. Definition 4.11.19), and let e p : e M → e M is anatural covering and (cid:16) e M , e F (cid:17) be the e p-lift of ( M , F ) (cf. Definition G.2.5). Then thefollowing conditions hold:(i) The category S C ∗ r ( M , F , x ) is good (cf. Definition 3.1.32). ii) The triple (cid:16) C ∗ r ( M , F ) , C ∗ r (cid:16) e M , e F (cid:17) , G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17)(cid:17) is the infinite noncommutativecovering of S C ∗ r ( M , F , x ) , (cf. Definition 3.1.34).Proof. From (7.3.49) it follows that the disconnected algebraical inverse noncom-mutative limit of S C ∗ r ( M , F , x ) is isomorphic to C ∗ r (cid:0) M , F (cid:1) which is the C ∗ -normcompletion of the algebraic direct sum L g ∈ J gC ∗ r (cid:16) e M , e F (cid:17) any connected compo-nent of the disconnected algebraical inverse noncommutative limit is isomorphicto C ∗ r (cid:16) e M , e F (cid:17) .(i) One needs check the conditions (a) - (c) of the Definition 3.1.32(a) For any λ ∈ Λ there is the natural injective *-homomorphism C ∗ r ( M λ , F λ ) ֒ → M (cid:16) C ∗ r (cid:16) e M , e F (cid:17)(cid:17) .(b) We already stated that C ∗ r (cid:0) M , F (cid:1) which is the C ∗ -norm completion of thealgebraic direct sum L g ∈ J gC ∗ r (cid:16) e M , e F (cid:17) (c) Follows from the surjective homomorphisms G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) → G ( M λ | M ) (ii) Follows from the proof of (i). 36768 hapter 8 Coverings of noncommutative tori
For all x = ( x , ..., x n ) ∈ R n one has k x k = | x | + ... + | x n | . So for any m ∈ N there is a polynomial map p m : R n → R such that k x k m = p m ( | x | , ..., | x n | ) .If f ∈ S ( R n ) then from the Equation (C.2.4) it follows that for all m ∈ N there is C ′ > | f ( x ) p m ( | x | , ..., | x n | ) | = | f ( x ) | k x k m < C ∀ x ∈ R n .If C ′′ = sup x ∈ R n | f ( x ) | then | f ( x ) | (cid:16) + k x k m (cid:17) < C ′ + C ′′ , i.e. ∀ m ∈ N ∃ C > | f ( x ) | < C (cid:16) + k x k m (cid:17) ∀ x ∈ R n .Similarly one can proof that ∀ f ∈ S ( R n ) ∀ m ∈ N ∃ C mf > | f ( x ) | < C mf (cid:0) + k x k m (cid:1) ∀ x ∈ R n (8.1.1)If R > y ∈ R N is such that k y k < R then from k x k > R if follows that k x + y k ≥ k x k − k y k . If k x k < max (
1, 2 R ) then f ( x + y ) < C ′′ otherwise f ( x + y ) < C mf (cid:0) + k x /2 k m (cid:1) .369f C m , Rf def = C ′′ ( + ( R )) m + C mf m then k y k < R ⇒ | f ( x + y ) | < C m , Rf (cid:0) + k x k m (cid:1) ∀ x ∈ R n . (8.1.2) C ∗ -algebra The finite-fold coverings of noncommutative tori are described in [16, 68]. Herewe would like to prove that these coverings comply with the general theory ofnoncommutative coverings described in 2.1.
Let C (cid:0) T n Θ (cid:1) be a noncommutative torus (cf. Definition F.1.2). Denote by { U k } k ∈ Z n ⊂ C (cid:0) T n Θ (cid:1) the set of unitary elements which satisfy to (F.1.1). We wouldlike to construct an inclusion π : C (cid:0) T n Θ (cid:1) ֒ → C (cid:16) T n e Θ (cid:17) where C (cid:16) T n e Θ (cid:17) is anothernoncommutative torus. Denote by n e U k o k ∈ Z n ⊂ C (cid:16) T n e Θ (cid:17) the set of unitary ele-ments which satisfy to (F.1.1), i.e. e U k e U p = e − π ik · e Θ p e U k + p (8.2.1)We suppose that there is a subgroup Γ ∈ Z n of maximal rank such that e U k ∈ π ( C ( T n Θ )) ⇔ k ∈ Γ . (8.2.2)Recall that, given a subgroup Γ $ Z n of maximal rank, there is an invertible n × n -matrix M with integer valued entries such that Γ = M Z n . It is proven in [68] thatthe inclusion π : C (cid:0) T n Θ (cid:1) ֒ → C (cid:16) T n e Θ (cid:17) exists if and only if the following conditionholds Θ ∈ M T e Θ M + M n ( Z ) . (8.2.3)From Γ = Γ = M Z n it turns out that C (cid:0) T n Θ (cid:1) is the C ∗ -norm completion of the C -linear span of of unitary elements given by U k = e U Mk ∈ C ( T n Θ ) ; k ∈ Z n . (8.2.4)For any j =
1, ..., n denote by e u j def = e U k j , where k j =
0, ..., 1 |{z} j th − place , ..., 0 ∈ C (cid:16) T n e Θ (cid:17) . (8.2.5)370here is the action R n × C (cid:16) T n e Θ (cid:17) → C (cid:16) T n e Θ (cid:17) ; s · u j e π js u j .If Γ ∨ the dual lattice Γ ∨ = { x ∈ Q n | x · Γ ∈ Z } = M − Z n (8.2.6)then the action R n × C (cid:16) T n e Θ (cid:17) → C (cid:16) T n e Θ (cid:17) can by restricted onto Γ ∨ such that ga = a ∀ a ∈ C ( T n Θ ) .If g ∈ Γ ∨ ∩ Z n then g e a = a for all e a ∈ C (cid:16) T n e Θ (cid:17) . It follows that there is the inclusion Γ ∨ / Z n ⊂ n g ∈ Aut (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ C ( T n Θ ) o (8.2.7)From Γ = M Z n and Z n = M Γ ∨ one has the isomorphism Γ ∨ / Z n ∼ = Z n / Γ and theinclusion. Z n / Γ ⊂ n g ∈ Aut (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ C ( T n Θ ) o (8.2.8) Definition 8.2.2.
In the described in 8.2.1 situation we say that the inclusion π : C (cid:0) T n Θ (cid:1) ֒ → C (cid:16) T n e Θ (cid:17) corresponds to the matrix M . Lemma 8.2.3.
If C (cid:0) T n Θ (cid:1) ⊂ C (cid:16) T n e Θ (cid:17) is a given by 8.2.1 inclusion then the groupG = n g ∈ Aut (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) (cid:12)(cid:12)(cid:12) ga = a ; ∀ a ∈ C ( T n Θ ) o (8.2.9) is finite.Proof. Let for any j =
1, ..., n the unitary element u j ∈ C (cid:16) T n e Θ (cid:17) is given by (8.2.5).From (8.2.2) it turns out that for any j ∈
1, ..., n there is l j ∈ N such that e U l j k j = e u l j j ∈ C (cid:0) T n Θ (cid:1) . C ( T n Θ ) j def = n a ∈ C ( T n Θ ) | a e u l j j = e u l j j a o ⊂ C ( T n Θ ) , C (cid:16) T n e Θ (cid:17) j = n e a ∈ C (cid:16) T n e Θ (cid:17) (cid:12)(cid:12)(cid:12) e aa = a e a ∀ a ∈ C ( T n Θ ) j o g ∈ G one has gC (cid:16) T n e Θ (cid:17) j ⊂ C (cid:16) T n e Θ (cid:17) j . Otherwise C (cid:16) T n e Θ (cid:17) j is a com-mutative C ∗ -algebra generated by u j . If e w = g e u j then e w l j = e u l j j , and since C (cid:16) T n e Θ (cid:17) j is commutative one has e w l j e u ∗ l j j = (cid:16) e w e u ∗ j (cid:17) l j =
1. (8.2.10)From (8.2.10) it turns out that there is 0 ≤ m j < l j such e w = e π imjlj e u ∗ j . Similarly onecan prove that there are l j ( j =
1, ..., n ) such that for any g ∈ G there are 0 ≤ m j < l j following condition holds g e U k = exp (cid:18) π i (cid:18) k l + ... + k n l n (cid:19)(cid:19) e U k ; where k = ( k , ..., k n ) . (8.2.11)From (8.2.11) it turns out that | G | = l · ... · l n , i.e. G is finite. For any x ∈ R n let us consider a continuous map f x : [
0, 1 ] → Aut ( C ( T n Θ )) ; t (cid:16) U k e π itk · x U k (cid:17) .and there is the π -lift e f x : [
0, 1 ] → Aut (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) of f x (cf. Definition 2.5.1) givenby e f x ( t ) e U k = e π it ( M − k ) · x e U k ;where M satisfies to the Equation .(8.2.4)From the Lemma 8.2.3 and the Corollary 2.5.6 it follows that e f x is the unique π -liftof f x . From the Lemma 2.5.4 it turns out that e f x commutes with the given by (8.2.9)group G . There is the continuous action R n × C ( T n Θ ) → C ( T n Θ ) , x • U k = f x ( ) U k = e π ik · x U k ; ∀ x ∈ R n (8.2.12)which can be uniquely lifted to the continuous action R n × C (cid:16) T n e Θ (cid:17) → C (cid:16) T n e Θ (cid:17) , x • e U k = e π i ( M − k ) · x e U k ; ∀ x ∈ R n (8.2.13)Clearly one has x • e a = e a ; ∀ e a ∈ C (cid:16) T n e Θ (cid:17) ⇔ x ∈ Γ (8.2.14)372 emma 8.2.5. If G is given by (8.2.9) then there is the natural group isomorphism G ∼ = Z n / Γ .Proof. If x ∈ Z n then clearly x • a = a for any a ∈ C (cid:0) T n Θ (cid:1) so there is the naturalgroup homomorphism φ : Z n → G . From (8.2.14) it turns out that ker φ = Γ ,i.e. there is the natural inclusion Z n / Γ → G . From the Lemma 2.5.4 it turnsout that the action of G commutes with the action of Z n / Γ on C (cid:16) T n e Θ (cid:17) it followsthat G is the direct product G = G ′ × Z n / Γ . Action of G on C (cid:16) T n e Θ (cid:17) induces theunitary action of G on the Hilbert space L (cid:16) C (cid:16) T n e Θ (cid:17) , e τ (cid:17) . It follows that there isthe following Hilbert direct sum L (cid:16) C (cid:16) T n e Θ (cid:17) , τ (cid:17) = ⊕ λ ∈ Λ G ′ H λ M ⊕ µ ∈ Λ Z n / Γ H µ (8.2.15)where Λ G ′ (resp. Λ Z n / Γ ) is the set of irreducible representations of G ′ (resp. Z n / Γ ), H λ (resp. H µ ) is the Hilbert subspace which corresponds to the representation λ (resp. µ ). The group Z n / Γ is and commutative group, hence any irreduciblerepresentation of Z n / Γ has dimension 1. For any x ∈ Γ ∨ / Z n the correspondingrepresentation ψ x : Z n / Γ × C → C is given by ψ (cid:16) k , z (cid:17) , χ x (cid:16) k (cid:17) = e π ikx ;where z ∈ C , k ∈ Z n and x ∈ Γ ∨ are representatives of k ∈ Z n / Γ and x ∈ Γ ∨ / Z n . (8.2.16)So there is a bijective map ϕ Z n / Γ : Z n / Γ ∨ ≈ −→ Λ Z n / Γ . From (8.2.13) it turns out that k ′ • e U k = χ M − k (cid:16) k ′ (cid:17) e U k ; where k ′ is representative of k ′ ∈ Z n / Γ ,hence one has C e U k ∈ H ϕ Z n / Γ ( M − k ) ⊂ M µ ∈ Λ Z n / Γ H µ The C -linear span of n e U k o k ∈ Z n is dense in L (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) , so there is the followingHilbert direct sum L (cid:16) C (cid:16) T n e Θ (cid:17) , τ (cid:17) = ⊕ µ ∈ Λ Z n / Γ H µ (8.2.17)Comparison of (8.2.15) and (8.2.17) gives ⊕ λ ∈ Λ G ′ H λ = { } it turns out that G ′ istrivial, hence G = Z n / Γ . 373 heorem 8.2.6. In the above situation the triple (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z n / Γ , π (cid:17) is anunital noncommutative finite-fold covering.Proof. Both C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) are unital C ∗ -algebras, so from the Definition 2.1.9one needs check the following conditions:(i) The quadruple (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z n / Γ , π (cid:17) is a noncommutative finite-foldpre-covering. (cf. Definition 2.1.5).(ii) C (cid:16) T n e Θ (cid:17) is a finitely generated C (cid:0) T n Θ (cid:1) -module.(i) One needs check (a), (b) of the Definition 2.1.5.(a) Follows from the Lemma 8.2.5.(b) Suppose e a = ∑ k ∈ Z n c k e U k ∈ C (cid:16) T n e Θ (cid:17) Z n / Γ ; c k ∈ C .If there is k ∈ Z n \ Γ such that c k = k ∈ Z n / Γ is represented by k then k e a = e a , so one has e a / ∈ C (cid:16) T n e Θ (cid:17) Z n / Γ . From this contradiction weconclude that c k = ⇔ k ∈ Γ . It follows that e a ∈ C (cid:0) T n Θ (cid:1) , hence one has C (cid:16) T n e Θ (cid:17) Z n / Γ = C (cid:0) T n Θ (cid:1) .(ii) Let e a ∈ C (cid:16) T n e Θ (cid:17) be any element given by e a = ∑ e x ∈ Z n e c e x e U e x .Let M ∈ M n ( Z ) be such that Γ = M Z n . The group Z n / Γ ∼ = Z n / M Z n , let N = | Z n / Γ | and { e x , . . . , e x N } ⊂ Z n is a set of representatives of Z n / Γ . If b j = | Z n / Γ | ∑ g ∈ Z n / Γ g (cid:16)e a e U ∗ e x j (cid:17) ∈ C ( T n Θ ) then from the definition of C (cid:0) T n Θ (cid:1) the series b j = ∑ y ∈ Z n d jy e U My d jy = e c My − e x j . It turns out that b j e U e x j = ∑ y ∈ Z n e c My + e x j e U My + e x j (8.2.18)Since { e x , . . . , e x N } ⊂ Z n is a set of representatives of Z n / Γ from (8.2.18) it followsthat n ∑ j = b j e U e x j = ∑ e x ∈ Z n e c e x e U e x = e a (8.2.19)The equation 8.2.19 means that C (cid:16) T n e Θ (cid:17) is a left C (cid:0) T n Θ (cid:1) -module generated by thefinite set n e U e x j o . Remark 8.2.7.
Let Θ = J θ where θ ∈ R \ Q and J = (cid:18) N − N (cid:19) .Denote by C (cid:0) T N θ (cid:1) def = C (cid:0) T N Θ (cid:1) . Let n ∈ N and n > e θ = θ / n . If u , ..., u N ∈ C (cid:0) T N θ (cid:1) are generators of C (cid:0) T N θ (cid:1) (cf. Definition F.1.3) and e u , ..., e u N ∈ C (cid:16) T N e θ (cid:17) are generators of C (cid:16) T N e θ (cid:17) then there is an injective *-homomorphism π : C (cid:16) T N θ (cid:17) ֒ → C (cid:16) T N e θ (cid:17) , u j e u nj j =
1, ..., 2 N .From the Theorem 8.2.6 it turns out that the quadruple (cid:16) C (cid:16) T N θ (cid:17) , C (cid:16) T N e θ (cid:17) , Z Nn , π (cid:17) is an unital noncommutative finite-fold covering. Lemma 8.2.8.
If the diagram with injective *-homomorphisms π , π . e C (cid:16) T n e Θ (cid:17) C (cid:16) T n e Θ (cid:17) C (cid:0) T n Θ (cid:1) π π π is commutative then the pair ( π , π ) is compliant (cf. Definition 3.1.1), roof. It is proven in that for the inclusions π , π there are invertible n × n -matrices M , M with integer valued entries such that Θ ∈ M T1 e Θ M + M n ( Z ) , Θ ∈ M T2 e Θ M + M n ( Z ) .If π exists then there are lattices Γ , Γ ⊂ Z n such that ∀ k ∈ Z n e U k ∈ π (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) ⇔ k ∈ Γ , ∀ k ∈ Z n e U k ∈ π (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) ⇔ k ∈ Γ . (8.2.20)Let us check conditions (a)-(d) of the Definition 3.1.1.(a) If there is an injective *-homomorphism π : C (cid:16) T n e Θ (cid:17) ֒ → C (cid:16) T n e Θ (cid:17) then from[68] it follows that there is an invertible n × n -matrices M with integer valuedentries such that Θ ∈ M T e Θ M + M n ( Z ) From the Theorem 8.2.6 it follows that π a noncommutative finite-fold covering(cf. Definition 2.1.17).(b) For any g ∈ G (cid:16) C (cid:16) T n e Θ (cid:17) (cid:12)(cid:12)(cid:12) C (cid:0) T n Θ (cid:1)(cid:17) from the equations (8.2.11) and (8.2.20) itfollows that k ∈ Γ ⇔ e U k ∈ π (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) AND ∃ c ∈ C g e U k = c e U k ⇒⇒ g e U k ∈ π (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) .so one has G (cid:16) C (cid:16) T n e Θ (cid:17)(cid:12)(cid:12)(cid:12) C ( T n Θ ) (cid:17) π (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) = π (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) .(c) From the Theorem 8.2.6 it follows that G (cid:16) C (cid:16) T n e Θ (cid:17)(cid:12)(cid:12)(cid:12) C ( T n Θ ) (cid:17) ∼ = Z n / Γ , G (cid:16) C (cid:16) T n e Θ (cid:17)(cid:12)(cid:12)(cid:12) C ( T n Θ ) (cid:17) ∼ = Z n / Γ where Γ ⊂ Γ the natural homomorphism φ : G (cid:16) C (cid:16) T n e Θ (cid:17)(cid:12)(cid:12)(cid:12) C (cid:0) T n Θ (cid:1)(cid:17) → G (cid:16) C (cid:16) T n e Θ (cid:17)(cid:12)(cid:12)(cid:12) C (cid:0) T n Θ (cid:1)(cid:17) is equivalent to the homo-morphism ϕ : Z n / Γ → Z n / Γ . The homomorphism ϕ is surjective, hence φ is376urjective.(d) If ρ : C (cid:16) T n e Θ (cid:17) ֒ → C (cid:16) T n e Θ (cid:17) is an injective *-homomorphism such that π = ρ ◦ π then both π and ρ can be regarded as *-isomorphisms from C (cid:16) T n e Θ (cid:17) ontotheir images. But from (8.2.4) it follows follows that the image of π coincides withthe image of ρ . It follows that there is g ∈ Aut (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) such that ρ = π ◦ g .From π = ρ ◦ π and π = π ◦ π it follows that ga = a for all a ∈ C (cid:16) T n e Θ (cid:17) , i.e. g ∈ G (cid:16) C (cid:16) T n e Θ (cid:17)(cid:12)(cid:12)(cid:12) C (cid:0) T n Θ (cid:1)(cid:17) . Lemma 8.2.9.
Let π : C (cid:0) T n Θ (cid:1) ֒ → C (cid:16) T n e Θ (cid:17) be an injective *-homomorphism which cor-responds to a noncommutative finite-fold covering (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z n / Γ (cid:17) . Thereis a diagonal covering (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n Θ / k (cid:17) , Z nk (cid:17) and the injective *-homomorphism ee π : C (cid:16) T n e Θ (cid:17) → C (cid:16) T n Θ / m (cid:17) such that following condition hold: • The homomorphism e π : C (cid:16) T n e Θ (cid:17) ֒ → C (cid:16) T n Θ / m (cid:17) corresponds an unial noncommu-tative finite-fold covering having unique map lifting (cid:16) C (cid:16) T n e Θ (cid:17) , C (cid:16) T n Θ / k (cid:17) , Z n / e Γ (cid:17) , • Following digramC (cid:16) T n e Θ (cid:17) C (cid:16) T n Θ / m (cid:17) C (cid:0) T n Θ (cid:1)e ππ ee π is commutative.Proof. From the construction 8.2.1 it turns out that: Γ = M Z n , Θ = M T e Θ M + P e Θ = (cid:16) M T (cid:17) − Θ M − + (cid:16) M T (cid:17) − PM − where P ∈ M n ( Z ) . There is m ∈ N such that N = mM − ∈ M n ( Z ) , or equiva-377ently MN = m . . . 0... . . . ...0 . . . m .If ee Θ = e Θ − (cid:0) M T (cid:1) − PM − m then Θ = mM T ee Θ Mm , e Θ = m ee Θ m − m (cid:16) M T (cid:17) − PM − m == m ee Θ m − N T PN ∈ m ee Θ m + M n ( Z ) .So there are noncommutative finite-fold coverings π ′ : C (cid:0) T n Θ (cid:1) ֒ → C (cid:16) T n ee Θ (cid:17) , π ′′ : C (cid:16) T n e Θ (cid:17) ֒ → C (cid:16) T n ee Θ (cid:17) , which correspond to triples (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n ee Θ (cid:17) , Z n / mM Z n (cid:17) , (cid:16) C (cid:16) T n e Θ (cid:17) , C (cid:16) T n ee Θ (cid:17) , Z nm (cid:17) From N = mM − it turns out Θ = mM T N T (cid:16) N T (cid:17) − ee Θ PN − NMm = m (cid:18)(cid:16) N T (cid:17) − ee Θ N − (cid:19) m , (cid:16) N T (cid:17) − ee Θ N − = Θ m , ee Θ = N T (cid:16) N T (cid:17) − ee Θ PN − N = N T Θ m N From the above equations it follows that there is a noncommutative finite-foldcovering e π : C (cid:16) T n ee Θ (cid:17) ֒ → C (cid:16) T n Θ / m (cid:17) which corresponds to the triple (cid:16) C (cid:16) T n ee Θ (cid:17) , C (cid:16) T n Θ / m (cid:17) , Z n / N Z n (cid:17) .The composition ee π = e π ◦ π : C (cid:0) T n Θ (cid:1) → C (cid:16) T n Θ / m (cid:17) corresponds to the diagonalcovering (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n Θ / m (cid:17) , Z nm (cid:17) . Corollary 8.2.10. If S C ( T n Θ ) is a family of all noncommutative finite-fold coverings givenby the Theorem 8.2.6 then following conditions hold: i) Let { p k } k ∈ N ∈ N is a sequence such that for every s ∈ N there are j , m ∈ N whichsatisfy to the following condition sm = j ∏ k = p j . If m j = ∏ jk = p k then the ordered set S N C ( T n Θ ) = (cid:26)(cid:18) C ( T n Θ ) , C (cid:18) T n Θ / m j (cid:19) , Z nm j (cid:19)(cid:27) j ∈ N is a cofinal subset (cf. Definition A.1.4) of S C ( T n Θ ) .(ii) The family S C ( T n Θ ) is a directed set with respect to the given by (3.1.4) order.Proof. (i) Let (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z n / Γ (cid:17) be a noncommutative finite-fold coveringhaving unique path lifting. From the Lemma 8.2.9 it turns out that there is anoncommutative finite-fold covering C (cid:16) T n e Θ (cid:17) ֒ → C (cid:16) T n Θ / s (cid:17) . Otherwise there is j ∈ N such m j = sm , so there are following noncommutative finite-fold coverings C (cid:16) T n Θ / s (cid:17) ֒ → C (cid:18) T n Θ / m j (cid:19) , C (cid:16) T n e Θ (cid:17) ֒ → C (cid:18) T n Θ / m j (cid:19) . (8.2.21)From (8.2.21) it turns out that (cid:18) C ( T n Θ ) → C (cid:18) T n Θ / m j (cid:19)(cid:19) ≥ (cid:16) C ( T n Θ ) → C (cid:16) T n e Θ (cid:17)(cid:17) ,i.e. (cid:26)(cid:18) C (cid:0) T n Θ (cid:1) , C (cid:18) T n Θ / m j (cid:19) , Z nm j (cid:19)(cid:27) j ∈ N is cofinal.(ii) Follows from (i). Let us consider an unital noncommutative finite-fold covering (cid:16) C ( T n Θ ) , C (cid:16) T n e Θ (cid:17) , Z n / Γ , π (cid:17) Z n / Γ is and commutative group, henceany irreducible representation of Z n / Γ has dimension 1. From the proof of theLemma 8.2.5 it follows that the moduli space of the irreducible representations Z n / Γ coincides with Γ ∨ / Z n and any irreducible representation corresponds to acharacter the given by (8.2.16) character χ x : Z n / Γ → C . It is known the orthogo-nality of characters (cf. [40]), i.e. ∑ g ∈ Z n / Γ χ x ( g ) χ x ( g ) = (cid:26) | Z n / Γ | x = x x = x . (8.2.22)Otherwise one has g e U k = χ M − k ( g ) e U k ; ∀ g ∈ Z n / Γ and taking into account (8.2.22) following condition holds D e U k , e U k E C ( T n Θ ) = ∑ g ∈ Z n / Γ g (cid:16) e U ∗ k e U k (cid:17) == ∑ g ∈ Z n / Γ χ M − k ( g ) χ M − k ( g ) e U ∗ k e U k == ∑ g ∈ Z n / Γ χ M − k ( g ) χ M − k ( g ) | e − π ik · e Θ k e U k − k == ( | Z n / Γ | e − π ik · e Θ k U M − ( k − k ) ∈ C (cid:0) T n Θ (cid:1) k − k ∈ Γ k − k / ∈ Γ . (8.2.23)From the equation (F.1.15) it follows that U M − ( k − k ) ξ l = e − π iM − ( k − k ) · Θ l ξ M − ( k − k )+ l == e − π i ( k − k ) · e Θ Ml ξ M − ( k − k )+ l (8.2.24)Let ρ : C (cid:0) T n Θ (cid:1) → B (cid:0) L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1)(cid:1) be the described in F.1.4 faithful GNS-representation. If e ρ : C (cid:16) T n e Θ (cid:17) → B (cid:16) e H (cid:17) is the induced by the pair (cid:16) ρ , (cid:16) C ( T n Θ ) , C (cid:16) T n e Θ (cid:17) , Z n / Γ , π (cid:17)(cid:17) representation (cf. Definition 2.3.1) then e H is the Hilbert norm completion of thetensor product C (cid:16) T n e Θ (cid:17) ⊗ C ( T n Θ ) L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) . Denote by ( · , · ) e H the scalar product380n e H given by (2.3.1). If ξ l , ξ l ∈ L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) then from (8.2.23), (8.2.24) and(F.1.14) it follows that (cid:16) e U k ⊗ ξ l , e U k ⊗ ξ l (cid:17) e H == ( e − π ik · e Θ k e π i ( k − k ) · e Θ Ml | Z n / Γ | k + Ml = k + Ml k + Ml = k + Ml . (8.2.25)From the Equation (8.2.25) it turns out that e H is a Hilbert direct sum e H = M p ∈ Z n e H p of one-dimensional Hilbert spaces such that e U k ⊗ ξ l ∈ e H k + Ml If k + Ml = k + Ml and e ξ = q | Z n / Γ | e − π ik · e Θ Ml e U k ⊗ ξ l , e ξ = q | Z n / Γ | e − π ik · e Θ Ml e U k ⊗ ξ l then from (8.2.25) it follows that (cid:16) e ξ , e ξ (cid:17) e H = e − π ik · e Θ k e π i ( k − k ) · e Θ Ml e π ik · e Θ Ml e − π ik · e Θ Ml = e π ix ,where x = − k · e Θ k + ( k − k ) · e Θ Ml + k · e Θ Ml − k · e Θ Ml == − k · e Θ k + k · ( Ml − Ml ) = − k · e Θ k + k · e Θ ( k − k ) Since e Θ is skew symmetric one has x = (cid:16) e ξ , e ξ (cid:17) e H = (cid:13)(cid:13)(cid:13) e ξ (cid:13)(cid:13)(cid:13) e H = (cid:13)(cid:13)(cid:13) e ξ (cid:13)(cid:13)(cid:13) e H = e ξ = e ξ . So there is the orthonormalbasis n e ξ p o p ∈ Z n such that e ξ p def = q | Z n / Γ | e − π ik · e Θ Ml e U k ⊗ ξ l where p = Ml + k . (8.2.26)and for all p ∈ Z n the element e ξ p does not depend on the representation p = Ml + k . Direct check shows that e U k e ξ l = e π ik · e Θ l e ξ Mk + l . (8.2.27)Using equations (8.2.26) and (F.1.15) one has the following lemma.381 emma 8.2.11. If e ρ : C (cid:16) T n e Θ (cid:17) → B (cid:16) e H (cid:17) is the induced by the pair (cid:16) ρ , (cid:16) C ( T n Θ ) , C (cid:16) T n e Θ (cid:17) , Z n / Γ , π (cid:17)(cid:17) then e ρ is equivalent to the described F.1.4 faithful GNS representation. Lemma 8.2.12.
Let (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z n / Γ , π (cid:17) be an unital noncommutative finite-fold covering given by the Theorem 8.2.6. Let { e x , . . . , e x N } ⊂ Z n is a set of representativesof Z n / Γ .(i) The finite family n e U e x j o j = N ⊂ C ∞ (cid:16) T n e Θ (cid:17) satisfies to the Lemma 2.7.3.(ii) C (cid:16) T n e Θ (cid:17) ∩ M N ( C ∞ ( T n Θ )) = C ∞ (cid:16) e M (cid:17) . (8.2.28) (iii) The given by the Theorem 8.2.6 unital noncommutative finite-fold covering (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z n / Γ , π (cid:17) is smoothly invariant (cf. Definition 2.7.2).Proof. (i) Let us prove that n e U e x j o satisfies to conditions (a) and (b) of the Lemma2.7.3.(a) Follows from D e U e x j , e U e x k E = e π ix j e Θ x k U M − ( x k − x j ) ∈ C ∞ (cid:0) T n Θ (cid:1) x j − x k ∈ Γ x j = x k (b) For any j =
1, ..., n and g ∈ Z n / Γ one has g e U e x j = e U e x j (cid:16) χ M − e x j ( g ) · C ∞ ( T n Θ ) (cid:17) .(ii) For any e a ∈ C (cid:16) T n e Θ (cid:17) from the Theorem 8.2.6 it follows that e a = N ∑ l = a l e U e x l where a l ∈ C ( T n Θ ) (8.2.29)On the other hand one has D e U e x j , e a e U e x k E = ∑ g ∈ Z n / Γ g (cid:16) e U ∗ x j e a e U e x k (cid:17) = ∑ g ∈ Z n / Γ N ∑ l = g (cid:16) e U e x j a l e U e x l e U e x k (cid:17) == ∑ g ∈ Z n / Γ N ∑ l = χ M − ( e x l + e x k − e x j ) g (cid:16) e U e x j a l e U e x l e U e x k (cid:17) M − (cid:0)e x l + e x k − e x j (cid:1) ∈ Γ ∨ / Z n (cf. (8.2.6)) represented by M − (cid:0)e x l + e x k − e x j (cid:1) ∈ Q n and χ M − ( e x l + e x k − e x j ) is a character which corresponds to χ M − ( e x l + e x k − e x j ) . Since theset { e x , . . . , e x N } ⊂ Z n is a set of representatives of Z n / Γ one has M − (cid:0)e x l + e x k − e x j (cid:1) = ⇔ M − (cid:0) e x l + e x k − e x j (cid:1) = ∑ g ∈ Z n / Γ χ M − ( e x l + e x k − e x j ) ( g ) = ( N M − (cid:0) e x l + e x k − e x j (cid:1) = M − (cid:0) e x l + e x k − e x j (cid:1) = e x l + e x k − e x j = a l = ∑ p ∈ Z n c lk U k where c lk ∈ C and a ′ l = e U e x j a l e U e x l e U e x k = ∑ p ∈ Z n c ′ lk U k then from (F.1.2) it follows that (cid:12)(cid:12) c ′ lk (cid:12)(cid:12) = (cid:12)(cid:12) c lk (cid:12)(cid:12) for all k ∈ Z . Hencesup k ∈ Z n ( + k k k ) s (cid:12)(cid:12)(cid:12) c lk (cid:12)(cid:12)(cid:12) = sup k ∈ Z n ( + k k k ) s (cid:12)(cid:12)(cid:12) c ′ lk (cid:12)(cid:12)(cid:12) and taking into account (C.2.2) one has a l = ∑ p ∈ Z n c lk U k ∈ C ∞ ( T n Θ ) ⇔ a ′ l = e U e x j a l e U e x l e U e x k = ∑ p ∈ Z n c ′ lk U k ∈ C ∞ ( T n Θ ) .So from (8.2.29) it turns out ∀ l =
1, ..., n a ′ l ∈ C ∞ ( T n Θ ) ⇒ e a ∈ C ∞ (cid:16) T n e Θ (cid:17) ,or, equivalently ∀ j , k =
1, ..., n D e U e x j , e a e U e x k E ∈ C ∞ ( T n Θ ) ⇒ e a ∈ C ∞ (cid:16) T n e Θ (cid:17) .Conversely if there are j , k ∈
1, ..., n such that D e U e x j , e a e U e x k E / ∈ C ∞ (cid:0) T n Θ (cid:1) and e x l = e x k − e x j then a l / ∈ C ∞ (cid:0) T n Θ (cid:1) . If a l = ∑ k ∈ Z n c k U k and e a = ∑ e k ∈ Z n e c e k e U e k then from(8.2.29) it follows that (cid:12)(cid:12)e c Mk + e x l (cid:12)(cid:12) = | c k | There is c ∈ R such that c > k k k > c ⇒ k Mk + e x l k > k k k ⇒ ( + k Mk + e x l k ) > + k k k .From a l / ∈ C ∞ (cid:0) T n Θ (cid:1) and (C.2.2) it follows that there is s ∈ N such thatsup k ∈ Z n ( + k k k ) s | c k | = ∞ .383f C > k ∈ Z such that k k k > c and ( + k k k ) s | c k | > C . It follows that ( + k Mk + e x l k ) s > C .So sup e k ∈ Z n (cid:16) + k e k k (cid:17) s (cid:12)(cid:12)e c e k (cid:12)(cid:12) = ∞ and from (C.2.2) it follows that e a / ∈ C ∞ (cid:16) T n e Θ (cid:17) . In result C ∞ (cid:16) T n e Θ (cid:17) = n e a ∈ C ∞ (cid:16) T n e Θ (cid:17)(cid:12)(cid:12)(cid:12) D e U e x j , e a e U e x k E ∈ C ∞ ( T n Θ ) j , k ∈
1, ..., n o .The Equation (8.2.29) follows from (i) of the Lemma 2.7.3.(iii) Follows from (ii) of the Lemma 2.7.3.The Hilbert space of relevant to spectral triple is e H m = e H ⊗ C m where m = [ n /2 ] and e H is described in 8.2.2 and the action of C (cid:16) T n e Θ (cid:17) on e H is given by (F.1.15). Theaction of C (cid:16) T n e Θ (cid:17) on e H m is diagonal. Let Ω D is the module of differential formsassociated with the spectral triple ( A , H , D ) (cf. Definition E.3.5). We suppose that Γ = M Z n where M ∈ M n ( Z ) is a diagonal matrix M = k . . . 0... . . . ...0 . . . k n k , ..., k n ∈ N Let us consider a map e ∇ : C ∞ (cid:16) T n e Θ (cid:17) → C ∞ (cid:16) T n e Θ (cid:17) ⊗ C ∞ ( T n Θ ) Ω D , e a n ∑ µ = ∂ a ∂ x µ ⊗ u ∗ µ (cid:2) D , u µ (cid:3) = n ∑ µ = ∂ a ∂ x µ ⊗ γ µ where γ µ Clifford (Gamma) matrices in M n ( C ) satisfying the relation (F.1.22)The map satisfies to the following equation ∇ e U e l = ( e l ,..., e l n ) = n ∑ µ = e l µ k µ e U e l ⊗ γ µ . (8.2.30)384f U l =( l ,..., l n ) ∈ C ∞ (cid:0) T n Θ (cid:1) , e l ′ = ( l k , . . . , l n k n ) , e l ′′ = e l + e l ′ then from (F.1.1) it turnsout e U e l U l e − π i e l · e Θ e l ′ U e l ′′ = e l + e l ′ and taking into account (8.2.30) one has ∇ (cid:16) e U e l U l (cid:17) = e − π i e l · e Θ e l ′ n ∑ µ = l µ k µ + e l µ k µ e U e l ′′ ⊗ γ µ == e − π i e l · e Θ e l ′ n ∑ µ = e l µ k µ e U e l ′′ ⊗ γ µ + e − π i e l · e Θ e l ′ n ∑ µ = l µ e U e l ′′ ⊗ γ µ == n ∑ µ = e l µ k µ e U e l U l ⊗ γ µ + n ∑ µ = l µ e U e l U l ⊗ γ µ == n ∑ µ = e l µ k µ (cid:16) e U e l ⊗ γ µ (cid:17) U l + e U e l ⊗ n ∑ µ = l µ U l ⊗ γ µ == (cid:16) ∇ e U e l (cid:17) U l + e U e l ⊗ [ D , U l ] = (cid:16) ∇ e U e l (cid:17) U l + e U e l ⊗ dU l .where dU l is given by (E.3.6). Right part of the above equation is a specific caseof (E.3.6), i.e. elements e U e l satisfy condition (E.3.6). Since C -linear span of n e U e l o (resp. { U l } ) is dense in C ∞ (cid:16) T n e Θ (cid:17) (resp. C ∞ (cid:0) T n Θ (cid:1) ) the map ∇ is a connection. If ( p , . . . , p n ) ∈ Z k × · · · × Z k then from ( p , . . . , p n ) e U e l = e π i p l k · · · · · e π i pnlnkn e U e l it turns out ∇ (cid:16) ( p , . . . , p n ) e U e l (cid:17) = e π i p l k · ... · e π i pnlnkn n ∑ µ = e l µ k µ e U e l ⊗ γ µ = ( p , . . . , p n ) (cid:16) ∇ e U e l (cid:17) ,i.e. ∇ is Z k × · · · × Z k -equivariant, i.e it satisfies to (2.7.6). If e ξ l = Ψ e Θ (cid:16) e U e l (cid:17) = e U l ⊗ C ( T n Θ ) then from h D , 1 C ( T n Θ ) i = e D (cid:16) e ξ e l ⊗ x (cid:17) = ∇ (cid:16) e U e l (cid:17) (cid:16) C ( T n Θ ) ⊗ x (cid:17) + e U e l D (cid:16) C ( T n Θ ) ⊗ x (cid:17) == n ∑ µ = e l µ k µ e ξ e l ⊗ γ µ x µ ; ∀ x = x . . . x m ∈ C m .385et us consider following objects • The spectral triple (cid:0) C ∞ ( T n Θ ) , L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) ⊗ C m , D (cid:1) given by (F.1.23), • An unital noncommutative finite-fold covering (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z k × ... × Z k n (cid:17) given by the Theorem 8.2.6.Summarize above constructions one has the following theorem. Theorem 8.2.13.
The (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z k × ... × Z k n (cid:17) -lift of the spectral triple (cid:0) C ∞ ( T n Θ ) , L ( C ( T n Θ ) , τ ) ⊗ C m , D (cid:1) is the spectral triple (cid:16) C ∞ (cid:16) T n e Θ (cid:17) , L (cid:16) C (cid:16) T n e Θ (cid:17)(cid:17) , e D (cid:17) . Lemma 8.3.1.
If C (cid:0) T n Θ (cid:1) is a noncommutative torus then the family S C ( T n Θ ) = (cid:8)(cid:0) C ( T n Θ ) , C (cid:0) T n Θ λ (cid:1) , Z n / Γ λ , π λ (cid:1)(cid:9) λ ∈ Λ (8.3.1) of all given by the Theorem 8.2.6 noncommutative finite-fold coverings is algebraical finitecovering category, i.e. S C ( T n Θ ) ∈ FinAlg (cf. Definition 3.1.4).Proof.
Let us check (a) and (b) of the Definition 3.1.4.(a) Follows from the Lemma 8.2.8.(b) The set of lattices on Z n is partially ordered and from (cid:16) C ( T n Θ ) , C (cid:16) T n Θ µ (cid:17) , Z n / Γ µ , π µ (cid:17) ≥ (cid:0) C ( T n Θ ) , C (cid:0) T n Θ ν (cid:1) , Z n / Γ ν , π ν (cid:1) ⇒ Γ µ j Γ ν it turns out that Λ has a partial order given by the Equation (3.1.4). The minimalelement λ min ∈ Λ corresponds to the C ∗ -algebra C (cid:0) T n Θ (cid:1) . From (ii) of the Corollary8.2.10 it follows that Λ is directed. Any given by the Theorem given by the Theorem 8.2.6 noncommutativefinite-fold covering uniquely defines a topological covering e T n → T n of the or-dinary tori. The Corollary 8.2.10 gives an infinite linearly ordered sequence ofcoverings T n ← T n ... ← T nj ← ... (8.3.2)386learly R n is the topological inverse limit of the sequence (8.3.2). The choice ofbase the point e x ∈ R n naturally gives the base point x j ∈ T nj for every j ∈ N since the sequence cofinal one has a base point x λ ∈ T n λ where T n λ is the toruswhich corresponds to C (cid:0) T n Θ (cid:1) ֒ → C (cid:16) T n Θ λ (cid:17) ∈ S C ( T n Θ ) . If µ ≥ ν then there is theunique pointed map (cid:16) T n µ , x µ (cid:17) → ( T n ν , x ν ) which uniquely defines the injective *-homomorphism π νµ : C (cid:16) T n Θ ν (cid:17) ֒ → C (cid:16) T n Θ µ (cid:17) . In result one has a pointed algebraicalfinite covering category S pt C ( T n Θ ) def = (cid:8) π λ : C ( T n Θ ) ֒ → C (cid:0) T n Θ λ (cid:1)(cid:9) λ ∈ Λ , n π νµ o µ , ν ∈ Λ µ > ν ! (8.3.3)(cf. Definition 3.1.6). Similarly to the Remark 8.2.7 denote by Let Θ def = J θ where θ ∈ R \ Q and J def = (cid:18) N − N (cid:19) .Denote by N def = n and C (cid:0) T N θ (cid:1) def = C (cid:0) T N Θ (cid:1) . Consider the given by the Corollary8.2.10 cofinal subcategory of S pt C ( T n Θ ) , i.e. S N C ( T n Θ ) = (cid:26)(cid:18) C (cid:16) T N θ (cid:17) , C (cid:18) T N θ / m j (cid:19) , Z nm j , π j (cid:19)(cid:27) j ∈ N , (cid:26) π jk : C (cid:18) T N θ / m j (cid:19) ֒ → C (cid:16) T N θ / m k (cid:17)(cid:27) j , k ∈ N j < k ⊂ S pt C ( T n Θ ) . (8.3.4) According to F.2.15 the algebra C b (cid:0) R N θ (cid:1) contains all plane waves. Remark 8.3.3. If { c k ∈ C } k ∈ N is such that ∑ ∞ k = | c k | < ∞ then from k exp ( ik · ) k op = k ∑ ∞ k = c k exp ( ik · ) k op < ∑ ∞ k = | c k | < ∞ , i.e. ∑ ∞ k = c k exp ( ik · ) ∈ C b (cid:0) R N θ (cid:1) . The equation (F.2.17) is similar to the equation (F.1.1) which defines C (cid:0) T N θ (cid:1) .From this fact and from the Remark 8.3.3 it follows that there is an injective *-homomorphism C ∞ (cid:0) T N θ (cid:1) ֒ → C b (cid:0) R N θ (cid:1) ; U k exp ( π ik · ) . An algebra C ∞ (cid:0) T N θ (cid:1) is dense in C (cid:0) T N θ (cid:1) so there is an injective *-homomorphism C (cid:0) T N θ (cid:1) ֒ → C b (cid:0) R N θ (cid:1) .387he faithful representation C b (cid:0) R N θ (cid:1) → B (cid:0) L (cid:0) R N (cid:1)(cid:1) gives a representation π : C (cid:0) T N θ (cid:1) → B (cid:0) L (cid:0) R N (cid:1)(cid:1) π : C (cid:16) T N θ (cid:17) → B (cid:16) L (cid:16) R N (cid:17)(cid:17) , U k exp ( π ik · ) (8.3.5)where U k ∈ C (cid:0) T n Θ (cid:1) is given by the Definition F.1.3. Denote by n U θ / m j k ∈ U (cid:16) C (cid:16) T N θ / m j (cid:17)(cid:17)o k ∈ Z N the basis of C (cid:16) T N θ / m j (cid:17) . Similarlyto (8.3.5) there is the representation π j : C (cid:16) T N θ / m j (cid:17) → B (cid:0) L (cid:0) R N (cid:1)(cid:1) given by π j (cid:16) U θ / m j k (cid:17) = exp (cid:18) π i km j · (cid:19) .If π λ : C (cid:0) T n Θ (cid:1) ֒ → C (cid:16) T n Θ λ (cid:17) ∈ S pt (cf. (8.3.3)) then since the given by (8.3.4) iscofinal in S pt C ( T n Θ ) there is j in N such that S pt C ( T n Θ ) that the contains the morphism C (cid:16) T n Θ λ (cid:17) ֒ → C (cid:16) T N θ / m j (cid:17) . If \ C (cid:0) T N θ (cid:1) def = C ∗ -lim −→ λ ∈ Λ C (cid:16) T n Θ λ (cid:17) then from the followingdiagram C (cid:16) T n Θ λ (cid:17) C (cid:16) T N θ / m j (cid:17) B (cid:0) L (cid:0) R N (cid:1)(cid:1) π λ π j yields a faithful representation b π : \ C (cid:0) T n Θ (cid:1) → B (cid:0) L (cid:0) R N (cid:1)(cid:1) . There is the action of Z N × R N → R N given by ( k , x ) k + x .The action naturally induces the action of Z N on both L (cid:0) R N (cid:1) and B (cid:0) L (cid:0) R N (cid:1)(cid:1) .Otherwise the action of Z N on B (cid:0) L (cid:0) R N (cid:1)(cid:1) induces the action of Z N on \ C (cid:0) T N θ (cid:1) .There is the following commutative diagram Z N G (cid:16) \ C (cid:0) T N θ (cid:1) | C (cid:0) T N θ (cid:1)(cid:17) G j def = G (cid:16) C (cid:16) T N θ / m j (cid:17) | C (cid:0) T N θ (cid:1)(cid:17) ≈ Z Nm j Z n G (cid:16) \ C (cid:0) T n Θ (cid:1) | C (cid:0) T n Θ (cid:1)(cid:17) G λ def = G (cid:16) C (cid:16) T n Θ λ (cid:17) | C (cid:0) T n Θ (cid:1)(cid:17) ≈ Z n / Γ λ If b G def = lim ←− λ ∈ Λ G λ then from the above diagram it follows that there is the naturalhomomorphism Z N ֒ → b G , and Z N is a normal subgroup. Let J ⊂ b G be a set ofrepresentatives of b G / Z N , and suppose that 0 b G ∈ J . Any g ∈ b G can be uniquelyrepresented as g = g J + g Z where g ∈ J , g Z ∈ Z N . For any g , g ∈ b G denote by Φ J ( g , g ) ∈ J , Φ Z ( g , g ) ∈ Z N , such that g + g = Φ J ( g , g ) + Φ Z ( g , g ) .Let us define an action of b G on L g ∈ J L (cid:0) R N (cid:1) given by g
0, ..., ξ |{z} g th2 − place , ..., 0, ... =
0, ..., Φ Z ( g , g ) ξ | {z } Φ J ( g + g ) th − place , ..., 0, ... . (8.3.6)Let X ⊂ L g ∈ J L (cid:0) R N (cid:1) be given by X = η ∈ M g ∈ J L (cid:16) R N (cid:17) | η =
0, ..., ξ |{z} th b G − place , ..., 0, ... .Taking into account that X ≈ L (cid:0) R N (cid:1) , we will write L (cid:0) R N (cid:1) ⊂ L g ∈ J L (cid:0) R N (cid:1) instead of X ⊂ L g ∈ J L (cid:0) R N (cid:1) . This inclusion and the action of b G on L g ∈ J L (cid:0) R N (cid:1) enable us write L g ∈ J gL (cid:0) R N (cid:1) instead of L g ∈ J L (cid:0) R N (cid:1) . If b π ⊕ : \ C (cid:0) T N θ (cid:1) → B (cid:16) L g ∈ J gL (cid:0) R N (cid:1)(cid:17) is given by b π ⊕ ( a ) ( g ξ ) = g (cid:16) b π (cid:16) g − a (cid:17) ξ (cid:17) ; ∀ a ∈ \ C (cid:0) T N θ (cid:1) , ∀ g ∈ J , ∀ ξ ∈ L (cid:16) R N (cid:17) . (8.3.7)389 .3.2 Inverse noncommutative limit Below we use the ’ × ’ symbol for the Moyal plane product using the describedin F.2.16 scaling construction. Lemma 8.3.6.
Let a , b ∈ S (cid:0) R N (cid:1) . For any ∆ ∈ R N let a ∆ ∈ S (cid:0) R N (cid:1) be such thata ∆ ( x ) def = a ( x + ∆ ) . (8.3.8) For any m , n ∈ N there is a constant A a , bm , n such that | a ∆ ⋄ b ( x ) | < A a , bm , n ( + k ∆ k ) m ( + k x k ) n ∀ x ∈ R N (8.3.9) where ⋄ means the given by (F.2.21) twisted convolution.Proof. From the definition of Schwartz functions it follows that for any f ∈ S (cid:0) R N (cid:1) and any l ∈ N there is C fl > | f ( u ) | < C fl ( + k u k ) l . (8.3.10)Denote by Ψ ( x ) def = |F ( a ∆ × b ) ( x ) | From (F.2.3) it follows that Ψ ( x , ∆ ) = a ∆ ⋄ b ( x ) = Z R N a ∆ ( x − t ) b ( t ) e ix · Jt dt == Z R N a ( x + ∆ − t ) b ( t ) e ix · Jt dt .Consider two special cases:(i) k x k ≤ k ∆ k ,(ii) k x k > k ∆ k . 390i) From (8.3.10) it turns out | Ψ ( x , ∆ ) | ≤ Z (cid:12)(cid:12)(cid:12) a ( x + ∆ − t ) b ( t ) e ix · Jt (cid:12)(cid:12)(cid:12) dt ≤≤ Z R N C aM ( + k t − ∆ − x k ) M C b M ( + k t k ) M dt == Z R N C aM ( + k t − ∆ − x k ) M ( + k t k ) M C b M ( + k t k ) M dt ≤≤ sup x ∈ R N , k x k≤ k ∆ k , s ∈ R N C aM C b M ( + k s − ∆ − x k ) M ( + k s k ) M ×× Z R N ( + k t k ) M dt . (8.3.11)If x , y ∈ R N then from the triangle inequality it follows that k x + y k > k y k − k x k ,hence ( + k x k ) M ( + k x + y k ) M ≥ ( + k x k ) M ( + max ( k y k − k x k )) M .If k x k ≤ k y k then k y k − k x k ≥ k y k and ( + k x k ) M ( + k x + y k ) M > (cid:18) k y k (cid:19) M . (8.3.12)Clearly if k x k > k y k then condition (8.3.12) also holds, hence(8.3.12) is always true.It turns out from k− x − ∆ k > k ∆ k and (8.3.12) thatinf x ∈ R N , k x k≤ k ∆ k , s ∈ R N ( + k s − ∆ − x k ) M ( + k s k ) M > (cid:13)(cid:13)(cid:13)(cid:13) ∆ (cid:13)(cid:13)(cid:13)(cid:13) M ,hence from (8.3.11) it turns out | Ψ ( x , ∆ ) | ≤ M C aM C b M k ∆ k M × Z R N ( + k t k ) M dt = C ′ k ∆ k M where C ′ = M C aM C b M Z R N ( + k t k ) M . (8.3.13)Otherwise from ( + k s − ∆ − x k ) M ( + k s k ) M > Ψ ( x , ∆ ) ≤ C ′ and taking into account (8.3.13) one has | Ψ ( x , ∆ ) | ≤ C ′ ( + k ∆ k ) M If M = M ′ then from k x k ≤ ∆ it follows that | Ψ ( x , ∆ ) | ≤ C ′ M ′ ( + k ∆ k ) M ′ ( + k x k ) M ′ . (8.3.14)If M is such that M > n and M > m then from (8.3.14) it follows that Ψ ( x , ∆ ) ≤ C ′ M ′ ( + k ∆ k ) m ( + k x k ) n . (8.3.15)(ii) If ( · , · ) L ( R N ) is the given by (C.2.6) scalar product then from (C.2.7) it turnsout | Ψ ( x , ∆ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) Z a ( t − ∆ − x ) b ( t ) e ix · Jt dt (cid:12)(cid:12)(cid:12)(cid:12) == (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) a ( • − ∆ − x ) , b ( • ) e ix · J • (cid:17) L ( R N ) (cid:12)(cid:12)(cid:12)(cid:12) == (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) F ( a ( • − ∆ − x )) , F (cid:16) b ( • ) e ix · J • (cid:17)(cid:17) L ( R N ) (cid:12)(cid:12)(cid:12)(cid:12) == (cid:12)(cid:12)(cid:12)(cid:12) Z R N F ( a ) ( • − ∆ − x ) ( u ) F (cid:16) b ( • ) e ix · J • (cid:17) ( u ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤≤ Z R N (cid:12)(cid:12)(cid:12) e − i ( − ∆ − x ) · u F ( a ) ( u ) F ( b ) ( u + Jx ) (cid:12)(cid:12)(cid:12) du ≤≤ Z R N C F ( a ) M ( + k u k ) M C F ( b ) M ( + k u − Jx k ) M du ≤≤ sup x ∈ R N , k x k > k ∆ k , s ∈ R N C F ( a ) M ( + k s k ) M C F ( b ) M ( + k s − Jx k ) M ×× ( + k u − Jx k ) M ( + k u k ) M Z R N ( + k u k ) M du .From (8.3.12) it follows that ( + k u k ) M ( + k u − Jx k ) M > k Jx k M ,inf x ∈ R N , k x k > k ∆ k , s ∈ R N ( + k s k ) M ( + k s − Jx k ) M > (cid:13)(cid:13)(cid:13)(cid:13) J ∆ (cid:13)(cid:13)(cid:13)(cid:13) M ,392nd taking into account ( + k u k ) M ( + k u − Jx k ) M ≥ ( + k s k ) M ( + k s − Jx k ) M > | Ψ ( x , ∆ ) | ≤ C F ( a ) M C F ( b ) M (cid:13)(cid:13) ∆ (cid:13)(cid:13) M k Jx k M Z R N ( + k u k ) M du , | Ψ ( x , ∆ ) | ≤ C F ( a ) M C F ( b ) M Z R N ( + k u k ) M du If M > N + m then the integral I = R R N ( + k u k ) M du is convergent and from | Ψ ( x , ∆ ) | ≤ C F ( a ) M C F ( b ) M I (cid:13)(cid:13)(cid:0) + ∆ (cid:13)(cid:13)(cid:1) M ( + k Jx k ) M , (8.3.16)From (F.1.20) and θ = k Jz k = k z k ; ∀ z ∈ R N and taking into account(8.3.16) there is C > | Ψ ( x , ∆ ) | ≤ C k ( + ∆ k ) M ( + k x k ) M , (8.3.17)If M ≥ m and M ≥ n then from (8.3.17) it turns out | Ψ ( x , ∆ ) | ≤ C k ( + ∆ k ) m ( + k x k ) n , (8.3.18)If A a , bm , n ∈ R is such that A a , bm , n ≥ C ′ M ′ and A a , bm , n ≥ C then from (8.3.15) and(8.3.18) it follows that A a , bm , n satisfies to (8.3.9). Corollary 8.3.7.
Let a , b ∈ S (cid:0) R N (cid:1) . For any ∆ ∈ R N let a ∆ ∈ S (cid:0) R N (cid:1) be such that ∆ ( x ) def = a ( x + ∆ ) . For any m , n ∈ N there is a constant C a , bm , n such that | ( a ∆ ⋄ b ) ( x ) | < C a , bm , n ( + k ∆ k ) m ( + k x k ) n , | ( b ⋄ a ∆ ) ( x ) | < C a , bm , n ( + k ∆ k ) m ( + k x k ) n , | ( a ∆ × b ) ( x ) | < C a , bm , n ( + k ∆ k ) m ( + k x k ) n , | ( b × a ∆ ) ( x ) | < C a , bm , n ( + k ∆ k ) m ( + k x k ) n , |F ( a ∆ ⋄ b ) ( x ) | < C a , bm , n ( + k ∆ k ) m ( + k x k ) n , |F ( b ⋄ a ∆ ) ( x ) | < C a , bm , n ( + k ∆ k ) m ( + k x k ) n , |F ( a ∆ × b ) ( x ) | < C a , bm , n ( + k ∆ k ) m ( + k x k ) n , |F ( b × a ∆ ) ( x ) | < C a , bm , n ( + k ∆ k ) m ( + k x k ) n . (8.3.19) where F means the Fourier transformation.Proof. From the Lemma 8.3.19 it follows that is a constant A a , bm , n such that | ( a ∆ ⋄ b ) ( x ) | < A a , bm , n ( + k ∆ k ) m ( + k x k ) n .Clearly | b ⋄ a ∆ ( x ) | = (cid:12)(cid:12) ( b ⋄ a ∆ ) ∗ ( x ) (cid:12)(cid:12) = and taking into account (F.2.25) one has | b ⋄ a ∆ ( x ) | = (cid:12)(cid:12) ( b ⋄ a ∆ ) ∗ ( x ) (cid:12)(cid:12) = | ( a ∗ ∆ ⋄ b ∗ ) ( x ) | < A b ∗ , a ∗ m , n ( + k ∆ k ) m ( + k x k ) n (8.3.20)From (F.2.23) it follows that a ∆ × b = a ∆ ⋄ e Fb ,hence from the Lemma 8.3.19 it follows that | a ∆ × b ( x ) | < A a , e Fbm , n ( + k ∆ k ) m ( + k x k ) n | b × a ∆ ( x ) | < A ( e Fb ) ∗ , a ∗ m , n ( + k ∆ k ) m ( + k x k ) n Applying (F.2.22) one has |F ( a ∆ ⋄ b ) ( x ) | < A F a , F e Fbm , n ( + k ∆ k ) m ( + k x k ) n , |F ( b ⋄ a ∆ ) ( x ) | < A F ( e Fb ) ∗ , F a ∗ m , n ( + k ∆ k ) m ( + k x k ) n , |F ( a ∆ × b ) ( x ) | < A F a , F bm , n ( + k ∆ k ) m ( + k x k ) n , |F ( b × a ∆ ) ( x ) | < .If C a , bm , n ∈ R is such that C a , bm , n > A a , bm , n , C a , bm , n > A b ∗ , a ∗ m , n , C a , bm , n > A a , e Fbm , n , C a , bm , n > A ( e Fb ) ∗ , a ∗ m , n , C a , bm , n > A F a , F bm , n , C a , bm , n > A F b ∗ , F a ∗ m , n , C a , bm , n > A F a , F e Fbm , n , C a , bm , n > A ( F e Fb ) ∗ , F a ∗ m , n then from the above construction it follows that C a , bm , n satisfies to the equations(8.3.19). Corollary 8.3.8.
Let a , b ∈ S (cid:0) R N (cid:1) . For any ∆ ∈ R N let a ∆ ∈ S (cid:0) R N (cid:1) be such thata ∆ ( x ) = a ( x + ∆ ) . For any m ∈ N there is a constant C a , bm such that for any x , ∆ ∈ R N following conditions hold k a ∆ × b k < C a , bm ( + k ∆ k ) m (8.3.21) where k·k is given by (F.2.7) .Proof. Follows from the equation 8.3.19.
Lemma 8.3.9.
If a ∈ S (cid:0) R N (cid:1) then for all j ∈ N the following conditions hold:(i) The series ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ga = a j (8.3.22) is convergent with respect to the weak*-topology (cf. Definition C.1.2) of the dualspace S ′ (cid:0) R N (cid:1) of S (cid:0) R N (cid:1) . Moreover a j is represented by infinitely differentiablemap R N → C , i.e. a j ∈ C ∞ (cid:0) R N (cid:1) . ii) One has a j = ∑ k ∈ Z N c k exp (cid:18) π i km j · (cid:19) (8.3.23) where { c k ∈ C } k ∈ Z N are rapidly decreasing coefficients given byc k = m Nj Z R N a ( x ) exp (cid:18) π i km j · x (cid:19) dx = m Nj F a (cid:18) km j (cid:19) . (8.3.24) Proof.
Let I def = (cid:2) m j (cid:3) N ⊂ R N be a cube. If R def = √ Nm j then k x k < R for all x ∈ I , hence from (8.1.2) it follows that | ( ga ) ( x ) | = | a ( x + g ) | < C m , Rf (cid:0) + k g k m (cid:1) ∀ x ∈ I .For all m > N the series ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) C m , Rf ( + k g k m ) is convergent. If C def = ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) C m , Rf (cid:0) + k g k m (cid:1) then for any subset G ⊂ ker (cid:16) Z N → Z Nm j (cid:17) and x ∈ I one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ g ∈ G ( ga ) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C . (8.3.25)Moreover the series ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ( ga ) | I is uniformly convergent, so the sum of the above series is continuous. If D β = ∂∂ x β ... ∂∂ x β nn then from the equation (C.2.4) it turns out that D β a ∈ S (cid:0) R N (cid:1) , so thesum of the series ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) (cid:16) gD β a (cid:17)(cid:12)(cid:12)(cid:12) I is also continuous. If ε > a ∈ S (cid:0) R N (cid:1) then there is a compact set V ⊂ R N such that | a ( x ) | < ε C for all R N \ V . There is a finite subset G ⊂ ker (cid:16) Z N → Z Nm j (cid:17) V ⊂ S g ∈ G gI . Denote by C a def = sup x ∈ R N | a ( x ) | . Otherwise there is afinite set G I ⊂ ker (cid:16) Z N → Z Nm j (cid:17) such that G I ⊂ G ⊂ ker (cid:16) Z N → Z Nm j (cid:17) ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ( ga ) | I − ∑ g ∈ G ( ga ) | I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε C a For any g ′ ∈ G one has gG I ⊂ G ⊂ ker (cid:16) Z N → Z Nm j (cid:17) ⇒⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ( ga ) | g ′ I − ∑ g ∈ g ′ G I ( ga ) | g ′ I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε C a If G def = S g ′ ∈ G g ′ G I then from above construction it follows that G ⊂ G ⊂ ker (cid:16) Z N → Z Nm j (cid:17) ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ga , a + − * ∑ g ∈ G ga , a +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε ,i.e. the series 8.3.22 is convergent with respect to the weak*-topology. The sum ofseries is periodic and since the sums ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ga | I and ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) (cid:16) gD β a (cid:17)(cid:12)(cid:12)(cid:12) I are continuous a j is represented by infinitely differentiable map R N → C .(ii) Follows from c k = m Nj Z I exp (cid:18) π i km j x (cid:19) ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ( ga ) ( x ) dx == m Nj ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) Z gI exp (cid:18) π i km j x (cid:19) a ( x ) dx == m Nj Z R N exp (cid:18) π i km j x (cid:19) a ( x ) dx = m Nj F a (cid:18) km j (cid:19) .397 emma 8.3.10. If a ∈ S (cid:0) R N θ (cid:1) is an element which correspond to positive elementof C b (cid:0) R N θ (cid:1) ⊂ B (cid:0) L (cid:0) R N θ (cid:1)(cid:1) (cf. Definition F.2.6) then for all j ∈ N there is a j ∈ C ∞ (cid:16) T N θ /2 m j (cid:17) such that the series ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ga = a j is strongly convergent in B (cid:0) L (cid:0) R N θ (cid:1)(cid:1) . Moreover the sum of the above series in sense ofthe strong topology coincides with the sum with respect to the weak*-topology of the dualspace S ′ (cid:0) R N (cid:1) (cf. Definition C.1.2).Proof. From the Lemma 8.3.9 it follows that there the series ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ga = a j ∈ S ′ (cid:16) R N (cid:17) (8.3.26)is convergent with respect to the weak*-topology (cf. Definition C.1.2). Since a j isa smooth m j - periodic function it can be regarded as the element of C ∞ (cid:16) T N θ /2 m j (cid:17) .On the other hand from 8.3.4 it follows that a j ∈ C b (cid:0) R N θ (cid:1) ⊂ B (cid:0) L (cid:0) R N θ (cid:1)(cid:1) . From(F.2.4) for any x ∈ S (cid:0) R N θ (cid:1) and g ∈ ker (cid:16) Z N → Z Nm j (cid:17) one has ( x , ( ga ) x ) L ( R N θ ) = (( ga ) x , x ) L ( R N θ ) = (( ga ) ⋆ θ x , x ) L ( R N θ ) == (cid:10) (( ga ) ⋆ θ x ) ∗ , x (cid:11) = h x ∗ ⋆ θ ( ga ) , x i = h ga , x ⋆ θ x ∗ i where ( · , · ) L ( R N θ ) is the scalar product of the Hilbert space and h· , ·i means thenatural pairing S ′ (cid:0) R N θ (cid:1) × S (cid:0) R N θ (cid:1) → C . Since ga is a positive operator of B (cid:0) L (cid:0) R N θ (cid:1)(cid:1) all summands of the series ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ( x , gax ) L ( R N θ ) = ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) h ga , x ⋆ θ x ∗ i are positive. Otherwise the series (8.3.26) is convergent with respect to the weak*-topology, so one has ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) h ga , x ⋆ θ x ∗ i = (cid:10) a j , x ⋆ θ x ∗ (cid:11) (8.3.27)From the positivity of summands it follows that ∑ g ∈ G ( x , gax ) L ( R N θ ) ≤ (cid:0) x , a j x (cid:1) L ( R N θ ) G ⊂ ker (cid:16) Z N → Z Nm j (cid:17) . Otherwise S (cid:0) R N θ (cid:1) is dense subset of L (cid:0) R N θ (cid:1) with respect to the Hilbert norm of L (cid:0) R N θ (cid:1) , so one has ∑ g ∈ G ′ ( ξ , ga ξ ) L ( R N θ ) ≤ (cid:0) ξ , a j ξ (cid:1) L ( R N θ ) for all ξ ∈ L (cid:0) R N θ (cid:1) , hence ∑ g ∈ G ′ ga ≤ a j .So from the Lemma D.1.25 it follows that the series ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ga is stronglyconvergent in B (cid:0) L (cid:0) R N θ (cid:1)(cid:1) . Moreover since S (cid:0) R N (cid:1) is dense in L (cid:0) R N (cid:1) andtaking into account the Equation (8.3.27) one has ∀ ξ ∈ L (cid:16) R N (cid:17) ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ( ξ , ga ξ ) L ( R N θ ) = (cid:0) ξ , a j ξ (cid:1) L ( R N θ ) ,it turns out that ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) ga = a j where the sum of the series means the strong convergence in B (cid:0) L (cid:0) R N θ (cid:1)(cid:1) . Lemma 8.3.11.
If m > n then there is C m , n > such that for any k ∈ N one hass k = k n ∑ x ∈ Z n (cid:0) + (cid:13)(cid:13) xk (cid:13)(cid:13)(cid:1) m < C m , n (8.3.28) Proof. If f : R n → R and φ k : R n → Z n are given by f ( x ) = ( + k x k ) m , φ k ( x , ..., x n ) = ( y , ..., y n ) ∈ Z n , y j ≤ kx j < y j + j =
1, ..., n then one has s k = Z R n f (cid:16) φ k (cid:16) xk (cid:17)(cid:17) dx .Otherwise from f (cid:16) φ k (cid:16) xk (cid:17)(cid:17) ≤ m f ( x ) ∀ x ∈ R n it follows that s k < Z R n m ( + k x k ) m dx .399rom m > n it follows that the integral R R n ( + k x k ) m dx is convergent so one canassume C m , n = Z R n m ( + k x k ) m dx Corollary 8.3.12.
Let m > N and e a ∈ S (cid:0) R N θ (cid:1) is such that kF e a ( x ) k < ( + k x k ) m For any j ∈ N following condition holdsa j = ∑ g ∈ ker (cid:16) Z N → Z Nmj (cid:17) e a ⇒ (cid:13)(cid:13) a j (cid:13)(cid:13) < C m , n where C m , n is given by the Lemma 8.3.11.Proof. From (8.3.23) and (8.3.24) it follows that (cid:13)(cid:13) a j (cid:13)(cid:13) < m Nj ∑ k ∈ Z N (cid:13)(cid:13)(cid:13)(cid:13) F e a (cid:18) km j (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) and taking into account the Lemma 8.3.11 one has (cid:13)(cid:13) a j (cid:13)(cid:13) < C m , n Corollary 8.3.13.
If m > n thens = ∑ x ∈ Z n \ k x k m < ∞ (8.3.29) Proof.
Follows from the Lemma 8.3.11 and the implication k x k > ⇒ k x k m < m ( + k x k ) m H ⊕ def = L g ∈ J gL (cid:0) R N (cid:1) and let b π ⊕ : \ C (cid:0) T N θ (cid:1) ֒ → B ( H ⊕ ) be givenby (8.3.7) representation. If e a ∈ S (cid:0) R N θ (cid:1) then from the Lemma 3.1.21 it turns out e a ∈ π ⊕ (cid:16) \ C (cid:0) T N θ (cid:1)(cid:17) ′′ . Since b π ⊕ is a faithful representation of \ C (cid:0) T N θ (cid:1) , one has aninjective homomorphism S (cid:0) R N θ (cid:1) ֒ → b π ⊕ (cid:16) \ C (cid:0) T N θ (cid:1)(cid:17) ′′ of involutive algebras. Forany e a ∈ S (cid:0) R N θ (cid:1) following condition holds ∑ g ∈ ker ( b G → G j ) g b π ⊕ ( e a ) = ∑ g ′ ∈ J g ′ ∑ g ′′ ∈ ker ( Z N → G j ) g ′′ b π ⊕ ( e a ) = ∑ g ∈ J gP .where P = ∑ g ∈ ker ( Z N → G j ) g b π ( e a ) .If J ⊂ b G is a set of representatives of b G / Z N and g ′ , g ′′ ∈ J are such that g ′ = g ′′ then operators g ′ P , g ′′ P act on mutually orthogonal Hilbert subspaces g ′ L (cid:0) R N (cid:1) , g ′′ L (cid:0) R N (cid:1) of the direct sum L g ∈ J gL (cid:0) R N (cid:1) , and taking into account k P k = k gP k one has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ ker ( b G → G j ) b π ⊕ ( e a ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ J gP (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k P k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ ker ( Z N → G j ) b π ( e a ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (8.3.30) Lemma 8.3.14.
If a in S (cid:0) R N θ (cid:1) is positive then following conditions hold:(i) For any j ∈ N the following seriesa j = ∑ g ∈ ker ( b G → G j ) ) ga , b j = ∑ g ∈ ker ( b G → G j ) ga are strongly convergent and the sums lie in C ∞ (cid:18) T N θ / m j (cid:19) , i.e. a j , b j ∈ C ∞ (cid:18) T N θ / m j (cid:19) ;(ii) For any ε > there is N ∈ N such that for any j ≥ N the following conditionholds (cid:13)(cid:13)(cid:13) a j − b j (cid:13)(cid:13)(cid:13) < ε .401 roof. (i) Follows from the Lemma 8.3.10.(ii) Denote by J j = ker (cid:0) Z N → G j (cid:1) = m j Z N . If a j = ∑ g ∈ J j ga , b j = ∑ g ∈ J j ga then a j − b j = ∑ g ∈ J j ga ∑ g ′ ∈ J j \{ g } g ′′ a . (8.3.31)From (8.3.8) it follows that ga = a g where a g ( x ) = a ( x + g ) for any x ∈ R N and g ∈ Z N . Hence the equation (8.3.31) is equivalent to a j − b j = ∑ g ∈ J j a g ∑ g ′ ∈ J j \{ g } a g ′ = ∑ g ∈ Z N a m j g ∑ g ′ ∈ Z N \{ g } a m j g ′ == ∑ g ′ ∈ J j g ′ a ∑ g ∈ Z N \{ } a m j g .Let m > M = N + + m . From the Corollary 8.3.7 it follows that there is C ∈ R such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ J j g ( aa ∆ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < C k ∆ k M .From the triangle inequality it follows that (cid:13)(cid:13)(cid:13) a j − b j (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ J j g a ∑ g ′ ∈ Z N \{ } a m j g ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤≤ ∑ g ′ ∈ Z N \{ } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ g ∈ J j g (cid:16) a a m j g ′ (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ∑ g ∈ Z N \{ } C (cid:13)(cid:13) m j g ′ (cid:13)(cid:13) M .From M > N it turns out that the series C ′ = ∑ g ′ ∈ Z N \{ } C k g ′ k M ∑ g ∈ Z N \{ } C (cid:13)(cid:13) m j g (cid:13)(cid:13) M = C ′ m Mj .If ε > N ∈ N is such m N > M q C ′ ε then from aboveequations it follows that for any j ≥ N the following condition holds (cid:13)(cid:13)(cid:13) a j − b j (cid:13)(cid:13)(cid:13) < ε . If { f nm } n , n ∈ N are elements, which satisfy to the Lemma F.2.11 then L (cid:0) R (cid:1) if the Hilbert space generated by orthogonal elements { f nm } n , m ∈ N . Denote by L (cid:0) R (cid:1) k def = M n ∈ N C f nn ⊂ L (cid:0) R (cid:1) , L (cid:16) R N (cid:17) k def = L (cid:0) R (cid:1) k O · · · O L (cid:0) R (cid:1) k | {z } N − times ⊂ L (cid:16) R N (cid:17) (8.3.32)where L and N mean the Hilbert direct sum and the Hilbert tensor productrespectively. From the Proposition F.2.13 it follows that any a ∈ C b (cid:0) R θ (cid:1) can berepresented by the matrix { a mn ∈ C } such that a = ∞ ∑ m , n = a mn f mn ,sup ξ ∈ L ( R ) k ξ k = k a ξ k < ∞ .Otherwise any b ∈ B (cid:16) L (cid:0) R (cid:1) k (cid:17) can represented by the matrix { b mn ∈ C } suchthat b = ∞ ∑ m , n = b mn f mn ,sup ξ ∈ L ( R ) k k ξ k = k b ξ k < ∞ .From L (cid:0) R (cid:1) k ⊂ L (cid:0) R (cid:1) it follows that C b (cid:0) R θ (cid:1) ⊂ B (cid:16) L (cid:0) R (cid:1) k (cid:17) . Similarly thereis the basis of L (cid:0) R N (cid:1) which contains all elements f j k ⊗ ... ⊗ f j N k N and a ∈ b (cid:0) R N θ (cid:1) can be represented by a = ∑ j k ... j N k N a j k ... j N k N f j k ⊗ ... ⊗ f j N k N (8.3.33)Otherwise there is the basis of L (cid:0) R N (cid:1) k which contains all elements f j j ⊗ ... ⊗ f j N j N and the given by (8.3.33) operator can be regarded as an element of B (cid:16) L (cid:0) R N (cid:1) k (cid:17) ,i.e. C b (cid:0) R N θ (cid:1) ∈ B (cid:16) L (cid:0) R N (cid:1) k (cid:17) . Let us enumerate the basis of L (cid:0) R N (cid:1) k i.e. (cid:8) ξ j (cid:9) j ∈ N = (cid:8) f j j ⊗ ... ⊗ f j N j N (cid:9) . (8.3.34)Any a ∈ B (cid:16) L (cid:0) R N (cid:1) k (cid:17) can be represented by the following way a = ∞ ∑ j , k = a jk ξ j ih ξ k . (8.3.35) Lemma 8.3.16.
Let us consider a dense inclusion S (cid:0) R θ (cid:1) ⊗ · · · ⊗ S (cid:0) R θ (cid:1)| {z } N − times ⊂ S (cid:16) R N θ (cid:17) of algebraic tensor product which follows from (F.2.14) . If a ∈ S (cid:0) R N θ (cid:1) is a positive suchthat • a = M ∑ j = k = c jk f jk ⊗ · · · ⊗ M N ∑ j = k = c Njk f jk where c ljk ∈ C and f jk are given by the Lemma F.2.11, • For any l =
1, . . . ,
N the sum ∑ M l j = k = c ljk f jk is a rank-one operator in B (cid:16) L (cid:0) R (cid:1) k (cid:17) .then a is π ⊕ -special (cf. Definition 3.1.19).Proof. Clearly a is a rank-one operator in B (cid:16) L (cid:0) R N (cid:1) k (cid:17) . Let j ∈ N , ε > z ∈ C (cid:18) T N θ / m j (cid:19) . Denote by G j def = ker (cid:16) b G → Z Nm j (cid:17) . One needs check conditions (a)and (b) of the Definition 3.1.19. 404a) From the Lemma 8.3.10 if follows that for any j ∈ N there is a ′ j ∈ C (cid:18) T N θ / m j (cid:19) such that ∑ g ∈ G j ga = π ⊕ (cid:0) a j (cid:1) (8.3.36)where the sum of the series implies the strong convergence in B ( H ⊕ ) . It followsthat the series ∑ g ∈ G j π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) = π ⊕ ( z ) ∗ π ⊕ (cid:16) a ′ j (cid:17) π ⊕ ( z ) ∈ C (cid:18) T N θ / m j (cid:19) . (8.3.37)is strongly convergent. If g ∈ G j then e g def = π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) is a positive rank-one operator. Also (cid:0) π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) (cid:1) and f ε (cid:0) π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) (cid:1) are rankone operator, so since z is G j invariant (for all j ≥ j ) one has k = (cid:13)(cid:13)(cid:13)(cid:0) π ⊕ ( z ) ∗ a π ⊕ ( z ) (cid:1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) π ⊕ ( z ) ∗ a π ⊕ ( z ) (cid:13)(cid:13) ⇒⇒ (cid:0) π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) (cid:1) = k π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) , k ε = (cid:13)(cid:13) f ε (cid:0) π ⊕ ( z ) ∗ a π ⊕ ( z ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) π ⊕ ( z ) ∗ a π ⊕ ( z ) (cid:13)(cid:13) ⇒⇒ f ε (cid:0) π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) (cid:1) = k ε π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) . (8.3.38)From the Equations (8.3.37) and (8.3.38) it turns out that both series ∑ g ∈ G j (cid:0) π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) (cid:1) = ∑ g ∈ G j k π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) == k π ⊕ (cid:0) z ∗ a j z (cid:1) ∈ C (cid:18) T N θ / m j (cid:19) , ∑ g ∈ G j f ε (cid:0) π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) (cid:1) = ∑ g ∈ G j k ε π ⊕ ( z ) ∗ ( ga ) π ⊕ ( z ) == k ε π ⊕ (cid:0) z ∗ a j z (cid:1) ∈ C (cid:18) T N θ / m j (cid:19) (8.3.39)are strongly convergent. The Equations (8.3.37) and (8.3.39) are specializations ofthe Equations (3.1.20).(b) Select ε > j ∈ N and for every z ∈ C (cid:18) T N θ / m j (cid:19) . Let a j ∈ C (cid:18) T N θ / m j (cid:19) is givenby (8.3.36) and denote by C def = (cid:13)(cid:13) a j (cid:13)(cid:13) . There is δ > δ + C δ < ε /2.405ince C ∞ (cid:18) T N θ / m j (cid:19) is dense in C (cid:18) T N θ / m j (cid:19) there is y ∈ C ∞ (cid:18) T N θ / m j (cid:19) such that k y − z k < δ . From y ∗ ay ∈ S (cid:0) R N θ (cid:1) and the Lemma 8.3.14 it follows that there is k > j such that ∀ m , n ∈ N n ≥ m ≥ k ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( y ∗ a m y ) − ∑ g ∈ Z Nmn / mm g ( y ∗ a n y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε k y − z k < δ one has where a λ ∈ A λ is such that ∀ m , n ∈ N n ≥ m ≥ k ⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( z ∗ a m z ) − ∑ g ∈ Z Nmn / mm g ( z ∗ a n z ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε . Corollary 8.3.17.
If A b π ⊕ is the disconnected inverse noncommutative limit of S θ with re-spect to b π ⊕ and A is the C ∗ -norm completion of the algebraic direct sum L g ∈ J gC (cid:0) R N θ (cid:1) then A b π ⊕ ⊂ AProof.
From the Lemma 8.3.16 it turns out that A b π ⊕ contains all elements a = f j k ⊗ · · · ⊗ f j N k N ∈ S (cid:0) R θ (cid:1) ⊗ · · · ⊗ S (cid:0) R θ (cid:1)| {z } N − times ⊂ S (cid:16) R N θ (cid:17) ⊂ C (cid:16) R N θ (cid:17) (8.3.40)where f j l k l ( l =
1, . . . , N ) are given by the Lemma F.2.11. However the C -linearspan of given by (8.3.40) elements is dense in C (cid:0) R N θ (cid:1) , hence C (cid:0) R N θ (cid:1) ⊂ A b π ⊕ .From the Equation (3.1.31) it turns out A b π ⊕ ⊂ A Denote byFin (cid:18) L (cid:16) R N (cid:17) k (cid:19) def = ( a ∈ B (cid:18) L (cid:16) R N (cid:17) k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∃ m ∈ N a = m ∑ j , k = a kj ξ j ih ξ k ) .406here the basis (cid:8) ξ j (cid:9) is given by (8.3.34). Any element of B (cid:16) L (cid:0) R N (cid:1) k (cid:17) fin corre-sponds to a finite C -linear combination of elements f j j ⊗ ... ⊗ f j N j N it turns outthat there is the natural inclusionFin (cid:18) L (cid:16) R N (cid:17) k (cid:19) ⊂ S (cid:16) R N θ (cid:17) .The C ∗ -norm completion of Fin (cid:16) L (cid:0) R N (cid:1) k (cid:17) equals to K (cid:16) L (cid:0) R N (cid:1) k (cid:17) and tak-ing into account (F.2.12) one concludes that S (cid:0) R N θ (cid:1) ⊂ K (cid:16) L (cid:0) R N (cid:1) k (cid:17) . Since C (cid:0) R N θ (cid:1) is the C ∗ -norm closure of S (cid:0) R N θ (cid:1) one has C (cid:16) R N θ (cid:17) ∼ = K (cid:18) L (cid:16) R N (cid:17) k (cid:19) (8.3.41)Note then Fin (cid:16) L (cid:0) R N (cid:1) k (cid:17) is a dense subspace of B (cid:16) L (cid:0) R N (cid:1) k (cid:17) with respect tothe strong topology. From (F.2.11) it follows that Z R N f j k ⊗ · · · ⊗ f j N k N = δ j k · ... · δ j N k N ,Hence if a ∈ Fin (cid:16) L (cid:0) R N (cid:1) k (cid:17) corresponds to f a ∈ S (cid:0) R N θ (cid:1) then one has Z R N f a dx = tr ( a ) . (8.3.42)Otherwise if a ∈ C (cid:0) T N θ (cid:1) is such that ∑ g ∈ b G ga = b π ⊕ ( a ) then from (C.2.9) it follows that τ ( a ) = Z R N f a dx = tr ( a ) (8.3.43)where τ is the given by (F.1.9) functional. If a ∈ B (cid:16) L g ∈ J gL (cid:0) R N (cid:1)(cid:17) is a specialelement and a j ∈ C (cid:18) T N θ / m j (cid:19) is such that ∑ g ∈ ker (cid:16) b G → Z Nmj (cid:17) ga = b π ⊕ (cid:0) a j (cid:1) a j corresponds to a family n b π ⊕ (cid:0) a j (cid:1)(cid:12)(cid:12) gL ( R N ) o g ∈ J . For any g ∈ J the sequence n b π ⊕ (cid:0) a j (cid:1)(cid:12)(cid:12) gL ( R N ) o j ∈ N is decreasing, so a corresponds to the family (cid:8) a g (cid:9) g ∈ J where a g equals to the strong limit a g = lim j → ∞ b π ⊕ (cid:0) a j (cid:1)(cid:12)(cid:12) gL ( R N ) . Lemma 8.3.19.
Let φ : C (cid:0) R N θ (cid:1) → B (cid:0) L (cid:0) R N (cid:1)(cid:1) the natural inclusion. Suppose thata ∈ B (cid:16) L g ∈ J gL (cid:0) R N (cid:1)(cid:17) is a special element. If a corresponds to a family (cid:8) a g (cid:9) g ∈ J thenfor any g ∈ J the positive operator a g ∈ g ϕ (cid:0) C (cid:0) R N θ (cid:1)(cid:1) .Proof. Let a ∈ C (cid:0) T N θ (cid:1) is such that ∑ g ∈ b G ga = b π ⊕ ( a ) Let Fin def = L g ∈ J Fin (cid:16) L (cid:0) R N (cid:1) k (cid:17) be the algebraic direct sum, denote by { a α } α ∈ A = n b ∈ Fin (cid:12)(cid:12)(cid:12) b ≤ a o α ∈ A .Let us define the order on A such that α ≥ β ⇔ a α ≥ a β for all α , β ∈ A . Any a α corresponds to the family (cid:8) a g α (cid:9) g ∈ J , and a corresponds to the family (cid:8) a g (cid:9) g ∈ J suchthat there is the strong limit a g = lim α ∈ A a g α . If a g α ∈ C (cid:0) T N θ (cid:1) is such that ∑ g ′ ∈ b G g ′ a g α = b π ⊕ (cid:0) a g α (cid:1) then from a g α ≤ a it follows that a g α < a and τ (cid:0) a g α (cid:1) ≤ τ ( a ) where τ is the givenby (F.1.9) functional. Since (cid:8) a g α (cid:9) α ∈ A is increasing (cid:8) τ (cid:0) a g α (cid:1)(cid:9) α ∈ A is also increasing, itfollows that there is the limit τ a = lim α ∈ A τ (cid:0) a g α (cid:1) < ∞ . Taking into account (8.3.43)one has the following limit τ a = lim α ∈ A tr (cid:0) a g α (cid:1) If (cid:8) ξ j (cid:9) j ∈ N ⊂ L (cid:0) R N (cid:1) k is the given by (8.3.34) topological basis of gL (cid:0) R N (cid:1) k then one has c m α = m ∑ j = (cid:0) ξ j , a g α , ξ j (cid:1) L ( R N ) k ≤ τ a and since (cid:8) a g α (cid:9) is weakly convergentlim α ∈ A m ∑ j = (cid:0) ξ j , a g α , ξ j (cid:1) L ( R N ) k = m ∑ j = (cid:0) ξ j , a g , ξ j (cid:1) gL ( R N ) k m → ∞ m ∑ k = ( ξ k , a g , ξ k ) L ( R N ) k = lim m → ∞ lim α ∈ A c m α = ∞ ∑ j = (cid:0) ξ j , a g , ξ j (cid:1) L ( R N ) k = τ a .Otherwise for all positive trace class operator b ∈ B (cid:16) L (cid:0) R N (cid:1) k (cid:17) one hastr (cid:16) b (cid:17) = ∞ ∑ j = (cid:16) ξ j , b , ξ j (cid:17) L ( R N ) < ∞ .it follows that tr ( a g ) = τ a < ∞ ,and since a g is positive it is trace-class. On the other hand any trace-class operatoris compact it turns out that a g ∈ g K (cid:16) gL (cid:0) R N (cid:1) k (cid:17) , hence from (8.3.41) it followsthat a g ∈ g ϕ (cid:0) C (cid:0) R N θ (cid:1)(cid:1) . Lemma 8.3.20.
If a ∈ B (cid:16) L g ∈ J gL (cid:0) R N (cid:1)(cid:17) is a special element, then it lies in theC ∗ -norm completion of the algebraic direct sum L g ∈ J gC (cid:0) R N θ (cid:1) Proof. If a ∈ C (cid:0) T N θ (cid:1) is such that ∑ g ∈ b G ga = b π ⊕ ( a ) then from the proof of the Lemma8.3.19 it follows that τ ( a ) = ∑ g ∈ b G tr ( a g ) It follows that for any ε > b G such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ ( a ) − ∑ g ∈ b G tr ( a g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε .it follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a − ∑ g ∈ b G a g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε .On the other hand from the Lemma 8.3.19 it turns out that ∑ g ∈ b G a g lies on thealgebraic direct sum L g ∈ J gC (cid:0) R N θ (cid:1) emma 8.3.21. If A b π ⊕ be the disconnected inverse noncommutative limit of S θ thenA b π ⊕ is isomorphic to the C ∗ -algebra A which is the C ∗ -norm completion of the direct sum L g ∈ J gC (cid:0) R N θ (cid:1) Proof.
From the Corollary 8.3.17 it turns out A ⊂ A b π ⊕ . Otherwise from the Lemma8.3.20 it follows that any special element lies in A , so one has A b π ⊕ ⊂ A . Let A b π ⊕ be the disconnected inverse noncommutative limit of S θ withrespect to b π ⊕ of S θ . From the Corollary 8.3.17 it follows that C (cid:16) R N θ (cid:17) ⊂ A b π ⊕ \ B (cid:16) L (cid:16) R N (cid:17)(cid:17) .From the Lemma 8.3.20 it follows that A b π ⊕ \ B (cid:16) L (cid:16) R N (cid:17)(cid:17) ⊂ C (cid:16) R N θ (cid:17) .In result we have A b π ⊕ \ B (cid:16) L (cid:16) R N (cid:17)(cid:17) = C (cid:16) R N θ (cid:17) . (8.3.44)Similarly for any g ∈ J on has A b π ⊕ \ B (cid:16) gL (cid:16) R N (cid:17)(cid:17) = gC (cid:16) R N θ (cid:17) .The algebra C (cid:0) R N θ (cid:1) is irreducible. Clearly C (cid:0) R N θ (cid:1) ⊂ A b π ⊕ is a maximal irre-ducible subalgebra. Theorem 8.3.23.
Following conditions hold:(i) The representation b π ⊕ is good,(ii) lim ←− b π ⊕ ↓ S θ = C (cid:16) R N θ (cid:17) ; G lim ←− b π ⊕ ↓ S θ | C (cid:16) T N θ (cid:17)! = Z N , (iii) The triple (cid:0) C (cid:0) T N θ (cid:1) , C (cid:0) R N θ (cid:1) , Z N (cid:1) is an infinite noncommutative covering of S θ with respect to b π ⊕ . roof. (i) There is the natural inclusion A b π ⊕ ֒ → ∏ g ∈ J B (cid:0) gL (cid:0) R N (cid:1)(cid:1) where ∏ means the Cartesian product of algebras. This inclusion induces the decompo-sition A b π ⊕ ֒ → ∏ g ∈ J (cid:16) A b π ⊕ \ B (cid:16) gL (cid:16) R N (cid:17)(cid:17)(cid:17) .From (8.3.44) it turns out A b π ⊕ T B (cid:0) gL (cid:0) R N (cid:1)(cid:1) = gC (cid:0) R N θ (cid:1) , hence there is theinclusion A b π ⊕ ֒ → ∏ g ∈ J gC (cid:16) R N θ (cid:17) .From the above equation it follows that C (cid:0) R N θ (cid:1) ⊂ A b π ⊕ is a maximal irreduciblesubalgebra. One needs check conditions (a)-(c) of the Definition 3.1.33(a) Clearly the map \ C (cid:0) T N θ (cid:1) → M (cid:0) C (cid:0) R N θ (cid:1)(cid:1) is injective,(b) From the Lemma 8.3.21 it turns out that algebraic direct sum L g ∈ J gC (cid:0) R N θ (cid:1) is a dense subalgebra of A b π ⊕ .(c) The homomorphism Z N → Z N / Γ λ is surjective,.(ii) and (iii) Follow from the proof of (i). The sequence of spectral triples
Let us consider following objects • A spectral triple of a noncommutative torus (cid:16) C ∞ ( T N θ ) , H = L (cid:0) C (cid:0) T N θ (cid:1) , τ (cid:1) ⊗ C N , D (cid:17) , • A good algebraical finite covering sequence given by S θ = (cid:26) C (cid:16) T N θ (cid:17) → C (cid:16) T N θ / m (cid:17) → C (cid:16) T N θ / m (cid:17) → ... → C (cid:18) T N θ / m j (cid:19) → ... (cid:27) ∈ FinAlg .given by (8.3.4).Otherwise from the Theorem 8.2.13 it follows that S ( C ∞ ( T θ ) , L ( C ( T N θ ) , τ ) ⊗ C N , D ) = { (cid:16) C ∞ ( T θ ) , L (cid:16) C (cid:16) T N θ (cid:17) , τ (cid:17) ⊗ C N , D (cid:17) , . . . , (cid:18) C ∞ (cid:18) T N θ / m j (cid:19) , L (cid:18) C (cid:18) T N θ / m j (cid:19) , τ j (cid:19) ⊗ C N , D j (cid:19) , . . . } ∈ CohTriple (8.3.45)411s a coherent sequence of spectral triples. We would like to proof that S ( C ∞ ( T θ ) , L ( C ( T N θ ) , τ ) ⊗ C N , D ) is axiomatically good (cf. Definition 3.5.11) and to find a (cid:0) C (cid:0) T N θ (cid:1) , C (cid:0) R N θ (cid:1) , Z N (cid:1) -lift of (cid:16) C ∞ ( T θ ) , L (cid:0) C (cid:0) T N θ (cid:1) , τ (cid:1) ⊗ C N (cid:17) . If ρ : C ( T θ ) → B (cid:0) L (cid:0) C (cid:0) T N θ (cid:1) , τ (cid:1)(cid:1) ⊗ C N is the natural representation then from the 8.3.1 it turns out that e ρ : C (cid:16) R N θ (cid:17) → B (cid:16) L (cid:16) R N θ (cid:17) ⊗ C N (cid:17) is induced by ( ρ , S θ , b π ⊕ ) . Let us consider a topological covering ϕ : R N → T N and a commutative spectral triple (cid:0) C ∞ (cid:0) T N (cid:1) , L (cid:0) T N , S (cid:1) , / D (cid:1) given by (F.1.24).Denote by e S = ϕ ∗ S and e / D = p − / D inverse image of the spinor bundle S (cf. A.3)and the p -inverse image Dirac operator / D (cf. Definition 4.7.11). Otherwise thereis a natural isomorphism of Hilbert spaces e Φ : L (cid:16) R N θ (cid:17) ⊗ C N ≈ −→ L (cid:16) R N (cid:17) ⊗ C N .Denote by e D = e Φ − ◦ e / D ◦ e Φ . (8.3.46) Axiomatic approach
Here we apply the developed in 3.5.2 approach. The given by (8.3.7) represen-tation b π ⊕ naturally induces the faithful representation e π : C (cid:16) R N θ (cid:17) ֒ → B (cid:16) L (cid:16) R N (cid:17)(cid:17) ,If e D is an unbounded operator then for any j ∈ N the given by Section 3.5.2construction yields the representation e π j : Ω ∗ C ∞ (cid:18) T N θ / m j (cid:19) → B (cid:16) L (cid:16) R N (cid:17)(cid:17) , e π j ( a da ... da n ) = a h e D , a i ... h e D , a n i ∀ a , ..., a n ∈ C ∞ (cid:18) T N θ / m j (cid:19) . (8.3.47) Lemma 8.3.24. If e D is given by the Equation (8.3.46) then e D is the axiomatic lift (cf.Definition 3.5.8) of the coherent set of spectral triples S ( C ∞ ( T θ ) , L ( C ( T N θ ) , τ ) ⊗ C N , D ) givenby (8.3.45) . roof. One needs check (a) and (b) of the Definition 3.5.8.(a) Both representations π T : C ∞ (cid:18) T N θ / m j (cid:19) → B (cid:0) L (cid:0) T N (cid:1)(cid:1) , π R : C ∞ (cid:18) T N θ / m j (cid:19) → B (cid:0) L (cid:0) R N (cid:1)(cid:1) are faithful so one has (cid:13)(cid:13)(cid:13) π T ( a ) ⊗ M s ( C ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) π R ( a ) ⊗ M s ( C ) (cid:13)(cid:13)(cid:13) = k a k ∀ a ∈ C ∞ (cid:18) T N θ / m j (cid:19) .(b) If π j : Ω ∗ C ∞ (cid:18) T N θ / m j (cid:19) → B (cid:0) L (cid:0) T N (cid:1)(cid:1) then (cid:13)(cid:13)(cid:13) π j (cid:16) a h e D , a i ... h e D , a n i(cid:17)(cid:12)(cid:12)(cid:12) = (cid:13)(cid:13)(cid:13) e π j (cid:16) a h e D , a i ... h e D , a n i(cid:17)(cid:12)(cid:12)(cid:12) ∀ a , ..., a n ,so one has ker π j = ker e π j for all j ∈ N . If a ∈ C (cid:0) R N θ (cid:1) is given by (8.3.40) then a satisfies to (a)-(c) of the Definition3.5.10, i.e. a is axiomatically smooth. On the other hand the C -linear span of givenby (8.3.40) elements is dense in C (cid:0) R N θ (cid:1) it follows that the coherent set of spectraltriples S ( C ∞ ( T θ ) , L ( C ( T N θ ) , τ ) ⊗ C N , D ) is good (cf. Definition 3.5.11 )41314 hapter 9 Isospectral deformations and theircoverings
Let M be Riemannian manifold which admits a spin c -structure (cf. DefinitionE.4.2). Suppose that there is a the smooth action of T × M → M . Let e x ∈ e M and x = π ( e x ) . Denote by ϕ : R → R / Z = T the natural covering. There are twoclosed paths ω , ω : [
0, 1 ] → M given by ω ( t ) = ϕ ( t , 0 ) x , ω ( t ) = ϕ ( t ) x .There are lifts of these paths, i.e. maps e ω , e ω : [
0, 1 ] → e M such that e ω ( ) = e ω ( ) = e x , π ( e ω ( t )) = ω ( t ) , π ( e ω ( t )) = ω ( t ) .Since π is a finite-fold covering there are N , N ∈ N such that if γ ( t ) = ϕ ( N t , 0 ) x , γ ( t ) = ϕ ( N t ) x .and e γ (resp. e γ ) is the lift of γ (resp. γ ) then both e γ , e γ are closed. Let us selectminimal values of N , N . If pr n : S → S is an n listed covering and pr N , N thecovering given by e T = S × S N × pr N −−−−−−→→ S × S = T e T × e M → e M such that e T × e M e M T . × M M pr N N × π π where e T ≈ T . Let e p = ( e p , e p ) be the generator of the associated with e T two-parameters group e U ( s ) so that e U ( s ) = exp ( i ( s e p + s e p )) .The covering e M → M induces an involutive injective homomorphism ϕ : C ∞ ( M ) → C ∞ (cid:16) e M (cid:17) .Suppose M → M / T is submersion, and suppose there is a weak fibration T → M → M / T (cf. [70]) There is the exact homotopy sequence of the weak fibration · · · → π n (cid:0) T , e (cid:1) i −→ π n ( M , e ) p −→ π n (cid:0) M / T , b (cid:1) ∂ −→ π n − (cid:0) T , e (cid:1) → . . . · · · → π (cid:0) M / T , b (cid:1) ∂ −→ π (cid:0) T , e (cid:1) i −→ π ( M , e ) p −→ π (cid:0) M / T , b (cid:1) ∂ −→ π (cid:0) T , e (cid:1) → . . .(cf. [70]) where π n is the n th homotopical group for any n ∈ N . If π : e M → M isa finite-fold regular covering then there is the natural surjective homomorphism π ( M , e ) → G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . If π : e M → M induces a covering π : e M / T → M / T then the homomorphism ϕ : π ( M , e ) → G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) can be included into thefollowing commutative diagram. π (cid:0) T , e (cid:1) ∼ = Z π ( M , e ) π (cid:0) M / T , b (cid:1) { e } G (cid:16) e T | T (cid:17) G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) G (cid:16) e M / e T | M / T (cid:17) { e } ϕ ′ ϕ ϕ ′′ i p i ∗ p ∗ Denote by G def = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) , G ′ def = G (cid:16) e T | T (cid:17) , G ′′ def = G (cid:16) e M / e T | M / T (cid:17) .From the above construction it turns out that G ′ = G (cid:16) e T | T (cid:17) = Z N × Z N .Otherwise there is an inclusion of Abelian groups G (cid:16) e T | T (cid:17) ⊂ e T . The action416 T × e M → e M is free, so the action G ′ × e M → is free, so the natural homomorphism G ′ → G is injective, hence there is an exact sequence of groups { e } → G ′ → G → G ′′ → { e } . (9.1.1)Let θ , e θ ∈ R be such that e θ = θ + nN N , where n ∈ Z .If λ = e π i θ , e λ = e π i e θ then λ = e λ N N . There are isospectral deformations C ∞ ( M θ ) , C ∞ (cid:16) e M e θ (cid:17) and C -linear isomorphisms l : C ∞ ( M ) → C ∞ ( M θ ) , e l : C ∞ (cid:16) e M (cid:17) → C ∞ (cid:16) e M e θ (cid:17) .These isomorphisms and the inclusion ϕ induces the inclusion ϕ θ : C ∞ ( M θ ) → C ∞ (cid:16) e M e θ (cid:17) , ϕ e θ ( C ∞ ( M θ )) n , n ⊂ C ∞ (cid:16) e M e θ (cid:17) n N , n N .Denote by G = G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) the group of covering transformations. Since e l isa C -linear isomorphism the action of G on C ∞ (cid:16) e M (cid:17) induces a C -linear action G × C ∞ (cid:16) e M e θ (cid:17) → C ∞ (cid:16) e M e θ (cid:17) . According to the definition of the action of e T on e M it follows that the action of G commutes with the action of e T . It turns out gC ∞ (cid:16) e M (cid:17) n , n = C ∞ (cid:16) e M (cid:17) n , n for any n , n ∈ Z and g ∈ G . If e a ∈ C ∞ (cid:16) e M (cid:17) n , n , e b ∈ C ∞ (cid:16) e M (cid:17) n ′ , n ′ then g (cid:16)e a e b (cid:17) =( g e a ) (cid:16) g e b (cid:17) ∈ C ∞ (cid:16) e M (cid:17) n + n ′ , n + n ′ . One has e l ( e a ) e l (cid:16)e b (cid:17) = e λ n ′ n e l (cid:16)e a e b (cid:17) , e λ n e p l (cid:16)e b (cid:17) = e λ n ′ n l (cid:16)e b (cid:17) e λ n e p , e l ( g e a ) e l (cid:16) g e b (cid:17) = g e a e λ n e p g e b e λ n ′ e p = e λ n ′ n g (cid:16)e a e b (cid:17) e λ ( n + n ′ ) e p .On the other hand g (cid:16)e l ( e a ) e l (cid:16)e b (cid:17)(cid:17) = g (cid:16)e λ n ′ n e l (cid:16)e a e b (cid:17)(cid:17) = e λ n ′ n g (cid:16)e a e b (cid:17) e λ ( n + n ′ ) e p .417rom above equations it turns out e l ( g e a ) e l (cid:16) g e b (cid:17) = g (cid:16)e l ( e a ) e l (cid:16)e b (cid:17)(cid:17) ,i.e. g corresponds to automorphism of C ∞ (cid:16) e M e θ (cid:17) . It turns out that G is thegroup of automorphisms of C ∞ (cid:16) e M e θ (cid:17) . From e a ∈ C ∞ (cid:16) e M e θ (cid:17) n , n it follows that e a ∗ ∈ C ∞ (cid:16) e M e θ (cid:17) − n , − n . One has g (cid:16)(cid:16)e l ( e a ) (cid:17) ∗ (cid:17) = g (cid:16)e λ − n e p e a ∗ (cid:17) = g (cid:16)e λ n n e a ∗ e λ − n e p (cid:17) = e λ n n g (cid:16)e l ( e a ∗ ) (cid:17) .On the other hand (cid:16) g e l ( e a ) (cid:17) ∗ = (cid:16) ( g e a ) e λ n e p (cid:17) ∗ = e λ − n e p ( ga ∗ ) = e λ n n (cid:16) ga ∗ e λ − n e p (cid:17) = e λ n n g (cid:16)e l ( e a ∗ ) (cid:17) ,i.e. g (cid:16)(cid:16)e l ( e a ) (cid:17) ∗ (cid:17) = (cid:16) g e l ( e a ) (cid:17) ∗ . It follows that g corresponds to the involutive au-tomorphism of C ∞ (cid:16) e M e θ (cid:17) . Since C ∞ (cid:16) e M e θ (cid:17) is dense in C (cid:16) e M e θ (cid:17) there is the uniqueinvolutive action G × C (cid:16) e M e θ (cid:17) → C (cid:16) e M e θ (cid:17) . For any y ∈ M / T there is a point x ∈ M mapped onto y and a connected submanifold U ⊂ M such that: • dim U = dim M − • U is transversal to orbits of T -action, • The fibration T → U × T → U × T / T is the restriction of the fibration T → M → M / T , • The image V y ∈ M / T of U × T in M / T is an open neighborhood of y , • V y is evenly covered by e M / e T → M / T .It is clear that M / T = [ y ∈ M / T V y .Since M / T is compact there is a finite subset I ∈ M / T such that M / T = [ y ∈ I V y .418bove equation will be rewritten as M / T = [ ι ∈ I V ι (9.1.2)where ι is just an element of the finite set I and we denote corresponding transver-sal submanifold by U ι . There is a smooth partition of unity subordinated to (9.1.2),i.e. there is a set (cid:8) a ι ∈ C ∞ (cid:0) M / T (cid:1)(cid:9) ι ∈ I of positive elements such that1 C ( M / T ) = ∑ ι ∈ I a ι , (9.1.3) a ι (cid:0)(cid:0) M / T (cid:1) \ V ι (cid:1) = { } .Denote by e ι def = √ a ι ∈ C ∞ (cid:0) M / T (cid:1) . (9.1.4)For any ι ∈ I we select an open subset e V ι ⊂ e M / T which is homeomorphicallymapped onto V ι . If e I = G ′′ × A then for any ( g ′′ , ι ) ∈ e I we define e V ( g ′′ , ι ) = g ′′ e V ι . (9.1.5)Similarly we select a transversal submanifold e U ι ⊂ e M which is homeomorphially mapped onto U ι . For any ( g ′′ , ι ) ∈ e I we define e U ( g ′′ , ι ) = g e U ι . (9.1.6)where g ∈ G is an arbitrary element mapped to g ′′ . The set V ι is evenly coveredby π ′′ : e M / e T → M / T , so one has g e V ι \ e V ι = ∅ ; for any nontrivial g ∈ G ′′ . (9.1.7)If e e ι ∈ C ∞ (cid:16) e M / e T (cid:17) is given by e e ι ( e x ) = ( e ι ( π ′′ ( e x )) e x ∈ e U ι e x / ∈ e U ι then from (9.1.3) and (9.1.7) it turns out1 C ( e M / e T ) = ∑ g ∈ G ′′ ∑ ι ∈ I e e ι , ( g e e ι ) e e ι =
0; for any nontrivial g ∈ G ′′ . (9.1.8)419f e I = G ′′ × I and e e ( g , ι ) = g e e ι for any ( g , ι ) ∈ G ′′ × I then from (9.1.8) it turns out1 C ( e M / e T ) = ∑ e ι ∈ e I e e e ι , ( g e e e ι ) e e e ι =
0; for any nontrivial g ∈ G ′′ ,1 C ( e M / e T ) = ∑ e ι ∈ e I e e e ι ih e e e ι . (9.1.9)It is known that C (cid:0) T (cid:1) is an universal commutative C ∗ -algebra generated by twounitary elements u , v , i.e. there are following relations uu ∗ = u ∗ u = vv ∗ = v ∗ v = C ( T ) , uv = vu , u ∗ v = vu ∗ , uv ∗ = v ∗ u , u ∗ v ∗ = v ∗ u ∗ . (9.1.10)If J = I × Z × Z then for any ( ι , j , k ) ∈ J there is an element f ′ ( ι , j , k ) ∈ C ∞ (cid:0) U ι × T (cid:1) given by f ′ ( ι , j , k ) = e ι u j v k (9.1.11)where e ι ∈ C ∞ (cid:0) M / T (cid:1) is regarded as element of C ∞ ( M ) . Let p : M → M / T .Denote by f ( ι , j , k ) ∈ C ∞ ( M ) an element given by f ( ι , j , k ) ( x ) = ( f ′ ( ι , j , k ) ( x ) p ( x ) ∈ V ι p ( x ) / ∈ V ι ,where the right part of the above equation assumes the inclusion U ι × T ֒ → M .(9.1.12)If we denote by e u , e v ∈ U (cid:16) C (cid:16) f T (cid:17)(cid:17) unitary generators of C (cid:16) f T (cid:17) then the covering π ′ : f T → T corresponds to a *-homomorphism C (cid:0) T (cid:1) → C (cid:16) f T (cid:17) given by u e u N , v e v N .There is the natural action of G (cid:16) e T | T (cid:17) ∼ = Z N × Z N on C (cid:0) T (cid:1) given by (cid:16) k , k (cid:17) e u = e π ik N e u , (cid:16) k , k (cid:17) e v = e π ik N e v (9.1.13)420here (cid:16) k , k (cid:17) ∈ Z N × Z N . If we consider C (cid:16) e T (cid:17) C ( T ) as a right Hilbert modulewhich corresponds to a finite-fold noncommutative covering then one has D e u j ′ e v k ′ , e u j ′′ e v k ′′ E C (cid:16)f T (cid:17) = N N δ j ′ j ′′ δ k ′ k ′′ C ( T ) ,1 C (cid:16) f T (cid:17) = N N j = N k = N ∑ j = k = e u j e v k ih e u j e v k . (9.1.14)If e J = e I × {
0, . . . , N − } × {
0, . . . , N − } then for any ( e ι , j , k ) ∈ e J there is anelement e f ′ ( e ι , j , k ) ∈ C ∞ (cid:16) e U e ι × e T (cid:17) given by e f ′ ( e ι , j , k ) = e e e ι e u j e v k (9.1.15)where e e e ι ∈ C ∞ (cid:16) e M / e T (cid:17) is regarded as element of C ∞ (cid:16) e M (cid:17) . Let p : e M → e M / e T .Denote by e f ( e ι , j , k ) ∈ C ∞ (cid:16) e M (cid:17) an element given by e f ( e ι , j , k ) ( e x ) = ( e f ′ ( e ι , j , k ) ( e x ) p ( e x ) ∈ e V e ι p ( e x ) / ∈ e V e ι .where right the part of the above equation assumes the inclusion e U e ι × e T ֒ → e M .(9.1.16)Any element e e e ι ∈ C (cid:16) e M / e T (cid:17) is regarded as element of C (cid:16) e M (cid:17) . From e e e ι ∈ C ∞ (cid:16) e M (cid:17) it turns out e l ( e e e ι ) = e e e ι , De l ( e e e ι ′ ) , e l ( e e e ι ′′ ) E C ( e M e θ ) = h e e e ι ′ , e e e ι ′′ i C ( e M ) .From (9.1.9)-(9.1.16) it follows that1 C ( e M ) = N N ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e f ( e ι , j , k ) ih e f ( e ι , j , k ) , (9.1.17) D e f ( e ι ′ , j ′ , k ′ ) , e f ( e ι ′′ , j ′′ , k ′′ ) E C ( e M ) = N N δ j ′ j ′′ δ k ′ k ′′ h e e e ι ′ , e e e ι ′′ i C ( e M ) ∈ C ∞ ( M ) , De l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17)E C ( e M e θ ) = N N δ j ′ j ′′ δ k ′ k ′′ h e e e ι ′ , e e e ι ′′ i C ( e M e θ ) ∈ C ∞ ( M θ ) .(9.1.18)421rom the (9.1.17) it turns out that C (cid:16) e M (cid:17) is a right C ( M ) module generated by fi-nite set of elements e f ( e ι , j , k ) where ( e ι , j , k ) ∈ e J , i.e. any e a ∈ C (cid:16) e M (cid:17) can be representedas e a = ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e f ( e ι , j , k ) a ( e ι , j , k ) ; where a ( e ι , j , k ) ∈ C ( M ) . (9.1.19)Moreover if e a ∈ C ∞ (cid:16) e M (cid:17) then one can select a ( e ι , j , k ) ∈ C ∞ ( M ) . However any a ( e ι , j , k ) ∈ C ∞ ( M ) can be uniquely written as a doubly infinite norm convergentsum of homogeneous elements, a ( e ι , j , k ) = ∑ n , n b T n , n ,with b T n , n of bidegree ( n , n ) and where the sequence of norms || b T n , n || is ofrapid decay in ( n , n ) . One has e l (cid:16) e f ( e ι , j , k ) a ( e ι , j , k ) (cid:17) = ∑ n , n e f ( e ι , j , k ) b T n , n e λ ( N n + j ) e p = ∑ n , n e f ( e ι , j , k ) e λ j e p e λ kN n b T n , n e λ N n e p (9.1.20)the sequence of norms || e λ kN n b T n , n || = || b T n , n || is of rapid decay in ( n , n ) itfollows that a ′ ( e ι , j , k ) = ∑ n , n e λ kN n b T n , n ∈ C ∞ ( M ) (9.1.21)From (9.1.19) - (9.1.21) it turns out e l ( e a ) = ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e l (cid:16) e f ( e ι , j , k ) (cid:17) l (cid:16) a ′ ( e ι , j , k ) (cid:17) ; where l (cid:16) a ′ ( e ι , j , k ) (cid:17) ∈ C ∞ ( M θ ) . (9.1.22)However C (cid:16) e M e θ (cid:17) is the norm completion of C ∞ (cid:16) e M θ (cid:17) , so from (9.1.22) it turns outthat C (cid:16) e M e θ (cid:17) is a right Hilbert C ( M θ ) -module generated by a finite set Ξ = ne l (cid:16) e f ( e ι , j , k ) (cid:17) ∈ C ∞ (cid:16) e M e θ (cid:17)o ( e ι , j , k ) ∈ e J (9.1.23)From the Lemma D.4.13 it follows that the module is projective. So one has thefollowing theorem. Theorem 9.1.1.
The triple (cid:16) C ( M θ ) , C (cid:16) e M e θ (cid:17) , G (cid:16) e M | M (cid:17)(cid:17) is an unital noncommuta-tive finite-fold covering. .1.2 Induced representation From (9.1.18) it turns out De l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17)E C ( e M e θ ) = δ j ′ j ′′ δ k ′ k ′′ De l ( e e e ι ′ ) , e l ( e e e ι ′′ ) E C ( e M e θ ) == δ j ′ j ′′ δ k ′ k ′′ h e e e ι ′ , e e e ι ′′ i C ( e M e θ ) = e e ι ′ e e ι ′′ ∈ C ∞ ( M θ ) . (9.1.24)Let g ∈ G be any element, and let e l (cid:16) e f ( e ι , j , k ) (cid:17) ∈ Ξ where Ξ is given by (9.1.23).From (9.1.15) and (9.1.16) it turns out e l (cid:16) e f ( e ι , j , k ) (cid:17) = e e e ι e l (cid:16) e u j e v k (cid:17) = e e e ι e u j e v k e λ k e p . (9.1.25)If g ∈ G be any element, then from the exact sequence (9.1.1) { e } → G ′ → G → G ′′ → { e } it follows that there is g ′′ ∈ G ′′ which is the image of g . For any e ι ∈ e I = G ′′ × I there is e ι ′ ∈ e I such that g ′′ transforms e ι to e ι ′ . If e V e ι ∈ e M / e T is givenby (9.1.5), and e U e ι ∈ e M is given by (9.1.6) then there is g ′ ∈ G ′ ∼ = Z N × Z N suchthat g ′′ e V e ι = e V e ι ′ , g e U e ι = g ′ e U e ι ′ .If g ′ corresponds to (cid:16) k , k (cid:17) ∈ Z N × Z N then from g ′′ e e e ι = e e e ι ′ and (9.1.13) it turnsout g e f ( e ι , j , k ) = g (cid:16)e e e ι e u j e v k (cid:17) = (cid:0) g ′′ e e e ι (cid:1) (cid:16) g ′ (cid:16) e u j e v k (cid:17)(cid:17) == e e e ι ′ (cid:18) e π ijk N e π ikk N e u j e v k (cid:19) = e π ijk N e π ikk N e f ( e ι ′ , j , k ) . (9.1.26)Form above equation it follows that for any e ι , e ι ∈ e I , g , g ∈ G , j ′ , j ′′ =
0, . . . N − k ′ , k ′′ =
0, . . . N − e ι ′ , e ι ′ ∈ e I , l , l ∈ Z such that D g e f ( e ι , j ′ , k ′ ) , g e f ( e ι , j ′′ , k ′′ ) E C ( e M ) = e π il N e π il N δ j ′ j ′′ δ k ′ k ′′ De e e ι ′ , e e e ι ′ E C ( e M ) ∈ C ∞ ( M ) , D g (cid:16)e l (cid:16) e f ( e ι , j ′ , k ′ ) (cid:17)(cid:17) , g (cid:16)e l (cid:16) e f ( e ι , j ′′ , k ′′ ) (cid:17)(cid:17)E C ( e M e θ ) == e π il N e π il N δ j ′ j ′′ δ k ′ k ′′ De e e ι ′ , e e e ι ′ E C ( e M e θ ) ∈ C ∞ ( M θ ) .(9.1.27)423rom (9.1.27) it turns out that the finite set e Ξ = G Ξ satisfies to the condition(a) of the Lemma 2.7.3, i.e. De l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17)E C ( e M e θ ) = C ∞ ( M θ ) ; ∀ e l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17) ∈ e Ξ .(9.1.28) Lemma 9.1.2.
For any e a ∈ C (cid:16) e M e θ (cid:17) following conditions are equivalent(a) e a ∈ C ∞ (cid:16) e M e θ (cid:17) ,(b) De l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , e a e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17)E C ( e M e θ ) ∈ C ∞ ( M θ ) ; ∀ e l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17) ∈ Ξ .Proof. (a) ⇒ (b) For any g ∈ G following condition holds g e l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , g e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17) ∈ C ∞ (cid:16) e M e θ (cid:17) , hence De l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , e a e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17)E C ( e M e θ ) = ∑ g ∈ G g (cid:16)e l (cid:16) e f ∗ ( e ι ′ , j ′ , k ′ ) (cid:17) , e a e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17)(cid:17) ∈ C ∞ (cid:16) e M e θ (cid:17) .Since De l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , e a e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17)E C ( e M e θ ) is G -invariant one has De l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) , e a e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17)E C ( e M e θ ) ∈ C ∞ ( M ) .(b) ⇒ (a) There is the following equivalence e e e ι e a e e e ι ∈ C ∞ (cid:16) e M e θ (cid:17) ⇔ h e e e ι , e a e e e ι i C ( e M e θ ) ∈ C ∞ ( M θ ) .Taking into account e e e ι = e l (cid:16) e f ( e ι ,0,0 ) (cid:17) one has a following logical equation ∀ e ι ∈ e I De l (cid:16) e f ( e ι ,0,0 ) (cid:17) , e a e l (cid:16) e f ( e ι ,0,0 ) (cid:17)E C ( e M e θ ) ∈ C ∞ ( M θ ) ⇔⇔ e l (cid:16) e f ( e ι ,0,0 ) (cid:17) e a e l (cid:16) e f ( e ι ,0,0 ) (cid:17) ∈ C ∞ (cid:16) e M e θ (cid:17) ⇒⇒ e a = ∑ e ι ∈ e I e e e ι e a e e e ι = ∑ e ι ∈ e I e l (cid:16) e f ( e ι ,0,0 ) (cid:17) e a e l (cid:16) e f ( e ι ,0,0 ) (cid:17) ∈ C ∞ (cid:16) e M e θ (cid:17) .424 orollary 9.1.3. Following conditions hold: • C (cid:16) e M e θ (cid:17) T M n ( C ∞ ( M θ )) = C ∞ (cid:16) e M e θ (cid:17) , • The unital noncommutative finite-fold covering (cid:16) C ( M θ ) , C (cid:16) e M e θ (cid:17) , G (cid:16) e M | M (cid:17)(cid:17) is smoothly invariant.Proof. Follows from (9.1.28), and Lemmas 2.7.3, 9.1.2.From (9.1.17) for any e a ∈ C (cid:16) e M e θ (cid:17) following condition holds e a = N N ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e l (cid:16) e f ( e ι , j , k ) (cid:17) De a , e l (cid:16) e f ( e ι , j , k ) (cid:17)E C ( e M e θ ) = ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) (9.1.29)where a ( e ι , j , k ) = N N De a , e l (cid:16) e f ( e ι , j , k ) (cid:17)E C ( e M e θ ) ∈ C ( M θ ) . If ξ ∈ L ( M , S ) then e a ⊗ ξ = ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) ⊗ ξ = ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e l (cid:16) e f ( e ι , j , k ) (cid:17) ⊗ a ( e ι , j , k ) ξ == ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e l (cid:16) e f ( e ι , j , k ) (cid:17) ⊗ ξ ( e ι , j , k ) ∈ C (cid:16) e M e θ (cid:17) ⊗ C ( M θ ) L ( M , S ) ,where ξ ( e ι , j , k ) = a ( e ι , j , k ) ξ ∈ L ( M , S ) .(9.1.30)Denote by e H = C (cid:16) e M e θ (cid:17) ⊗ C ( M θ ) L ( M , S ) and let ( · , · ) e H is the given by (2.3.1)Hilbert product. If ξ , η ∈ L ( M , S ) then from (9.1.24) it turns out (cid:16)e l (cid:16) e f ( e ι ′ , j ′ , k ′ ) (cid:17) ⊗ ξ , e l (cid:16) e f ( e ι ′′ , j ′′ , k ′′ ) (cid:17) ⊗ η (cid:17) e H = N N δ j ′ j ′′ δ k ′ k ′′ ( ξ , e e ι ′ e e ι ′′ η ) L ( M , S ) . (9.1.31)425rom (9.1.31) it turns out the orthogonal decomposition e H = j = N − k = N − M j = k = e H jk ,where e H jk = ( e ξ ∈ e H | e ξ = ∑ e ι ∈ e I e l (cid:16) e f ( e ι , j , k ) (cid:17) ⊗ ξ ( e ι , j , k ) ∈ C (cid:16) e M e θ (cid:17) ⊗ C ( M θ ) L ( M , S ) ) .From (9.1.31) it turns out than for any 0 ≤ j ′ , j ′′ < N and 0 ≤ k ′ , k ′′ < N there isan isomorphism of Hilbert spaces given by Φ j ′ k ′ j ′′ k ′′ : e H j ′ k ′ ≈ −→ e H j ′′ k ′′ , e l (cid:16) e f ( e ι , j ′ , k ′ ) (cid:17) ⊗ ξ e l (cid:16) e f ( e ι , j ′′ , k ′′ ) (cid:17) ⊗ ξ . (9.1.32)Similarly if e H comm = C (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) then there is the decomposition e H comm = j = N − k = N − M j = k = e H comm jk ,where e H comm jk = ( e ξ ∈ e H comm | e ξ = ∑ e ι ∈ e I e f ( e ι , j , k ) ⊗ ξ ( e ι , j , k ) ∈ C (cid:16) e M (cid:17) ⊗ C ( M ) L ( M , S ) ) .From the Lemma (4.10.13) it turns out e H comm = L (cid:16) e M , e S (cid:17) the induced represen-tation is given by the natural action of C ( M ) on L (cid:16) e M , e S (cid:17) . Similarly to (9.1.32)for any 0 ≤ j ′ , j ′′ < N and 0 ≤ k ′ , k ′′ < N there is an isomorphism of Hilbertspaces given by Ψ j ′ k ′ j ′′ k ′′ : e H comm j ′ k ′ ≈ −→ e H comm j ′′ k ′′ , e f ( e ι , j ′ , k ′ ) ⊗ ξ e f ( e ι , j ′′ , k ′′ ) ⊗ ξ . (9.1.33)From e l (cid:16) e f ( e ι ,0,0 ) (cid:17) = e f ( e ι ,0,0 ) = e e e ι it turns out the natural ismomorphism e H comm0,0 ∼ = e H (9.1.34)426e would like to define the isomorphism ϕ : e H ≈ C (cid:16) e M e θ (cid:17) ⊗ C ( M θ ) L ( M , S ) ≈ −→ e H comm such that for any e a ∈ C (cid:16) e M e θ (cid:17) , and ξ ∈ L ( M , S ) following condition holds ϕ ( e a ⊗ ξ ) = e a (cid:16) C ( M ) ⊗ ξ (cid:17) (9.1.35)where the right part of the above equation assumes the action of C (cid:16) e M e θ (cid:17) on L (cid:16) e M , e S (cid:17) by operators (H.5.3). From the decomposition (9.1.29) it turns the equa-tion (9.1.35) is true if and only if it is true for any e a = e l (cid:16) e f ( e ι , j , k ) (cid:17) a where a ∈ C ( M θ ) ,i.e. ϕ (cid:16)e l (cid:16) e f ( e ι , j , k ) (cid:17) a ⊗ ξ (cid:17) = e l (cid:16) e f ( e ι , j , k ) (cid:17) a (cid:16) C ( M ) ⊗ ξ (cid:17) ,or equivalently ϕ (cid:16)e l (cid:16) e f ( e ι , j , k ) (cid:17) ⊗ a ξ (cid:17) = e l (cid:16) e f ( e ι , j , k ) (cid:17) (cid:16) C ( M ) ⊗ a ξ (cid:17) (9.1.36)From (9.1.33) the right part of (9.1.36) is given by e l (cid:16) e f ( e ι , j , k ) (cid:17) (cid:16) C ( M ) ⊗ a ξ (cid:17) = Ψ jk (cid:16)e λ kp (cid:16) e f ( e ι ,0,0 ) ⊗ a ξ (cid:17)(cid:17) From the above equation it turns out that if η ∈ e H j , k then ϕ ( η ) = ϕ | e H j , k ( η ) = Ψ jk (cid:16)e λ kp Φ jk ( η ) (cid:17) (9.1.37)and ϕ = j = N − k = N − M j = k = ϕ e H j , k : e H ≈ −→ e H comm then from above construction it turns out that ϕ satisfies to (9.1.35). In result wehave the following lemma. Lemma 9.1.4. If e ρ : C (cid:16) e M e θ (cid:17) → B (cid:16) e H (cid:17) is induced by (cid:16) ρ , (cid:16) C ( M θ ) , C (cid:16) e M e θ (cid:17) , G (cid:16) e M , M (cid:17)(cid:17)(cid:17) then e ρ can be represented by action of C (cid:16) e M e θ (cid:17) on L (cid:16) e M , e S (cid:17) by operators (H.5.3) . .1.3 Coverings of spectral triples If f ( ι , j , k ) ∈ C ∞ ( M ) is given by (9.1.12) then from then from (9.1.11) it turns out f ′ ( ι , j , k ) = e ι u j v k ∈ C ( U ι ) = C (cid:0) V ι × T (cid:1) In the above formula the product u j v k can be regarded as element of both C (cid:0) T (cid:1) and C b (cid:0) V ι × T (cid:1) = C b ( W ι ) , where W ι ⊂ M is the homeomorphic image of V ι × T . Since the Dirac operator / D is invariant with respect to transformations u e i ϕ u u , v e i ϕ v v one has [ / D , u ] = d ι u u , [ / D , v ] = d ι v v (9.1.38)where d ι u , d ι v : V ι → M dim S ( C ) are continuous matrix-valued functions. I wouldlike to avoid functions in C b ( U ι ) , so instead (9.1.38) the following evident conse-quence of it will be used h / D , au j v k i = [ / D , a ] u j v k + a ( jd ι u + kd ι v ) u j v k ; a ∈ C (cid:0) M / T (cid:1) , supp a ⊂ V ι .(9.1.39)In contrary to (9.1.38) the equation (9.1.39) does not operate with C b ( U ι ) , it oper-ates with C ( U ι ) . Let π : e M → M and let e / D = p − / D be the π -inverses image of / D (cf. Definition 4.7.11). Suppose e V e ι ⊂ e M / e T is mapped onto V ι ⊂ M / T . Then weset d e ι u def = d ι u , d e ι v def = d ι v . The covering π maps e V e ι × e T onto V ι × T . If e u , e v ∈ C (cid:16) e T (cid:17) are natural generators, then the covering e T → T is given by C (cid:0) T (cid:1) → C (cid:16) e T (cid:17) , u e u N , v e v N .From the above equation and taking into account (9.1.38) one has h e / D , e u i = d e ι u N e u , h e / D , e v i = d e ι v N e v , h e / D , e a e u j e v k i = h e / D , e a i e u j e v k + e a (cid:18) j e ι u N + k e ι v N (cid:19) e u j e v k ; e a ∈ C (cid:16) e M / e T (cid:17) , supp e a ⊂ e V e ι . (9.1.40)For any e a ∈ C ∞ (cid:16) e M (cid:17) such that supp e a ⊂ f W e ι following condition holds h e / D , e a i = lift f W e ι ([ / D , desc ( e a )]) (9.1.41)428f b ∈ C ∞ (cid:0) M / T (cid:1) ⊂ C ∞ ( M ) then from (9.1.40) and (9.1.41) it turns out h e / D , e f ( e ι , j , k ) bu j ′ v k ′ i = pe e e ι e u j e v k u j ′ v k ′ ⊗ [ / D , √ e e ι b ] ++ b pe e e ι e u j e v k u j ′ v k ′ ⊗ [ / D , √ e e ι ] ++ b pe e e ι e u j e v k u j ′ v k ′ ⊗ √ e e ι (cid:18) j ′ d u + jd u N + k ′ d v + kd v N (cid:19) and taking into account h e / D , e l i = h e / D , e l (cid:16) e f ( e ι , j , k ) bu j ′ v k ′ (cid:17)i = pe e e ι e l (cid:16) e u j e v k u j ′ v k ′ (cid:17) ⊗ [ / D , √ e e ι b ] ++ b pe e e ι e l (cid:16) e u j e v k u j ′ v k ′ (cid:17) ⊗ [ / D , √ e e ι ] ++ b pe e e ι e l (cid:16) e u j e v k u j ′ v k ′ (cid:17) ⊗ √ e e ι (cid:18) j ′ d u + jd u N + k ′ d v + kd v N (cid:19) (9.1.42)Taking into account that e l (cid:0) e u j e v k (cid:1) e l (cid:16) u j ′ v k ′ (cid:17) = e λ j ′ N k e l (cid:16) e u j e v k u j ′ v k ′ (cid:17) the equation (9.1.42)is equivalent to h e / D , e l (cid:16) e f ( e ι , j , k ) (cid:17) b e l (cid:16) u j ′ v k ′ (cid:17)i = pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ [ / D , √ e e ι b ] ++ b pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ [ / D , √ e e ι ] ++ b pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ √ e e ι (cid:18) j ′ d u + jd u N + k ′ d v + kd v N (cid:19) (9.1.43)For any e a ∈ C (cid:16) e M e θ (cid:17) there is the decomposition given by (9.1.29), i.e. e a = ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) Let Ω D be the module of differential forms associated with the spectral triple429 C ∞ ( M θ ) , L ( M , S ) , / D (cid:1) (cf. Definition E.3.5). Let us define a C -linear map ∇ : C ∞ (cid:16) e M e θ (cid:17) → C ∞ (cid:16) e M e θ (cid:17) ⊗ C ∞ ( M θ ) Ω D , ∑ e ι ∈ e I j = N − k = N − ∑ j = k = e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) ∑ e ι ∈ e I j = N − k = N − ∑ j = k = pe e e ι e l (cid:16) e u j e v k (cid:17) ⊗ h / D , √ e e ι a ( e ι , j , k ) i ++ ∑ e ι ∈ e I j = N − k = N − ∑ j = k = pe e e ι e l (cid:16) e u j e v k (cid:17) a ( e ι , j , k ) ⊗ [ / D , √ e e ι ] ++ ∑ e ι ∈ e I j = N − k = N − ∑ j = k = pe e e ι e l (cid:16) e u j e v k (cid:17) a ( e ι , j , k ) ⊗ √ e e ι (cid:18) jd u N + kd v N (cid:19) . (9.1.44)For any a ∈ C ∞ ( M θ ) following condition holds ∇ (cid:16)e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) a (cid:17) = pe e e ι e l (cid:16) e u j e v k (cid:17) ⊗ h / D , √ e e ι a ( e ι , j , k ) a i ++ pe e e ι e l (cid:16) e u j e v k (cid:17) a ( e ι , j , k ) a ⊗ [ / D , √ e e ι ] + pe e e ι e l (cid:16) e u j e v k (cid:17) a ( e ι , j , k ) a ⊗ √ e e ι (cid:18) jd u N + kd v N (cid:19) == pe e e ι e l (cid:16) e u j e v k (cid:17) ⊗ h / D , √ e e ι a ( e ι , j , k ) i a + e e e ι e l (cid:16) e u j e v k (cid:17) a ( e ι , j , k ) ⊗ [ / D , a ] ++ pe e e ι e l (cid:16) e u j e v k (cid:17) a ( e ι , j , k ) a ⊗ [ / D , √ e e ι ] + pe e e ι e l (cid:16) e u j e v k (cid:17) a ( e ι , j , k ) a ⊗ √ e e ι (cid:18) jd u N + kd v N (cid:19) == ∇ (cid:16)e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) (cid:17) a + e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) [ / D , a ] .(9.1.45)From (9.1.44), (9.1.45) and taking into account (E.3.8) one concludes that ∇ is aconnection (cf. Definition E.3.8). If (cid:16) l , l (cid:17) ∈ Z N × Z N and α = e π il jN e π il kN ∈ C then following condition holds ∇ (cid:16)(cid:16) l , l (cid:17) (cid:16)e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) (cid:17)(cid:17) = pe e e ι α e l (cid:16) e u j e v k (cid:17) ⊗ h / D , √ e e ι a ( e ι , j , k ) i ++ pe e e ι α e l (cid:16) e u j e v k (cid:17) a ( e ι , j , k ) ⊗ [ / D , √ e e ι ] + pe e e ι α e l (cid:16) e u j e v k (cid:17) a ( e ι , j , k ) ⊗ √ e e ι (cid:18) jd u N + kd v N (cid:19) == α ∇ (cid:16)e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) (cid:17) = (cid:16) l , l (cid:17) ∇ (cid:16)e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j , k ) (cid:17) ,430.e. the connection ∇ is Z N × Z N equivariant (cf. (2.7.6)). If a ( e ι , j ′ , k ′ ) ∈ C ∞ ( M θ ) j ′ , k ′ is an element of bidegree ( j ′ , k ′ ) then there is b ∈ C ∞ ( M θ ) such that e e e ι a ( e ι , j ′ , k ′ ) = e e e ι bu j ′ v k ′ . (9.1.46)From (9.1.45) it it follows that ∇ (cid:16)e l (cid:16) e f ( e ι , j , k ) a ( e ι , j ′ , k ′ ) (cid:17)(cid:17) = pe e e ι e l (cid:16) e u j e v k (cid:17) ⊗ h / D , √ e e ι b e l (cid:16) u j ′ v k ′ (cid:17)i ++ b pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ [ / D , √ e e ι ] ++ b pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ √ e e ι (cid:18) jd u N + kd v N (cid:19) ,and taking into account pe e e ι e l (cid:16) e u j e v k (cid:17) ⊗ h / D , √ e e ι b e l (cid:16) u j ′ v k ′ (cid:17)i = pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ [ / D , √ e e ι b ] ++ b pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ √ e e ι (cid:0) j ′ d u + k ′ d v (cid:1) one has ∇ (cid:16)e l (cid:16) e f ( e ι , j , k ) a ( e ι , j ′ , k ′ ) (cid:17)(cid:17) = pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ [ / D , √ e e ι b ] ++ b pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ [ / D , √ e e ι ] ++ b pe e e ι e l (cid:16) e u j e v k (cid:17) e l (cid:16) u j ′ v k ′ (cid:17) ⊗ √ e e ι (cid:18) j ′ d u + jd u N + k ′ d v + kd v N (cid:19) (9.1.47)From (9.1.47) and (9.1.43) it turns out that ∇ (cid:16)e l (cid:16) e f ( e ι , j , k ) a ( e ι , j ′ , k ′ ) (cid:17)(cid:17) = h e / D , e l (cid:16) e f ( e ι , j , k ) (cid:17) a ( e ι , j ′ , k ′ ) i .Any e a ∈ C ∞ (cid:16) e M e θ (cid:17) is an infinite sum of elements e f ( e ι , j , k ) a ( e ι , j ′ , k ′ ) it turns out ∇ ( e a ) = h e / D , e a i . (9.1.48)Taking into account the Corollary 9.1.3 one has the following theorem. Theorem 9.1.5.
The noncommutative spectral triple (cid:16) C ∞ (cid:16) e M e θ (cid:17) , L (cid:16) e M , e S (cid:17) , e / D (cid:17) is a (cid:16) C ( M θ ) , C (cid:16) e M e θ (cid:17) , G (cid:16) e M | M (cid:17)(cid:17) -lift of (cid:0) C ∞ ( M θ ) , L ( M , S ) , / D (cid:1) . .1.4 Unoriented twisted spectral triples Suppose that M is unoreintable manifold which satisfies to (H.5.1), i.e. T ⊂ Isom ( M ) ,Suppose that the natural 2-fold covering p : e M → M is such that e M is a Rieman-nian manifold which admits a spin c structure (cf. Definition E.4.2), so there is anoriented spectral triple (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S (cid:17) , e / D (cid:17) . Assume that there is a Hermi-tian bundle S → M on M such that e S is the p -inverse image of S (cf. DefinitionA.3.7). From 4.10.4 it turns out that there is an unoriented spectral triple given by(4.10.27), i.e. (cid:18) C ∞ ( M ) , L (cid:16) e M , e S (cid:17) Z , / D (cid:19) .Otherwise from (H.5.6) it follows that there is an oriented twisted spectral triple (cid:16) l (cid:16) C ∞ (cid:16) e M (cid:17)(cid:17) , L (cid:16) e M , e S (cid:17) , e / D (cid:17) .. Action of G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ∼ = Z on e M induces an action of Z on both C ∞ (cid:16) e M (cid:17) and lC ∞ (cid:16) e M (cid:17) such that C ∞ ( M ) = C ∞ (cid:16) e M (cid:17) Z , lC ∞ ( M ) = lC ∞ (cid:16) e M (cid:17) Z ,From the above construction we have an unoriented twisted spectral triple (cid:18) lC ∞ ( M ) , L (cid:16) e M , e S (cid:17) Z , / D (cid:19) .which satisfies to the Definition (2.9.1).432 hapter 10 The double covering of thequantum group SO q ( ) The covering of the quantum SO ( ) is described in [30, 60]. Here we wouldlike to prove that the covering complies with the general theory of noncommu-tative coverings described in 2.1. Moreover we prove that the covering gives anunoriented spectral triple (cf. Definition 2.9.1). Denote by G = (cid:8) g ∈ Aut (cid:0) C (cid:0) SU q ( ) (cid:1)(cid:1) (cid:12)(cid:12) ga = a ; ∀ a ∈ C (cid:0) SO q ( ) (cid:1)(cid:9) (10.1.1)Denote by C (cid:0) SU q ( ) (cid:1) N def = n e a ∈ C (cid:0) SU q ( ) (cid:1) (cid:12)(cid:12) e a ββ ∗ = q N ββ ∗ e a o ; ∀ N ∈ Z (10.1.2)and let us prove that the Equation (10.1.2) yields the Z -grading of C (cid:0) SU q ( ) (cid:1) .Really following conditions hold: • e a ∈ C (cid:0) SU q ( ) (cid:1) N ⇔ e a = α N ∑ ∞ j = k = b jk β j β ∗ k N ≥ ( α ∗ ) − N ∑ ∞ j = k = b jk β j β ∗ k N < From the Theorem H.3.4 it follows that the linear span of given by the Equa-tion (H.3.11) elements is dense in C (cid:0) SU q ( ) (cid:1) , hence C (cid:0) SU q ( ) (cid:1) is the C ∗ -norm completion of the following direct sum M N ∈ Z C (cid:0) SU q ( ) (cid:1) N . (10.1.4)From ββ ∗ ∈ C (cid:0) SO q ( ) (cid:1) it follows that the grading is G -invariant, i.e. gC (cid:0) SU q ( ) (cid:1) N = C (cid:0) SU q ( ) (cid:1) N ; ∀ g ∈ G , ∀ N ∈ N . (10.1.5) C (cid:0) SU q ( ) (cid:1) is a commutative C ∗ -algebra generated generated by β and β ∗ i.e. e a ∈ C (cid:0) SU q ( ) (cid:1) ⇒ e a = ∞ ∑ j = k = c jk β j β ∗ k ; c jk ∈ C .It follows that g β ∈ C (cid:0) SU q ( ) (cid:1) , i.e. g β = ∑ ∞ j = k = c jk β j β ∗ k , and taking into account β ∈ C (cid:0) SO q ( ) (cid:1) ⇒ ( g β ) = β one concludes g β = ± β . (10.1.6)From α ∈ C (cid:0) SU q ( ) (cid:1) and the equations (10.1.3), (10.1.5) it turns out that for any g ∈ G one has g α ∈ C (cid:0) SU q ( ) (cid:1) ⇒ g α = α ∞ ∑ j = k = b jk β j β ∗ k ; b jk ∈ C (10.1.7)Otherwise from α ∈ SO q ( ) it turns out α = g α = ( g α ) = α ∞ ∑ j = k = b jk β j β ∗ k == α ∞ ∑ j = k = ∞ ∑ l = m = q k + m b jk b lm β j + l ( β ∗ ) k + m = α ∞ ∑ j = k = c rs β r β ∗ s ;where c rs = r ∑ l = s ∑ m = q k + m b r − l , s − m b lm (10.1.8)434t turns out that c = c rs = ( r , s ) = (
0, 0 ) . Otherwise c = b it turns out that b = ǫ = ±
1. Suppose that there are j , k ∈ N such that b jk = j + k >
0. If j and k are such that b jk = j + k = min b mn = m + n > m + n then c jk = ǫ b jk (cid:0) + q j + k (cid:1) =
0. There is a contradiction with c rs = r + s >
0. It follows that if j + k > b jk =
0, hence one has g α = εα = ± α .In result we have g α = ± α , g β = ± β If g α = α and g β = − β then g ( αβ ) = − αβ , it is impossible because αβ ∈ SO q ( ) .It turns out that G = Z and if g ∈ G is not trivial then g α = − α and g β = − β . Sowe proved the following lemma Lemma 10.1.1.
If G = (cid:8) g ∈ Aut (cid:0) C (cid:0) SU q ( ) (cid:1)(cid:1) | ga = a ; ∀ a ∈ C (cid:0) SO q ( ) (cid:1)(cid:9) thenG ≈ Z . Moreover if g ∈ G is the nontrivial element theng (cid:16) α k β n β ∗ m (cid:17) = ( − ) k + m + n α k β n β ∗ m , g (cid:16) α ∗ k β n β ∗ m (cid:17) = ( − ) k + m + n α ∗ k β n β ∗ m . C ∗ -algebra Lemma 10.2.1. C (cid:0) SU q ( ) (cid:1) is a finitely generated projective C (cid:0) SO q ( ) (cid:1) module.Proof. Let A f be given by the Theorem H.3.4. If A Z f = A f T C (cid:0) SO q ( ) (cid:1) then from(H.3.9) and the Theorem H.3.4 it turns out that given by (H.3.11) elements α k β n β ∗ m and α ∗ k ′ β n β ∗ m with even k + m + n or k ′ + m + n is the basis of A Z f . If e a = α k β n β ∗ m / ∈ A Z f k + m + n is odd. If m > e a = α k β n β ∗ m − β ∗ = a β ∗ where a ∈ A Z f .If m = n > e a = α k β n − β = a β where a ∈ A Z f If m = n = k > e a = α k − α = a α where a ∈ A Z f .From e a = α ∗ k ′ β n β ∗ m / ∈ A Z f it follows that k ′ + m + n is odd. Similarly to the above proof one has e a = a α or e a = a α ∗ or e a = a β or e a = a β ∗ where a ∈ A Z f .From the above equations it turns out that A f is a left A Z f -module generated by α , α ∗ , β , β ∗ . Algebra A Z f (resp. A f ) is dense in C (cid:0) SO q ( ) (cid:1) (resp. C (cid:0) SU q ( ) (cid:1) )it follows that C (cid:0) SU q ( ) (cid:1) is a left C (cid:0) SO q ( ) (cid:1) -module generated by α , α ∗ , β , β ∗ .From the Remark Kasparov stabilization theorem it turns out that C (cid:0) SU q ( ) (cid:1) is afinitely generated projective C (cid:0) SO q ( ) (cid:1) module. Corollary 10.2.2.
The triple (cid:0) C (cid:0) SO q ( ) (cid:1) , C (cid:0) SU q ( ) (cid:1) , Z (cid:1) is an unital noncommuta-tive finite-fold covering.Proof. Follows from C (cid:0) SO q ( ) (cid:1) = C (cid:0) SU q ( ) (cid:1) Z and Lemmas 10.1.1, 10.2.1. Corollary 10.2.3.
The triple (cid:0) C (cid:0) SO q ( ) (cid:1) , C (cid:0) SU q ( ) (cid:1) , Z × Z , π (cid:1) is an unital non-commutative finite-fold covering. SO q ( ) as an unoriented spectral triple Let h : C (cid:0) SU q ( ) (cid:1) be a given by the Equation (H.3.3) state, and L (cid:0) C (cid:0) SU q ( ) (cid:1) , h (cid:1) is the Hilbert space of the corresponding GNS representation (cf. Section D.2.1).The state h is Z -invariant, hence there is an action of Z on L (cid:0) C (cid:0) SU q ( ) (cid:1) , h (cid:1) Z on C ∞ (cid:0) SU q ( ) (cid:1) . From the aboveconstruction it follows that the unital orientable spectral triple (cid:16) C ∞ (cid:0) SU q ( ) (cid:1) , L (cid:0) C (cid:0) SU q ( ) (cid:1) , h (cid:1) , e D (cid:17) can be regarded as the triple given by condition 4 of the Definition 2.9.1. Also onesees that all conditions of the the Definition 2.9.1 hold, so one has an unorientedspectral triple (cid:16) C ∞ (cid:0) SO q ( ) (cid:1) , L (cid:0) C (cid:0) SU q ( ) (cid:1) , h (cid:1) Z , D (cid:17) where C ∞ (cid:0) SO q ( ) (cid:1) = C (cid:0) SO q ( ) (cid:1) T C ∞ (cid:0) SU q ( ) (cid:1) and D = e D | L ( C ( SU q ( ) ) , h ) Z .43738 cknowledgment I am very grateful to Prof. Joseph C Varilly and Arup Kumar Pal for advisingme on the properties of Moyal planes and equivariant spectral triples respectively.Author would like to acknowledge members of the Moscow State University Sem-inars "Noncommutative geometry and topology", "Algebras in analysis" leaded byprofessors A. S. Mishchenko and A. Ya. Helemskii for a discussion of this work.43940 ppendices ppendix A
Topology
A.1 General topology
Definition A.1.1. [53] If X is a set, a basis for a topology on X is a collection B ofsubsets of X (called basis elements ) such that(a) For each x ∈ X , there is at least one basis element B containing x .(b) If x belongs to the intersection of two basis elements B and B , then there isa basis element B containing x such that B ⊂ B ∩ B .If B satisfies these two conditions, then we define the topology T generated by B as follows: A subset U of X is said to be open in X (that is, to be an element of T ) if for each x ∈ U , there is a basis element B ∈ B such that x ∈ B and B ⊂ U . Remark A.1.2.
It is proven that given by the Definition A.1.1 collection T is atopology (cf. [53]). Definition A.1.3. [22] Let Λ be a set and ≤ is relation on Λ . We say that ≤ directs Λ or Λ is directed by ≤ , if ≤ has following properties:(a) If λ ≤ µ and µ ≤ ν , then λ ≤ ν ,(b) For every λ ∈ Λ , λ ≤ λ ,(c) For any µ , ν ∈ Λ there exists a λ ∈ Λ such that µ ≤ λ and ν ≤ λ . Definition A.1.4. [22] A subset Ξ ⊂ Λ is said to be cofinal in Λ if for every λ ∈ Λ there is χ ∈ Ξ such that λ ≤ χ . 443 efinition A.1.5. [22] A net in topological space X is an arbitrary function from anon-empty directed set Λ to the space X . Nets will be denoted by S = { x λ ∈ X } λ ∈ Λ . Definition A.1.6. [22] A point x ∈ X is called a limit of a net S = { x λ ∈ X } λ ∈ Λ if for every neighborhood U of x there exists λ ∈ Λ such that x λ ∈ U for every λ ≥ λ ; we say that { x λ } converges to x . The limit will be denoted by x = lim S , or x = lim λ ∈ Λ x λ , or lim x λ . Remark A.1.7.
A net can converge to many points, however if X is Hausdorff thenthe net can have the unique limit. Definition A.1.8. [53] If a space X has a countable basis for its topology, then X is said to satisfy the second countability axiom , or to be second-countable . Definition A.1.9. [53] Suppose that one-point sets are closed in X . Then X issaid to be regular if for each pair consisting of a point x and a closed set B disjointfrom x , there exist disjoint open sets containing x and B respectively. The space X is said to be normal if for each pair A , B of disjoint closed sets of X there existdisjoint open sets containing A and B respectively. Definition A.1.10. [53] A topological space X is completely regular if one-point setsare closed in X and for each point x and each closed U ⊂ X not containing x ,there is a continuous function f : X → [
0, 1 ] such that f ( x ) = f ( Y ) = { } . Theorem A.1.11. [53] Every regular space with a countable basis is normal.
Exercise A.1.12. [53] Show that every locally compact Hausdorff space is com-pletely regular.
Theorem A.1.13. [53] Every compact Hausdorff space is normal.
Theorem A.1.14. [53]
Urysohn lemma.
Let X be a normal space, let A , B be discon-nected closed subsets of X . Let [ a , b ] be a closed interval in the real line. Then there exista continuous map f : X → [ a , b ] such that f ( A ) = { a } and f ( B ) = { b } . Theorem A.1.15. [53]
Tietze extension theorem.
Let X be a normal space; let A be aclosed subspace of X .(a) Any continuous map of A into the closed interval [ a , b ] of R may be extended to acontinuous map of all X into [ a , b ] (a) Any continuous map of A into R may be extended to a continuous map of all X into R . efinition A.1.16. [53] If φ : X → C is continuous then the support of φ is definedto be the closure of the set φ − ( C \ { } ) Thus if x lies outside the support, thereis some neighborhood of x on which φ vanishes. Denote by supp φ the support of φ . Theorem A.1.17. [14] For a locally compact Hausdorff space X , the following are equiv-alent:(a) The Abelian C ∗ -algebra C ( X ) is separable;(b) X is σ -compact and metrizable;(c) X is second-countable. Definition A.1.18. [53]. A space X is said to be locally connected at x if for everyneighborhood U of x there is a connected neighborhood V of x contained in U . If X is locally connected at each of its points, it is said simply to be locally connected . Theorem A.1.19. [53] A space X is locally connected if and only if for every open set U ,each component of U is open in X . There are two equivalent definitions of C ( X ) and both of them are used inthis book. Definition A.1.20.
An algebra C ( X ) is the norm closure of the algebra C c ( X ) ofcompactly supported continuous complex-valued functions. Definition A.1.21. A C ∗ -algebra C ( X ) is given by the following equation C ( X ) = { ϕ ∈ C b ( X ) | ∀ ε > ∃ K ⊂ X ( K is compact ) & ∀ x ∈ X \ K | ϕ ( x ) | < ε } ,i.e. (cid:13)(cid:13) ϕ | X \ K (cid:13)(cid:13) < ε . Definition A.1.22. [53] An indexed family of sets { A α } of topological space X issaid to be locally finite if each point x in X has a neighborhood that intersects foronly finite many values of α . Definition A.1.23. [53] Let {U α ∈ X } α ∈ A be an indexed open covering of X . Anindexed family of functions φ α : X → [
0, 1 ] is said to be a partition of unity , dominated by {U α } , if:445. φ α ( X \ U α ) = { }
2. The family supp φ α is locally finite.3. ∑ α ∈ A φ α ( x ) = x ∈ X . Definition A.1.24. [53] A space X is paracompact if every open covering X = ∪ U α has a locally finite open refinement X = ∪ V β . Theorem A.1.25. [53] Let X be a paracompact Hausdorff space; let {U α ∈ X } α ∈ A bean indexed open covering of X . Then there exists a partition of unity, dominated by {U α } . Proposition A.1.26. [12] A differential manifold M admits a (smooth) partition of unityif and only if it is paracompact.
Theorem A.1.27.
Every paracompact Hausdorff space X is normal Definition A.1.28. [22] We say that a topological space X is a Lindelöf space , or hasa
Lindelöf property , if X is regular and every open cover of X contains a countablesubcover. Theorem A.1.29. [22] Every regular second-countable space in Lindelöf.
Theorem A.1.30. [53] Every regular Lindelöf space is paracompact.
Definition A.1.31. [53] A compactification of a space X is a compact Hausdorffspace Y containing X as a subspace and the closure X of X is Y , i.e X = Y . Theorem A.1.32. [53] Let X be a completely regular space. There exists a compactifica-tion Y of X having the property that every bounded continuous map f : X → R extendsuniquely to a continuous map of Y into R . Definition A.1.33.
For each completely regular space X , let us choose, once andfor all, a compactification of X satisfying the extension condition of TheoremA.1.32. We will denote this compactification of by β X and call it the Stone-Cechcompactification of X . It is characterized by the fact that any continuous map X → Y of X into a compact Hausdorff space Y extends uniquely to a contin-uous map β X → Y . Definition A.1.34. [56] Let p : Y → X (A.1.1)be a continuous surjection of compact Hausdorff spaces, in particular, a closedmap. Let us consider the map (A.1.1) and a certain point x of X , which has a446nite number of pre-images y , . . . , y m . Then a neighborhood U of x is said to be regular if p − ( U ) = V ⊔ · · · ⊔ V m , (A.1.2)where V i are some neighborhoods of y i , i =
1, . . . , m . Lemma A.1.35. [56] Let p : Y → X be a continuous closed map of Hausdorff spaces.Then any point x of X with a finite number of pre-images has a regular neighborhood. Theorem A.1.36. [45] Theorem on iterated limits. Let D be a directed set, let E m bea directed set for each m in D, let F be the product D × ∏ m ∈ D E m , and for ( m , f ) in Flet R ( m , f ) = ( m , f ( m )) . If S ( m , n ) is a member of a topological space for each m in Dand each n in E n , then S ◦ R converges to lim m lim n S ( m , n ) whenever this iterated limitexists. Definition A.1.37. [67] Suppose next that X is a set and F is a nonempty family ofmappings f : X → Y f , where each Y f is a topological space. (In many importantcases, Y f is the same for all f ∈ F .) Let τ be the collection of all unions of finiteintersections of sets f − ( V ) with f ∈ F and V open in Y f . Then τ is a topology on X , and it is in fact the weakest topology on X that makes every f ∈ F continuous:If τ ′ is any other topology with that property, then τ ⊂ τ ′ . This τ is called the weak topology on X induced by F , or, more succinctly, the F - topology of X . A.2 Coverings
Results of this section are copied from [70]. The covering projection word isreplaced with covering outside this section.
Definition A.2.1. [70] Let e π : e X → X be a continuous map. An open subset
U ⊂X is said to be evenly covered by e π if e π − ( U ) is the disjoint union of open subsetsof e X each of which is mapped homeomorphically onto U by e π . A continuousmap e π : e X → X is called a covering projection if each point x ∈ X has an openneighborhood evenly covered by e π . e X is called the covering space and X the basespace of the covering. Definition A.2.2. [70] A topological space X is said to be locally path-connected ifthe path components of open sets are open.447enote by π the functor of fundamental group [70]. Definition A.2.3. [70] Let p : e X → X be a covering. A self-equivalence is a home-omorphism f : e X → e X such that p ◦ f = p . This group of such homeomorphismsis said to be the group of covering transformations of p or the covering group . Denoteby G (cid:16) e X | X (cid:17) this group.
Theorem A.2.4. [70] A fibration has unique path lifting if and only is every fiber has nononconstant paths.
Theorem A.2.5. [70] If X is locally connected, a continuous map p : e X → X is acovering projection if and only if for each component Y of X the mapp | p − ( Y ) : p − ( Y ) → Y is a covering projection. Corollary A.2.6. [70] Consider a commutative triangle e X e X X pp p where X is locally connected and p , p are covering projections. If p is a surjection thenp is a covering projection. Theorem A.2.7. [70] If p : e X → X is a covering projection onto locally connected basespace, then for any component e Y of e X the mapp | e Y : e Y → p (cid:16) e Y (cid:17) is a covering projection onto some component of e X . Theorem A.2.8. [70] Let p : e X → X , p : e X → X be objects in the category ofconnected covering spaces of a connected locally path-connected space X . The followingare equivalent(a) There is a coveting projection f : e X → e X such that p ◦ f = p .(b) For all e x ∈ e X and e x ∈ e X such that p ( e x ) = p ( e x ) , π ( p ) (cid:16) π (cid:16) e X , e x (cid:17)(cid:17) is conjugate to a subgroup of π ( p ) (cid:16) π (cid:16) e X , e x (cid:17)(cid:17) . c) There exist e x ∈ e X and e x ∈ e X such that π ( p ) (cid:16) π (cid:16) e X , e x (cid:17)(cid:17) is conjugate toa subgroup of π ( p ) (cid:16) π (cid:16) e X , e x (cid:17)(cid:17) . Lemma A.2.9. [70] A local homeomorphism is an open map.
A.2.1 Unique path lifting
A.2.10.
There is a significant problem in the algebraic topology, called the liftingproblem. Let p : E → B and f : X → B continuous maps of topological spaces.The lifting problem [70] for f is to determine whether there is a continuous map f ′ : X → E such that f = p ◦ f ′ -that is, whether the dotted arrow in the diagram EX B f pf ′ corresponds to a continuous map making the diagram commutative. If there issuch map f ′ , then f can be lifted to E , and we call f ′ a lifting or lift of f . If p is acovering projection and X = [
0, 1 ] ⊂ R then f can be lifted. Definition A.2.11. [70] A continous map p : E → B is said to have the unique pathlifting if, given paths ω and ω ′ in E such that p ◦ ω = p ◦ ω ′ and ω ( ) = ω ′ ( ) ,then ω = ω ′ . Theorem A.2.12. [70] Let p : e X → X be a covering projection and let f , g : Y → e X beliftings of the same map (that is, p ◦ f = p ◦ g). If Y is connected and f agrees with g forsome point of Y then f = g. Remark A.2.13.
From theorem A.2.12 it follows that a covering projection hasunique path lifting.
A.2.2 Regular and universal coverings
Definition A.2.14. [70] A fibration p : e X → X with unique path lifting is said tobe regular if, given any closed path ω in X , either every lifting of ω is closed ornone is closed. Theorem A.2.15. [70] Let p : e X → X be a fibration with unique path lifting andassume that a nonempty e X is a locally path-connected space. Then p is regular if and onlyif for some e x ∈ e X , π ( p ) π (cid:16) e X , e x (cid:17) is a normal subgroup of π ( X , p ( e x )) . heorem A.2.16. [70] Let G be a properly discontinuous group of homeomorphisms ofspace Y . Then the projection of Y to the orbit space Y / G is a covering projection. If Y is connected, this covering is regular and G is its group of covering transformations, i.e.G = G ( Y | Y / G ) . Lemma A.2.17. [70] Let p : e X → X be a fibration with a unique path lifting. If e X is connected and locally path-connected and e x ∈ e X then p is regular if and only ifG (cid:16) e X | X (cid:17) transitively acts on each fiber of p, in which case ψ : G (cid:16) e X | X (cid:17) ≈ π ( X , p ( e x )) / π ( p ) π (cid:16) e X , e x (cid:17) . Remark A.2.18. [70] If e X is simply connected, any fibration p : e X → X is regular,and we also have the next result.
Corollary A.2.19. [70] Let p : e X → X be a fibration with a unique path lifting where e X is simply connected locally path-connected and nonempty. Then the group of self-equivalences of p is isomorphic to the fundamental group of X , i.e. π ( X ) ≈ G (cid:16) e X (cid:12)(cid:12)(cid:12) X (cid:17) . Definition A.2.20. [70] A universal covering space of a connected space X is anobject p : e X → X of the category of connected covering spaces of X such that forany object p ′ : e X ′ → X of this category there is a morphism e X e X ′ X fp p ′ in the category. Lemma A.2.21. [70] A connected locally path-connected space X has a simply connectedcovering space if and only if X is semilocally 1-connected. Lemma A.2.22. [70] A simply connected covering space of a connected locally path-connected space X is an universal covering space of X . A.3 Vector bundles
Let k be the field of real or complex numbers, and let X be a topological space. Definition A.3.1. [43] A quasi-vector bundle with base X is given by:450a) A finite dimensional k -vector space E x for every point x of X .(b) A topology on the disjoint union E = F E x which induces the natural topol-ogy on each E x , such that the obvious projection π : E → X is continuous.The quasi-vector bundle with base will be denoted by ξ = ( E , π , X ) . The space E is the total space of ξ and E x is the fiber of ξ at the point x . A.3.2.
Let V be a finite dimensional vector space over k , E x = V and the totalspace may be identified with X × V with the product topology then the quasi-vector bundle ( X × V , π , X ) is called a trivial vector bundle . A.3.3.
Let ξ = ( E , π , X ) be a quasi-vector bundle, and let let X ′ ⊂ X be a subspaceof X . The triple ξ ′ = (cid:16) π − ( X ′ ) , π | π − ( X ′ ) , X ′ (cid:17) is called the restriction of ξ to X ′ .The fibers of ξ ′ are just fibers of ξ over the subspace ξ . One has X ′′ ⊂ X ′ ⊂ X ⇒ ( ξ | X ′ ) | X ′′ = ξ | X ′′ . (A.3.1) Definition A.3.4.
Let ξ = ( E , π , X ) be quasi-vector bundle. Then ξ is said to be locally trivial or a vector bundle if for every point x in X , there exists a neighborhoodof x such that ξ | U is isomorphic to a trivial bundle. Definition A.3.5.
Let ξ = ( E , π , X ) be a vector bundle. Then a continuous section of ξ is a continuous map s : X → E such that π ◦ s = Id X , Remark A.3.6.
In [43] the vector bundles over fields R and C are considered. Herewe consider complex vector bundles only. Definition A.3.7. [43] Let f : X ′ → X be a continuous map. For every point x ′ of X ′ , let E ′ x ′ = E f ( x ′ ) . Then the set E ′ = F x ′ ∈X ′ may be identified with the fiberproduct X ′ × X E formed by the pairs ( x ′ , e ) such that f ( x ′ ) = π ( e ) . If π ′ : E ′ → X ′ is defined by π ′ ( x ′ , e ) = x ′ , it is clear that the triple ξ = ( E ′ , π ′ , X ′ ) defines a quasi-vector bundle over X ′ , when we provide E ′ with the topology induced by X ′ × E .We write ξ ′ = f ∗ ( ξ ) or f ∗ ( E ) : this is the inverse image of ξ by f . Definition A.3.8. If ( E , π , X ) is a quasi-vector bundle then a continuous map s : X → E is said to be a continuous section if π ◦ s = Id X . The space Γ ( X , E ) , ofcontinuous sections can be regarded as both left and right C b ( X ) -module. Theorem A.3.9. [43]. Theorem (Serre, Swan). Let A = C k ( X ) be the ring of continuousfunctions on a compact space X with values in k. Then the section functor Γ induces anequivalence of categories of vector bundles over X and finitely generated projecive A-modules. .3.10. Let X be a compact topological space and S the complex vector bundleon X , such for any x ∈ X the fiber S x of S is a Hilbert space, i.e. there is a scalarproduct. ( · , · ) x : S x × S x → C . (A.3.2)If Γ ( M , S ) is the space of continuous sections of S then we suppose that for any ξ , η ∈ Γ ( M , S ) the map X → C given by x ( ξ x , η x ) x is continuous. If µ X isa measure on X then there is the scalar product ( · , · ) : Γ ( M , S ) × Γ ( M , S ) → C given by ( ξ , η ) def = Z X ( ξ x , η x ) x d µ X (A.3.3)Denote by L ( X , S , µ X ) or L ( X , S ) the Hilbert norm completion of Γ ( M , S ) , anddenote by ( · , · ) L ( X , S , µ X ) or ( · , · ) L ( X , S ) the given by (A.3.3) scalar product. Thereis the natural representation C ( X ) → B (cid:0) L ( X , S ) (cid:1) . (A.3.4) Definition A.3.11.
In the situation of A.3.10 we say that S is Hermitian vector bundle .452 ppendix B
Algebra
B.1 Algebraic Morita equivalence
Definition B.1.1. A Morita context ( A , B , P , Q , ϕ , ψ ) or, in some authors (e.g. Bass[5]) the pre-equivalence data is a generalization of Morita equivalence between cat-egories of modules. In the case of right modules, for two associative k -algebras(or, in the case of k = Z , rings) A and B , it consists of bimodules A P B , B Q A andbimodule homomorphisms ϕ : P ⊗ B Q → A , ψ : Q ⊗ A P → B satisfying mixedassociativity conditions, i.e. for any p , p ′ ∈ P and q , q ′ ∈ Q following conditionshold: ϕ ( p ⊗ q ) p ′ = p ψ (cid:0) q ⊗ p ′ (cid:1) , ψ ( q ⊗ p ) q ′ = q ϕ (cid:0) p ⊗ q ′ (cid:1) . (B.1.1)A Morita context is a Morita equivalence if both ϕ and ψ are isomorphisms ofbimodules. Remark B.1.2.
The Morita context ( A , B , P , Q , ϕ , ψ ) is a Morita equivalence if andonly if A -module P is a finitely generated projective generator (cf. [5] II 4.4) B.2 Finite Galois coverings
Here I follow to [4]. Let A ֒ → e A be an injective homomorphism of unital alge-bras, such that • e A is a projective finitely generated A -module,453 There is an action G × e A → e A of a finite group G such that A = e A G = ne a ∈ e A | g e a = e a ; ∀ g ∈ G o .Let us consider the category M G e A of G − e A modules, i.e. any object M ∈ M G e A is a e A -module with equivariant action of G , i.e. for any m ∈ M a following conditionholds g ( e am ) = ( g e a ) ( gm ) for any e a ∈ e A , g ∈ G .Any morphism ϕ : M → N in the category M G e A is G - equivariant, i.e. ϕ ( gm ) = g ϕ ( m ) for any m ∈ M , g ∈ G .Let e A [ G ] be an algebra such that e A [ G ] ≈ e A × G as an Abelian group and amultiplication law is given by ( a , g ) ( b , h ) = ( a ( gb ) , gh ) .The category M G e A is equivalent to the category M e A [ G ] of e A [ G ] modules. Otherwisein [4] it is proven that if e A is a finitely generated, projective A -module then there isan equivalence between a category M A of A -modules and the category M e A [ G ] . Itturns out that the category M G e A is equivalent to the category M A . The equivalenceis given by mutually inversed functors ( − ) ⊗ e A : M A → M G e A and ( − ) G : M G e A → M A . B.3 Profinite and residually finite groups
B.3.1. [21] Let G be a group and { G α } be the set of all normal subgroups of finiteindex in G . Then the set (cid:8) G / G α , φ αβ (cid:9) of finite quotients G / G α , of G together withthe canonical projections φ αβ : G / G α → G / G β whenever G α ⊂ G β is an inversesystem. The inverse limit lim ←− G / G α of this system is called the profinite completion of G and is denoted by b G . The group b G can also be described as the closure ofthe image of G under the the diagonal mapping ∆ : G → ∏ ( G / G α ) where G / G α ,is given the discrete topology and ∏ ( G / G α ) has the product topology. In thisdescription the elements of b G are the elements ( g α ) ∈ ∏ ( G / G α ) which satisfy φ αβ ( g α ) = β whenever G α ⊂ G β . Definition B.3.2. [9] A group G is said to be residually finite if for each nontrivialelement from G there exists a finite group K and a homomorphism ϕ : G → K such that ϕ ( g ) =
1. 454 efinition B.3.3. [44] A subgroup H is called a normal subgroup (or invariantsubgroup ) of the group G , if the left-sided partition of the group G with respect tothe subgroup H coincides with the right-sided partition, i.e. if for every g ∈ G theequation gH = Hg (B.3.1)holds (understood in the sense that the two subsets coincide in G ). Thus we canspeak simply of the partition of the group G with respect to the normal subgroup H .45556 ppendix C Functional analysis
C.1 Weak topologies
Definition C.1.1. [67] Suppose X is a topological vector space (with topology τ )whose dual X ∗ separates points on X . The X ∗ -topology (cf. Definition A.1.37) of X is called the weak topology of X . Definition C.1.2. [67] Let X again be a topological vector space whose dual is X ∗ .For the definitions that follow, it is irrelevant whether X ∗ separates points on Xor not. The important observation to make is that every x ∈ X induces a linearfunctional f x on X ∗ , defined by f x Λ = Λ x The X -topology of X ∗ (cf. Definition A.1.37) is called the weak *- topology of X ∗ . Remark C.1.3. [67] In the above situation the net { x α } α ∈ A ∈ X is convergentwith respect to weak topology if the net { Λ x α } ∈ C is convergent for all Λ ∈ X ∗ .Similarly the net { Λ α } α ∈ A ∈ X ∗ is convergent with respect to weak*-topology ifthe net { Λ α x } ∈ C is convergent for all x ∈ X . C.2 Fourier transformation
There is a norm on Z n given by k ( k , ..., k n ) k = q k + ... + k n . (C.2.1)The space of complex-valued Schwartz functions on Z n is given by S ( Z n ) def = n a = { a k } k ∈ Z n ∈ C Z n | sup k ∈ Z n ( + k k k ) s | a k | < ∞ , ∀ s ∈ N o . (C.2.2)457et T n be an ordinary n -torus. We will often use real coordinates for T n ,that is, view T n as R n / Z n . Let C ∞ ( T n ) be an algebra of infinitely differentiablecomplex-valued functions on T n . There is the bijective Fourier transformations F T : C ∞ ( T n ) ≈ −→ S ( Z n ) ; f b f given by b f ( p ) = F T ( f )( p ) = Z T n e − π ix · p f ( x ) dx (C.2.3)where dx is induced by the Lebesgue measure on R n and · is the scalar producton the Euclidean space R n . The Fourier transformation carries multiplication toconvolution, i.e. c f g ( p ) = ∑ r + s = p b f ( r ) b g ( s ) .The inverse Fourier transformation F − T : S ( Z n ) ≈ −→ C ∞ ( T n ) ; b f f is given by f ( x ) = F − T b f ( x ) = ∑ p ∈ Z n b f ( p ) e π ix · p .There is the C -valued scalar product on C ∞ ( T n ) given by ( f , g ) = Z T n f gdx = ∑ p ∈ Z n b f ( − p ) b g ( p ) .Denote by S ( R n ) be the space of complex Schwartz (smooth, rapidly decreasing)functions on R n . S ( R n ) == n f ∈ C ∞ ( R n ) : k f k α , β ) < ∞ ∀ α = ( α , ..., α n ) , β = ( β , ..., β n ) ∈ Z n + o , k f k α , β = sup x ∈ R n (cid:12)(cid:12)(cid:12) x α D β f ( x ) (cid:12)(cid:12)(cid:12) (C.2.4)where x α = x α · ... · x α n n , D β = ∂∂ x β ... ∂∂ x β n n .The topology on S ( R n ) is given by seminorms k · k α , β .Let F and F − be the ordinary and inverse Fourier transformations given by ( F f ) ( u ) = Z R N f ( t ) e − π it · u dt , (cid:16) F − f (cid:17) ( u ) = Z R N f ( t ) e π it · u dt (C.2.5)458hich satisfy following conditions F ◦ F − | S ( R n ) = F − ◦ F | S ( R n ) = Id S ( R n ) .There is the C -valued scalar product on S ( R n ) given by ( f , g ) = Z R n f gdx = Z R n F f F gdx . (C.2.6)which if F -invariant, i.e. ( f , g ) L ( R n ) = ( F f , F g ) L ( R n ) . (C.2.7)There is the action of Z n on R n such that gx = x + g ; x ∈ R n , g ∈ Z n and T n ≈ R n / Z n . Any f ∈ C ∞ ( T n ) can be regarded as Z n - invariant and smoothfunction on R n . On the other hand if f ∈ S ( R n ) then the series h = ∑ g ∈ Z n g f is point-wise convergent and h is a smooth Z n - invariant function. So we canassume that h ∈ C ∞ ( T n ) . This construction provides a map S ( R n ) → C ∞ ( T n ) , f h = ∑ g ∈ Z n g f . (C.2.8)If b f = F f , b h = F T h then for any p ∈ Z n a following condition holds b h ( p ) = b f ( p ) . (C.2.9)45960 ppendix D Operator algebras
D.1 C ∗ -algebras and von Neumann algebras In this section I follow to [54, 57].
Definition D.1.1. [54] A
Banach *-algebra is a *-algebra A together with a completesubmultiplicative norm such that k a ∗ k = k a k ∀ a ∈ A . If, in addition, A has a unitsuch that k k =
1, we call A a unital Banach *-algebra .A C ∗ -algebra is a Banach *-algebra such that k a ∗ a k = k a k ∀ a ∈ A . (D.1.1) Example D.1.2. [54] If H is a Hilbert space, then the algebra of bounded operators B ( H ) is a C ∗ -algebra. Definition D.1.3. [61] A two sided ideal I in a C ∗ -algebra A is essential if I hasnonzero intersection with every other nonzero ideal A .Alternatively the essential ideal can be given by the following lemma. Lemma D.1.4. [61] An ideal I is essential if and only if aI = { } implies a = . Example D.1.5. [61] Let A def = C ( X ) and let U be an open subset of X. Then I def = ( f ∈ A | f ( x ) = ∀ x ∈ X \ U ) is an essential ideal in A if and only if U is dense in X . Definition D.1.6. [61] A unitization of a C ∗ -algebra A is a C ∗ -algebra B withidentity and an injective *-homomorphism ι : A ֒ → B such that ι ( A ) is an essentialideal of B . 461 xample D.1.7. Suppose A is C ∗ -algebra which has no identity. Then A + = A ⊕ C is a *-algebra with ( a ⊕ λ ) ( b ⊕ µ ) = ( ab + λ b + µ a ) ⊕ λµ , ( a ⊕ λ ) ∗ = a ∗ ⊕ λ .It is proven in [61] that there is the natural unique C ∗ -norm k·k A + on A + such that k a ⊕ k A + = k a ⊕ k A where k·k A is the C ∗ -norm on A . Thus A + is an unital C ∗ -algebra, and the naturalmap A ֒ → A + is a unitization. Definition D.1.8.
Let A be a C ∗ -algebra. The described in the Example D.1.7unitization ι : A ֒ → B is called minimal . Definition D.1.9. [61] A unitization ι : A ֒ → B is called maximal if for everyembedding j : A ֒ → C of A as an essential ideal of a C ∗ -algebra φ : C → B suchthat φ ◦ j = ι . Remark D.1.10.
It is proven in [61] that for any C ∗ -algebra A there unique maximalunitization. Definition D.1.11.
We say that the maximal unitization of A is the multiplier algebra of A and denote it by M ( A ) . Definition D.1.12. [57] Let A be a C ∗ -algebra. The strict topology on the multi-plier algebra M ( A ) is the topology generated by seminorms ||| x ||| a = k ax k + k xa k ,( a ∈ A ). If Λ is a directed set and { a λ ∈ M ( A ) } λ ∈ Λ is a net the we denote by β - lim λ ∈ Λ a λ the limit of { a λ } with respect to the strict topology. If x ∈ M ( A ) anda sequence of partial sums ∑ ni = a i ( n =
1, 2, ...), ( a i ∈ A ) tends to x in the stricttopology then we shall write x = β - lim ∞ ∑ i = a i . Definition D.1.13. [57] Let A be a C ∗ -algebra. A net { u λ } λ ∈ Λ in A + with k u λ k ≤ λ ∈ Λ is called an approximate unit for A if λ < µ implies u λ < u µ and iflim k x − xu λ k = x in A . Then, of course, lim k x − u λ x k = Theorem D.1.14. [57] Each C ∗ -algebra contains an approximate unit. Proposition D.1.15. [1] If B is a C ∗ -subalgebra of A containing an approximate unit forA, then M ( B ) ⊂ M ( A ) (regarding B ′′ as a subalgebra of A ′′ ). efinition D.1.16. [57] A cone M in the positive part of C ∗ -algebra A is said tobe hereditary if 0 ≤ x ≤ y , y ∈ M implies x ∈ M for each x ∈ A . A *-subalgebra B of A is hereditary if B + is hereditary in A + . Lemma D.1.17. [54] Let B be a C ∗ -subalgebra of C ∗ -algebra A. Then B is hereditary inA if and only if bab ′ ∈ B for all b , b ′ ∈ B and a ∈ A. Lemma D.1.18. [54] Let A be a C ∗ -algebra.(i) If L is a closed left ideal in A then L ∩ L ∗ is a hereditary C ∗ -subalgebra of A. Themap L L ∩ L ∗ is the bijection from the set of closed left ideals of A onto the theset of hereditary C ∗ -subalbebras of A.(ii) If L , L are closed left ideals, then L ⊆ L is and only if L ∩ L ∗ ⊂ L ∩ L ∗ .(iii) If B is a hereditary C ∗ -subalgebra of A, then the setL ( B ) = { a ∈ A | a ∗ a ∈ B } is the unique closed left ideal of A corresponding to B. Definition D.1.19. [54] If A is a C ∗ -algebra, its center C is the set of elements of A commuting with every of A . Definition D.1.20. [57] Let H be a Hilbert space. The strong topology on B ( H ) isthe locally convex vector space topology associated with the family of seminormsof the form x
7→ k x ξ k , x ∈ B ( H ) , ξ ∈ H . Definition D.1.21. [57] Let H be a Hilbert space. The weak topology on B ( H ) isthe locally convex vector space topology associated with the family of seminormsof the form x
7→ | ( x ξ , η ) | , x ∈ B ( H ) , ξ , η ∈ H . Theorem D.1.22. [57] Let M be a C ∗ -subalgebra of B ( H ) , containing the identity oper-ator. The following conditions are equivalent: • M = M ′′ where M ′′ is the bicommutant of M; • M is weakly closed; • M is strongly closed.
Definition D.1.23. [57] Any C ∗ -algebra M is said to be a von Neumann algebra ora W ∗ - algebra if M satisfies to the conditions of the Theorem D.1.22.463 efinition D.1.24. [57] We say that a von Neumann algebra M is a factor if thecenter of consists only of scalar multiplies of 1 M . Lemma D.1.25. [57] Let Λ be an increasing in the partial ordering. Let { x λ } λ ∈ Λ be anincreasing of self-adjoint operators in B ( H ) , i.e. λ ≤ µ implies x λ ≤ x µ . If k x λ k ≤ γ for some γ ∈ R and all λ then { x λ } is strongly convergent to a self-adjoint elementx ∈ B ( H ) with k x λ k ≤ γ . For each x ∈ B ( H ) we define the range projection of x (denoted by [ x ] ) as pro-jection on the closure of x H . If M is a von Neumann algebra and x ∈ M then [ x ] ∈ M . Proposition D.1.26. [57] For each element x in a von Neumann algebra M there is aunique partial isometry u ∈ M and positive | x | ∈ M + with uu ∗ = [ | x | ] and x = | x | u. Definition D.1.27.
The formula x = | x | u in the Proposition D.1.26 is said to be the polar decomposition . Definition D.1.28. [57] We say that an element h is a C ∗ -algebra is strictly positive if φ ( h ) > φ on A . Theorem D.1.29. [57] Let A be a C ∗ -algebra. The following conditions are equivalent:(i) There is a strictly positive element h in A + .(ii) There is an element h in A + such that [ h ] = in A ′′ .(iii) There is a countable approximate unit for A. Proposition D.1.30. [6] Let A be a C ∗ -algebra, and h ∈ A + . Then h is strictly positiveif and only if hA is dense in A. Definition D.1.31. [57] A C ∗ -algebra A is said to be simple if 0 and A are itsonly closed ideals. In this book we consider only simple C ∗ -algebras of compactoperators K ( H ) where H = C n ( n ∈ N ) or H = ℓ ( N ) . Theorem D.1.32. [57] For each C ∗ -algebra A there is a dense hereditary ideal K ( A ) ,which is minimal among dense ideals. Definition D.1.33.
The ideal K ( A ) from the theorem D.1.32 is said to be the Ped-ersen’s ideal of A . Theorem D.1.34. [1] Let π be a surjective morphism between separable C ∗ -algebras Aand B. Then π extends to a surjective morphism of M ( A ) onto M ( B ) . emark D.1.35. The Theorem D.1.34 can be regarded as noncommutative Tietze’sextension theorem A.1.15, see [1] for details.
D.1.36. [57] Let H be a Hilbert space and M a subset of B ( H ) sa (where B ( H ) sa ⊂ B ( H ) is the R -space of self-adjoint operators). The monotone sequential closure of M is defined as the smallest class B ( M ) in B ( H ) sa that contains M and containsthe strong limit of each monotone (increasing or decreasing) sequence of elementsfrom B ( M ) . Lemma D.1.37. [57] Each countable subset of B ( M ) lies in the monotone sequentialclosure of a separable subset of M. Theorem D.1.38. [57] Let A be a C ∗ -subalgebra of B ( H ) . Then B ( A sa ) is the self-adjoint part of a C*-algebra. Definition D.1.39. [57] Let A be a C ∗ -algebra. We define the enveloping Borel*-algebra of A to be the C ∗ -algebra B ( A ) = B ( A sa ) + i B ( A sa ) (D.1.2)the monotone sequential closure being taken on the universal Hilbert space for A . D.2 States and representations
Definition D.2.1. [57] A state of a C ∗ -algebra A is a positive functional of normone. The set of states of A is denoted by SA (or just S if no confusion can arise). Definition D.2.2. [54] We say a state τ on a C ∗ -algebra A is pure if it has theproperty that whenever p is a positive linear functional on A such that p ≤ τ ,necessarily there is a number t ∈ [
0, 1 ] such that p = t τ . The set of pure states on A is denoted by PS ( A ) . D.2.1 GNS construction
Any state τ of C ∗ -algebra A induces a faithful GNS representation [54]. Thereis a C -valued product on A given by ( a , b ) def = τ ( a ∗ b ) .This product induces a product on A / I τ where I τ def = { a ∈ A | τ ( a ∗ a ) = } . So A / I τ is a pre-Hilbert space. Denote by L ( A , τ ) the Hilbert completion of A / I τ .465he Hilbert space L ( A , τ ) is a space of a GNS representation A → B (cid:0) L ( A , τ ) (cid:1) (or equivalently A × L ( A , τ ) → ( L ( A , τ ) ) which comes from the Hilbert normcompletion of the natural action A × A / I τ → A / I τ . The natural map A → A / I τ induces the homomorphism of left A -modules f τ : A → L ( A , τ ) , a a + I τ (D.2.1)such that f τ ( A ) is a dense subspace of L ( A , τ ) . Theorem D.2.3. [57] For each positive functional τ on a C ∗ -algebra A there is a cyclicrepresentation π τ : A → L ( A , τ ) with a cyclic vector ξ τ ∈ L ( A , τ ) such that ( π τ ( x ) ξ τ , ξ τ ) = τ ( a ) for all x ∈ A. Definition D.2.4.
The given by the Theorem D.2.3 representation is said to be a
GNS reprerentation . D.2.5. If X is a second-countable locally compact Hausdorff space then from theTheorem A.1.17 it follows that C ( X ) is a separable algebra. Therefore C ( X ) hasa state τ such that associated with τ GNS representation [54] is faithful. From [8]it follows that the state τ can be represented as the following integral τ ( a ) = Z X a d µ (D.2.2)where µ is a positive Radon measure. In analogy with the Riemann integration,one can define the integral of a bounded continuous function a on X . There is a C -valued product on C ( X ) given by ( a , b ) = τ ( a ∗ b ) = Z X a ∗ b d µ ,hence C ( X ) is a phe-Hilbert space. Denote by L ( C ( X ) , τ ) or L ( X , µ ) theHilbert space completion of C ( X ) . D.2.2 Irreducible representations
Theorem D.2.6. [57] Let π : A → B ( H ) be a nonzero representation of C ∗ -algebra A.The following conditions are equivalent:(i) There are no non-trivial A-subspaces for π .(ii) The commutant of π ( A ) is the scalar multipliers of 1. iii) π ( A ) is strongly dense in B ( H ) .(iv) For any two vectors ξ , η ∈ H with ξ = there is a ∈ A such that π ( a ) ξ = η .(v) Each nonzero vector in H is cyclic for π ( A ) .(vi) A → B ( H ) is spatially equivalent to a cyclic representation associated with a purestate of A. Definition D.2.7.
Let A → B ( H ) be a nonzero representation of C ∗ -algebra A .The representation is said to be irreducible if it satisfies to the Theorem D.2.6. Remark D.2.8.
From the condition (i) of the Theorem D.2.6 it turns out that ir-reducibility is the categorical property, i.e. if there is the equivalence betweencategory of representations of C ∗ -algebras then any irreducible representation ismapped to irreducible one. The equivalence between category of representationscorresponds to the strong Morita equivalence (cf. D.6). Definition D.2.9. [57] Let A be a C ∗ -algebra. We say that two representations π : A → B ( H ) and π : A → B ( H ) are spatially equivalent (or unitary equivalent )if there is an isometry u of H onto H such that u π ( a ) u ∗ = π ( a ) for all a ∈ A .By the spectrum of A we understand the set ˆ A of spatially equivalence classes ofirreducible representations. For any x ∈ ˆ A we denote by rep x : A → B ( H ) OR rep Ax : A → B ( H ) (D.2.3)a representation which corresponds to x . Sometimes we use alternative notationof the spectrum A ∧ def = ˆ A . Definition D.2.10. [57] Let A be a C ∗ -algebra, and let S be the state space of A . For any s ∈ S there is an associated representation π s : A → B ( H s ) . Therepresentation L s ∈ S π s : A → L s ∈ S B ( H s ) is said to be the universal representation .The universal representation can be regarded as A → B ( L s ∈ S H s ) . Definition D.2.11. [57] Let A be a C ∗ -algebra, and let A → B ( H ) be the universalrepresentation A → B ( H ) . The strong closure of π ( A ) is said to be the envelop-ing von Neumann algebra or the enveloping W ∗ -algebra of A . The enveloping vonNeumann algebra will be denoted by A ′′ . Proposition D.2.12. [57] The enveloping von Neumann algebra A ′′ of a C ∗ -algebra Ais isomorphic, as a Banach space, to the second dual of A, i.e. A ′′ ≈ A ∗∗ . heorem D.2.13. [57] For each non-degenerate representation π : A → B ( H ) of aC ∗ -algebra A there is a unique normal morphism of A ′′ onto π ( A ) ′′ which extends π . Definition D.2.14.
Let A be a C ∗ -algebra. The primitive spectrum is the space of theprimitive ideals of A , we shall denote the primitive spectrum by ˇ A . Remark D.2.15.
There is the natural surjective mapˆ A → ˇ A (D.2.4)which maps an irreducible representation to its kernel. D.2.16.
For each set F in ˇ A define a closed idealker ( F ) = \ t ∈ F t . (D.2.5)For each subset I of A define a sethull ( I ) = (cid:8) t ∈ ˇ A | I ⊂ t (cid:9) (D.2.6)Using a surjective map ˆ A → ˇ A one can define the hull ( I ) ⊂ ˆ A as the preimage ofhull ( I ) in ˇ A Theorem D.2.17.
The class { hull ( I ) | I ⊂ A } form the closed sets for a topology on ˇ A.There is a bijective order preserving isomorphism between open sets in this topology andthe closed ideals in A.
Definition D.2.18.
The topology on ˇ A defined in the Theorem D.2.17 is called bethe Jacobson topology . The topology on ˆ A induced by the surjecive map ˆ A → ˇ A and the Jacobson topology on ˇ A is also called to be the Jacobson topology . Remark D.2.19.
In the following text we consider the only Jacobson topology onboth ˆ A and ˇ A . Proposition D.2.20. [57] If B is a hereditary C ∗ -subalbebra of A then there is the naturalhomeomorphism between ˆ A \ hull ( B ) and ˆ B, where hull ( B ) = (cid:8) x ∈ ˆ A | rep x ( B ) = { } (cid:9) . Corollary D.2.21. [54] Every closed ideal of C ∗ -albebra is a hereditary C ∗ -subalbebra. Remark D.2.22.
From the Theorem D.2.17, the Proposition D.2.20 and the Corol-lary D.2.21 any closed ideal I corresponds of C ∗ -algebra A corresponds to the opensubset U ⊂ ˆ A such that there is the natural homeomorphism ˆ I ∼ = U . Moreover if I is an essential ideal then U is dense subset of ˆ A .468 roposition D.2.23. [54] Let B be a C ∗ -algebra. For each positive functional φ on Bthere is a norm preserving extension of φ to a positive functional on A. If B is hereditarythis extension is unique. Proposition D.2.24. [57] If B is a C ∗ -subalgebra of A then each pure state of B can beextended to a pure state of A. Proposition D.2.25. [57] If B is a C ∗ -subalgebra of A then for each irreducible repre-sentation ρ : B → B ( H ′ ) of B there is an irreducible representation A → B ( H ) of Awith a closed subspace H ⊂ H such that ρ B : B → B ( H ) is spatially equivalent to ρ : B → B ( H ′ ) . Theorem D.2.26. (Dauns Hofmann) [57] For each C ∗ -algebra A there is the naturalisomorphism from the center of M ( A ) onto the class of bounded continuous functions on ˇ A. Theorem D.2.27. [61] Suppose that A is a C ∗ -algebra and that a ∈ A.(a) The function x
7→ k rep x ( a ) k is lower semi-continuous on ˆ A; that is (cid:8) x ∈ ˆ A | k rep x ( a ) k ≤ k (cid:9) is closed for all k ≥ .(b) For each k > , (cid:8) x ∈ ˆ A | k rep x ( a ) k ≥ k (cid:9) is compact. Proposition D.2.28. [57] The subset n ˇ A of ˇ A corresponding to irreducible represen-tations of A with finite dimension less or equal to n is closed. The set n ˇ A \ n − ˇ A ofn-dimensional representations is a Hausdorff space in its relative topology.
Definition D.2.29. [62] Let A be a bounded self-adjoint operator and Ω a Borelset of R . If 1 Ω is the characteristic function of Ω then P Ω = Ω ( A ) is called the spectral projection of A . We also write χ Ω instead 1 Ω . D.2.30. [61] Let Prim A be the space of primitive ideals of A . The topology onPrim A always determines the ideal structure of A : the open sets U in Prim A arein one-to-one correspondence with the ideals A | U def = \ { P ∈ Prim A | P / ∈ U } (D.2.7)and there are natural homeomorphisms P P ∩ U of U onto Prim A | U , and P P / A U of ( Prim A ) \ U onto Prim A / A U . When Prim A is a (locally compact)469ausdorff space T , we can localize at a point t ‚ that is, examine behavior in aneighbourhood of t ‚ either by looking at the ideal A U corresponding to an openneighbourhood U of t , or by passing to the quotient A | F def = A / (cid:0) A Prim A \ F (cid:1) (D.2.8)corresponding to a compact neighborhood F of t . Remark D.2.31.
Clearly there are natural injective A | U → A and surjective A → A | F *-homomorphisms, Remark D.2.32.
The notations D.2.7 and D.2.8 slightly differs from [61]. Here wewrite A | U and A | F instead of A U and A F respectively. Definition D.2.33. [57] Let A be a C ∗ -algebra with the spectrum ˆ A . We choosefor any t ∈ ˆ A a pure state φ t and associated representation π t : A → B ( H t ) . Therepresentation π a = M t ∈ ˆ A π t on M t ∈ ˆ A H t (D.2.9)is called the (reduced) atomic representation of A . Any two atomic representationsare unitary equivalent and any atomic representation of A is faithful (cf. [57]). Remark D.2.34. [57] The atomic representation π a : A → B ( H a ) is faithful. Corollary D.2.35. [57] For each C ∗ -algebra A the atomic representation is faithful onthe enveloping Borel *-algebra B ( A ) of A (cf. Definition D.1.39). D.3 Inductive limits of C ∗ -algebras Definition D.3.1.
An injective *-homomorphism φ : A ֒ → B of unital C ∗ -algebrasis said to be unital if and only if φ ( A ) = B . (cf. [71]). Remark D.3.2.
In the cited text [71] the word "principal" used instead "unital". Inthis book all entries of the word "principal *-homomorphism" are replaced with"unital *-homomorphism".
Definition D.3.3. [71] Let Λ be an increasingly directed set and A λ be a C ∗ -algebrahaving an identity 1 λ associated with λ ∈ Λ If there exists a C ∗ -algebra A with theidentity 1 and a unital isomorphism f λ of A λ into A for every λ ∈ Λ such that f µ (cid:0) A µ (cid:1) ⊂ f ν ( A ν ) if ν < µ ; µ , ν ∈ Λ and that the join of f λ ( A λ ) ( λ ∈ Λ ) is uniformly dense in A , A is called the C ∗ - inductive limit of A λ , and is denoted by C ∗ - lim −→ Λ A λ or C ∗ - lim −→ A λ heorem D.3.4. [71] Let { A λ ) } λ ∈ Λ be a family of C ∗ -algebras where Λ denotes anincreasingly directed set. If, for every µ , ν with µ < ν , there exists a unital injective*-homomorphism f µν : A µ → A ν satisfyingf µν = f µλ ◦ f λν where µ < λ < ν . then there exists the C ∗ -inductive limit of A λ . D.3.5.
Let Ω and Ω λ be state spaces of A and A λ respectively. When A is a C ∗ - inductive limit of { A λ } , every state τ of A defines a state τ λ on A λ . Then,for every µ , ν ∈ Λ such as µ < ν , we put f ∗ µν the conjugate mapping of theprincipal isomorphism f µν of A µ into A ν which maps Ω ν onto Ω ν has the followingproperties τ µ = f ∗ µν ( τ ν ) (D.3.1) f ∗ µν = f ∗ µλ ◦ f ∗ λν if µ < λ < ν (D.3.2)Conversely, a system of states { τ λ ∈ Ω λ } λ ∈ Λ which satisfies the condition (D.3.1)defines a state on A since every positive bounded linear functional on the algebraicinductive limit A of { A λ } , is uniquely extended over A . Proposition D.3.6. [71] Let a C ∗ -algebra A be a C ∗ -inductive limit of { A λ ) } λ ∈ Λ , and Λ ′ be a cofinal subset in Λ , then A is the C ∗ -inductive limit of { A λ ′ ) } λ ′ ∈ Λ ′ . Theorem D.3.7. [71] If a C ∗ -algebra A is a C ∗ -inductive limit of A λ ( λ ∈ Λ ), the statespace Ω of A is homeomorphic to the projective limit of the state spaces Ω λ of A λ . Corollary D.3.8. [71] If a commutative C ∗ -algebra A is a C ∗ -inductive limit of thecommutative C ∗ -algebras A λ ( λ ∈ Λ ), the spectrum X of A is the projective limit ofspectrums X λ of A λ ( λ ∈ Λ ). D.4 Hilbert modules and compact operators
Definition D.4.1. [61] A left A -module X Banach A - module if X is a Banach spaceand k a · x k ≤ k a k k x k for all a ∈ A and x ∈ X . A Banach A -module is nondegenerate is nondegenerate if span { a · x | a ∈ A x ∈ X } is dense in X . We then have a λ · x → x whenever x ∈ X and { a λ } is a bounded approximate identity for A . Proposition D.4.2. [61] Suppose that X is a nondegenerate Banach A-module. Thenevery element of X is of the form a · x for some a ∈ A and x ∈ X . efinition D.4.3. [Paschke [55], Rieffel [63]] Let A be a C ∗ -algebra. A pre-HilbertA-module is a right B -module X (with a compatible C -vector space structure),equipped with a conjugate-bilinear map (linear in the second variable) h− , −i A : X × X → A satisfying(a) h x , x i A ≥ x ∈ X ;(b) h x , x i A = x = h x , y i A = h y , x i ∗ A for all x , y ∈ X ;(d) h x , y · a i A = h x , y i B · a for all x , y ∈ X , a ∈ A .The map h− , −i A is called a A-valued inner product on X . Following equation k x k = k h x , x i A k (D.4.1)defines a norm on X . If X is complete with respect to this norm, it is called a Hilbert A-module . Definition D.4.4. [61] A Hilbert A -module X A is a full Hilbert A -module if theideal I def = span { h ξ , η i A | ξ , η ∈ X A } is dense in A . Remark D.4.5.
Suppose that A is a C ∗ -algebra and that p is a projection in A (or M ( A ) ). Following facts are proven in the Example 2.12 of [61]. Then Ap def = { ap | a ∈ A } is a Hilbert pAp module with inner product h ap , bp i pAp def = pa ∗ bp (D.4.2)This Hilbert module is full. Similarly, pA is a Hilbert A -module which is full overthe ideal ApA def = span { apb | a , b ∈ A } generated by p , and Ap is itself a full leftHilbert ApA -module.
Remark D.4.6.
For any C ∗ -pre-Hilbert X module, or more more precisely, for anysesquilinear form h· , ·i the polarization equality h ξ , η i = ∑ k = i k D ξ + i k η , ξ + i k η E (D.4.3)is obviously satisfied for all ξ , η ∈ X . If H is a Hilbert space then there is thefollowing analog of the identity (D.4.3) ( ξ , η ) H = ∑ k = i k (cid:13)(cid:13) ξ + i k η (cid:13)(cid:13) ∀ ξ , η ∈ X (D.4.4)472 efinition D.4.7. An A -linear map L : X → X is said to be adjointable if there is L ∗ : X → X such that h ξ , L η i A = h L ∗ ξ , η i A ∀ ξ , η ∈ X A Denote by L A ( X ) the C ∗ -algebra of adjointable maps. Definition D.4.8. [57] If X is a C ∗ Hilbert A -rigged module then denote by θ ξ , ζ ∈L B ( X ) such that θ ξ , ζ ( η ) = ζ h ξ , η i X , ( ξ , η , ζ ∈ X ) (D.4.5)Norm closure of a generated by such endomorphisms ideal is said to be the algebraof compact operators which we denote by K ( X ) . The K ( X ) is an ideal of L A ( X ) .Also we shall use following notation ξ ih ζ def = θ ξ , ζ . Theorem D.4.9. [6] Let X A is a Hilbert A-module. The C ∗ -algebra of adjointable maps L A ( X A ) is naturally isomorphic to the algebra M ( K ( X A )) of multiplies of compactoperators K ( X A ) . Remark D.4.10.
The text of the Theorem D.4.9 differs from the text of the Theorem13.4.1 [6]. It is made for simplicity reason.
Definition D.4.11. (cf. [51]) The direct sum of countable number of copies of aHilbert module X is denoted by ℓ ( X ) . The Hilbert module ℓ ( A ) is said tobe the standard Hilbert A-module over A . If A is unital then ℓ ( A ) possesses thestandard basis (cid:8) ξ j (cid:9) j ∈ N . ℓ ( A ) = ( { a n } n ∈ N ∈ A N | ∞ ∑ n = a ∗ n a n < ∞ ) , h{ a n } , { b n }i ℓ ( A ) = ∞ ∑ n = a ∗ n b n . (D.4.6) Theorem D.4.12.
Kasparov Stabilization or Absorption Theorem. [6] If X A is acountably generated Hilbert A-module, then X A ⊕ ℓ ( A ) ∼ = ℓ ( A ) . Lemma D.4.13. [51] Let X be a finitely generated Hilbert submodule in a Hilbert moduleY over a unital C ∗ -algebra. Then Y is an orthogonal direct summand in X. Definition D.4.14. [69] Let A be an unital C ∗ -algebra a bounded A operator K : ℓ ( A ) → ℓ ( A ) is compact (in the sense of Mishchenko) if we havelim n → ∞ (cid:13)(cid:13)(cid:13) K | L ⊥ n (cid:13)(cid:13)(cid:13) = L n = ⊕ nj = A ⊂ ℓ ( A ) = ⊕ ∞ j = A , L ⊥ n = ⊕ ∞ j = n + A ⊂ ℓ ( A ) = ⊕ ∞ j = A .473 emark D.4.15. [69] Any compact (in the sense of Mishchenko) operator is com-pact in the sense of the Definition D.4. Lemma D.4.16. [69] Let K : ℓ ( A ) → ℓ ( A ) be compact operator (in the sense ofMishchenko) Let p n be the projection on L n along L ⊥ n . The following properties are equiv-alent.(i) lim n → ∞ (cid:13)(cid:13)(cid:13) K | L ⊥ n (cid:13)(cid:13)(cid:13) = .(ii) lim n → ∞ k K p n − K k = .(iii) lim n → ∞ k p n K − K k = . D.5 Hermitian modules and functors
In this section we consider an analogue of the A ⊗ B − : B M → A M functor oran algebraic generalization of continuous maps. Following text is in fact a citationof [64]. Definition D.5.1. [64] Let B be a C ∗ -algebra. By a (left) Hermitian B-module wewill mean the Hilbert space H of a non-degenerate *-representation A → B ( H ) .Denote by Herm ( B ) the category of Hermitian B -modules. D.5.2. [64] Let A , B be C ∗ -algebras. In this section we will study some generalmethods for construction of functors from Herm ( B ) to Herm ( A ) . Definition D.5.3. [64] Let A and B be C ∗ -algebras. By a Hermitian B-rigged A-module we mean a C ∗ -Hilbert B module, which is a left A -module by means of*-homomorphism of A into L B ( X ) . D.5.4.
Let X be a Hermitian B -rigged A -module. If V ∈ Herm ( B ) then we canform the algebraic tensor product X ⊗ B alg V , and equip it with an ordinary pre-inner-product which is defined on elementary tensors by h x ⊗ v , x ′ ⊗ v ′ i = hh x ′ , x i B v , v ′ i V . (D.5.1)Completing the quotient X ⊗ B alg V by subspace of vectors of length zero, we obtainan ordinary Hilbert space, on which A acts by a ( x ⊗ v ) = ax ⊗ v (D.5.2)474o give a *-representation of A . We will denote the corresponding Hermitian mod-ule by X ⊗ B V . The above construction defines a functor X ⊗ B − : Herm ( B ) → Herm ( A ) if for V , W ∈ Herm ( B ) and f ∈ Hom B ( V , W ) we define f ⊗ X ∈ Hom A ( V ⊗ X , W ⊗ X ) on elementary tensors by ( f ⊗ X )( x ⊗ v ) = x ⊗ f ( v ) . Wecan define action of B on V ⊗ X which is defined on elementary tensors by b ( x ⊗ v ) = ( x ⊗ bv ) = xb ⊗ v .The complete proof of above facts is contained in the Proposition 2.66 [61] Definition D.5.5.
Let us consider the situation D.5.4. The functor X ⊗ B − : Herm ( B ) → Herm ( A ) is said to be the Rieffel correspondence . If π : B → B ( H B ) a representationthen the representation ρ : A → B ( H A ) given by the functor X ⊗ B − : Herm ( B ) → Herm ( A ) is said to be the induced representation . We use following notation ρ = X − Ind AB π or ρ = Ind AB π or ρ = X − Ind π (D.5.3)(cf. Chapter 2.4 of [61]). Remark D.5.6. If A is a C ∗ -algebra and A → B ( H ) is a representation then ( a ξ , η ) H = ( ξ , a ∗ η ) H for each a ∈ A and ξ , η . Taking into account (c) of theDefinition D.4.3 the equation (D.5.1) can be rewritten by following way h x ⊗ v , x ′ ⊗ v ′ i = h v , h x , x ′ i B v ′ i V . (D.5.4)The equation (D.5.4) is more convenient for our purposes than (D.5.1) one. D.6 Strong Morita equivalence for C ∗ -algebras The notion of the strong Morita equivalence was introduced by Rieffel.
Definition D.6.1. [Rieffel [63–65]] Let A and B be C ∗ -algebras. By an A-B-equivalencebimodule (or
A-B-imprimitivity bimodule ) we mean an ( B , A ) -bimodule which isequipped with A - and B -valued inner products with respect to which X is a rightHilbert A -module and a left Hilbert B -module such that(a) h x , y i B z = x h y , z i A for all x , y , z ∈ X ;(b) h X , X i A spans a dense subset of A and h X , X i B spans a dense subset of B ,i.e. X is a full Hilbert A -module and a full Hilbert B -module.We call A and B strongly Morita equivalent if there is an A - B -equivalence bimodule.475 xample D.6.2. [61] Let p be a projection in M ( A ) . We saw in the Remark D.4.5that Ap is a full right Hilbert pAp -module and a full left Hilbert ApA -module.(Recall that
ApA denotes the ideal generated by p , which is the closed span of theset ApA .) Thus Ap is an ApA - pAp -imprimitivity bimodule. Theorem D.6.3. [61, 63] Let X be an A-B equivalence bimodule. Then given by D.5.5functor X ⊗ B − induces an equivalence between the category of Hermitian B modules andthe category of Hermitian A-modules, the inverse being given by ˜ X ⊗ A − . Remark D.6.4. If X is an A – B -imprimitivity bimodule and e X is the conjugatevector space then e X is a B – A -imprimitivity bimodule (cf. [61]). Theorem D.6.5. [61] Suppose that X is an A–B-imprimitivity bimodule, and π , ρ are nondegenerate representations of B and A, respectively. Then e X − Ind ( X − Ind π ) is naturally unitarily equivalent to π , and X − Ind (cid:16) e X − Ind ρ (cid:17) is naturally unitarilyequivalent to ρ . Corollary D.6.6. [61] If X is an A–B-imprimitivity bimodule, then the inverse of theRieffel correspondence X − Ind is e X − Ind . Corollary D.6.7. [61] Suppose that X is an A–B-imprimitivity bimodule, and that π is a nondegenerate representation of B. Then e X − Ind π is irreducible if and only if π isirreducible. Corollary D.6.8.
If X is A-B-imprimitivity bimodule then the Rieffel correspondencerestricts to a homeomorphism h X : Prim B ≈ −→ Prim A. Definition D.6.9.
In the situation of the Corollary D.6.8 the homeomorphism h X :Prim B ≈ −→ Prim A is said to be the Rieffel homeomorphism . D.7 Operator spaces and algebras
Definition D.7.1. [7] A concrete operator space is a (usually closed) linear subspace X of B ( H , H ) , for Hilbert spaces H , H (indeed the case H = H usuallysuffices, via the canonical inclusion B ( H , H ) ⊂ B ( H ⊕ H ) . Definition D.7.2. [7] A concrete operator algebra is a closed subalgebra of B ( H ) , forsome Hilbert space H . Remark D.7.3.
There are abstract definitions of operator algebras and and an oper-ator spaces . It is proven that the classes of operator algebras and operator spacesare equivalent to the class of concrete operator algebras and spaces respectively(cf. [7].) 476 efinition D.7.4. [7] (Unital operator spaces). We say that an operator space X is unital if it has a distinguished element usually written as e or 1, called the identity of X , such that there exists a complete isometry u : X ֒ → A into a unital C ∗ -algebrawith u ( e ) = A . A unital-subspace of such X is a subspace containing e . D.7.5. [7] (Completely bounded maps). Suppose that X and Y are operator spacesand that u : X → Y is a linear map. For a positive integer n , we write u n for theassociated map [ x jk ] (cid:2) u (cid:0) x jk (cid:1)(cid:3) from M n ( X ) to M n ( Y ) . This is often called the( n th ) amplification of u , and may also be thought of as the map Id M n ( C ) ⊗ u on M n ( C ) ⊗ X . Similarly one may define u m , n : M m , n ( X ) → M m , n ( Y ) . If each matrixspace M n ( X ) and M n ( Y ) has a given norm k·k n , and if un is an isometry for all n ∈ N , then we say that u is completely isometric , or is a complete isometry . Similarly, u is completely contractive (resp. is a complete quotient map ) if each u n is a contraction(resp. takes the open ball of M n ( X ) onto the open ball of M n ( Y ) ). A map u iscompletely bounded if k u k cb def = sup (cid:8) (cid:13)(cid:13)(cid:2) u (cid:0) x jk (cid:1)(cid:3)(cid:13)(cid:13) (cid:12)(cid:12) (cid:13)(cid:13)(cid:2) x jk (cid:3)(cid:13)(cid:13) < ∀ n ∈ N (cid:9) Compositions of completely bounded maps are completely bounded, and one hasthe expected relation k u ◦ v k cb ≤ k u k cb k v k cb . If u : X → Y is a completelybounded linear bijection, and if its inverse is completely bounded too, then we saythat u is a complete isomorphism. In this case, we say that X and Y are completelyisomorphic and we write X ≈ Y . Remark D.7.6. [7] Any *-homomorphism u : A → B of C ∗ -algebras is injective ifand only if u is completely isometric (cf. [7]). D.7.7. [7] (Extensions of operator spaces). An extension of an operator space X isan operator space Y , together with a linear completely isometric map j : X ֒ → Y .Often we suppress mention of j , and identify X with a subspace of Y . We say that Y is a rigid extension of X if Id Y is the only linear completely contractive map Y → Y which restricts to the identity map on j ( X ) . We say Y is an essential extension of X if whenever u : Y → Z is a completely contractive map into another operatorspace Z such that u ◦ j is a complete isometry, then u is a complete isometry. Wesay that ( Y , j ) is an injective envelope of X if Y is injective, and if there is no injectivesubspace of Y containing j ( X ) . Lemma D.7.8.
Let ( Y , j ) be an extension of an operator space X such that Y is injective.The following are equivalent:(i) Y is an injective envelope of X, ii) Y is a rigid extension of X,(iii) Y is an essential extension of X. Definition D.7.9. [7] Thus we define a C ∗ - extension of a unital operator space X tobe a pair ( A , j ) consisting of a unital C ∗ -algebra A , and a unital complete isometry j : X → A , such that j ( X ) generates A as a C ∗ -algebra. Corollary D.7.10. (i) If X is a unital operator space (resp. unital operator algebra, approximately unitaloperator algebra), then there is an injective envelope ( I ( X ) , j ) for X, such that I ( X ) is a unital C ∗ -algebra and j is a unital map (resp. j is a unital homomorphism, j isa homomorphism).(ii) If A is an approximately unital operator algebra, and if ( Y , j ) is an injective envelopefor A + , then ( Y , j | A ) is an injective envelope for A.(iii) If A is an approximately unital operator algebra which is injective, then A is a unitalC ∗ -algebra. Theorem D.7.11. [7] (Arveson‚ Hamana) If X is a unital operator space, then thereexists a C ∗ -extension ( B , j ) of X with the following universal property: Given any C ∗ -extension ( A , k ) of X, there exists a (necessarily unique and surjective) ∗ -homomorphism π : A → B, such that π ◦ k = j. Definition D.7.12. [7] If X is a unital operator space, then the given by the Theo-rem D.7.11 pair ( B , j ) is said to be the C ∗ - envelope of X . Denote by C ∗ e ( X ) def = B . (D.7.1) Definition D.7.13. [7] The C ∗ - envelope of a nonunital operator algebra A is a pair ( B , j ) , where B is the C ∗ -subalgebra generated by the copy j ( A ) of A inside a C ∗ -envelope ( C ∗ e ( A + ) , j ) of the unitization A + of A . Denote by C ∗ e ( A ) def = B . (D.7.2) Theorem D.7.14. (Haagerup, Paulsen, Wittstock). Suppose that X is a subspace of a C ∗ -algebra B, that H and H are Hilbert spaces, and that u : X → B ( H , H ) is a completelybounded map. Then there exists a Hilbert space H , a ∗ -representation π : B → B ( H ) (which may be taken to be unital if B is unital), and bounded operators S : H → H andT : H → H , such that u ( x ) = S π ( x ) T for all x ∈ X. Moreover this can be done with k S k k T k = kk cb In particular, if ϕ ∈ Ball ( X ∗ ) , and if B is as above, then there exist H ,p as above, and unit vectors ξ , η ∈ H , with ϕ = ( π ( x ) ξ , η ) H on X. .8 C ∗ -algebras of type I D.8.1 Basic facts
Let A be a C ∗ -algebra. For each positive x ∈ A + and irreducible representation π : A → B ( H ) the (canonical) trace of π ( x ) depends only on the equivalence classof π , so that we may define a functionˆ x : ˆ A → [ ∞ ] ;ˆ x ( t ) = tr ( π ( x )) (D.8.1)where ˆ A is the spectrum of A (the space of equivalence classes of irreduciblerepresentations) and tr is the trace of the operator. From Proposition 4.4.9 [57]it follows that ˆ x is lower semicontinuous function in the Jacobson topology (cf.Definition D.2.18). Lemma D.8.1.
Let A be a C ∗ -algebra whose spectrum X is paracompact (and henceHausdorff). Suppose that {U α } α ∈ A is a locally finite cover of X = ∪ α ∈ A U α by relativelycompact open sets, { φ α } is a partition of unity subordinate to {U α } , and { a α } is a setof elements in A parameterized by A . If there is a function f ∈ C ( X ) such that k rep x ( a α ) k ≤ f ( x ) for all x and α , then there is a unique element a of A such that rep x ( a ) = ∑ α ∈ A φ α rep x ( a α ) for every x ∈ X ; we write a = ∑ α ∈ A φ α a α . Definition D.8.2. [57] We say that element x ∈ A + has continuous trace if ˆ x ∈ C b ( ˆ A ) . We say that C ∗ -algebra has continuous trace if a set of elements with con-tinuous trace is dense in A + . Remark D.8.3.
If a C ∗ -algebra A has continuous trace then for simplicity we saythat A is continuous-trace C ∗ - algebra ,There are alternative equivalent definitions continuous-trace C ∗ -algebras. Oneof them is presented below. Definition D.8.4. [61] A continuous-trace C ∗ - algebra is a C ∗ -algebra A with Haus-dorff spectrum X such that, for each x ∈ X there are a neighborhood U of t and a ∈ A such that rep x ( a ) is a rank-one projection for all x ∈ U . Definition D.8.5. [57] A positive element in C ∗ - algebra A is Abelian if subalgebra xAx ⊂ A is commutative. Definition D.8.6. [57] We say that a C ∗ -algebra A is of type I if each non-zeroquotient of A contains a non-zero Abelian element. If A is even generated (as C ∗ -algebra) by its Abelian elements we say that it is of type I .479 roposition D.8.7. [57] A positive element x in C ∗ -algebra A is Abelian if dim π ( x ) ≤ for every irreducible representation π : A → B ( H ) . Definition D.8.8. [57] A C ∗ -algebra is called liminaly (or CCR ) if ρ ( A ) = K foreach irreducible representation ρ : A → B ( H ) . Theorem D.8.9. [57] Let A be a C ∗ -algebra of type I. Then(i) K ⊂ π ( A ) for each irreducible representation π : A → B ( H ) of A.(ii) ˆ A = ˇ A. Proposition D.8.10. [57] Each hereditary C ∗ -subalgebra and each quotient of a C ∗ -algebra which is of type I or has continuous trace is of type I or has continuous trace,respectively. Corollary D.8.11. [57] Any CCR C ∗ -algebra is of type I. Proposition D.8.12.
Suppose that A and B are C ∗ -algebras and X is an A − B-imprimitivitybimodule.(a) If h X : Prim B ≈ −→ Prim
A is the Rieffel homeomorphism and f ∈ C b ( Prim A ) thenf · x = x · ( f ◦ h X ) for all x ∈ X .(b) If A and B have Hausdorff spectrum X , then X is an imprimitivity bimodule over X if and only if f · x = x · f for all x ∈ X and f ∈ C ( X ) . Proposition D.8.13. [61] A C ∗ -algebra A with Hausdorff spectrum X has continuoustrace if and only if A is locally Morita equivalent to C ( X ) , in the sense that each x ∈ X has a compact neighborhood V such that A | V is Morita equivalent to C ( V ) over V . D.8.14.
For our research it is important the proof of the Proposition D.8.13. Hereis the fragment of the proof described in [61]. Conversely, suppose that A hascontinuous trace, and fix x ∈ X . Choose a compact neighbourhood V of x and p ∈ A such that rep x ( p ) is a rank-one projection for all x ∈ V Then p | V is a projection in A | V , and A | V p | V is an A | V − p | V A | V p | V -imprimitivity bi-module (cf. Example D.6.2). Note that the map f f p | V is an isomorphismof C ( V ) into p | V A | V p | V . On the other hand, if a ∈ A and x ∈ V , then rep x ( p ) is a rank-one projection in the algebra rep x ( A ) of compact operators, and rep x ( pap ) = rep x ( p ) rep x ( a ) ( pap ) rep x ( p ) must be a scalar multiple f a ( x ) rep x ( p ) of x rep x ( p ) . We claim that f a is continuous, so that f f p | V is an isomor-phism of C ( V ) onto p | V A | V p | V . Well, for x , y ∈ V we have | f a ( x ) − f a ( y ) | = k rep x ( f a ( x ) − f a ( y )) p k = k rep x ( pap − f a ( y ) p ) k .480ince ˆ A is Hausdorff, for fixed y the right-hand side is a continuous function of x by Lemma D.2.27; since it vanishes at y , the left-hand side goes to 0 as x → y In other words, f a is continuous, as claimed, and A | V p | V is an p | V A | V p | V - C ( V ) -imprimitivity V -bimodule. Because the actions of C ( V ) on the left and right of A | V p | V coincide, it is actually an p | V A | V p | V - C ( V ) -imprimitivity bimodule byProposition D.8.12. Proposition D.8.15. [61]Suppose that A is a continuous-trace C ∗ -algebra with paracom-pact spectrum X . Then there are compact subsets {V α ⊂ X } α ∈ A whose interiors form acover {U α } of X , such that for each α ∈ A , there is an A | V α − C ( V α ) -imprimitivitybimodule X α . Proposition D.8.16. [57] If A is a C ∗ -algebra with continuous trace there is for eachx ∈ K ( A ) + and n < ∞ such that dim π ( A ) ≤ n for every irreducible representation π : A → B ( H ) . Moreover, the map x ˆ x (cf. (D.8.1) ) is a faithful, positive linearsurjection of K ( A ) onto K (cid:0) ˆ A (cid:1) . Lemma D.8.17. [57] If A is a separable algebra then ˇ A is second-countable.
Proposition D.8.18. [57] Let A be a C ∗ - algebra with continuous trace. Then(i) A is of type I ;(ii) ˆ A is a locally compact Hausdorff space;(iii) For each t ∈ ˆ A there is an Abelian element x ∈ A such that ˆ x ∈ K ( ˆ A ) and ˆ x ( t ) = .The last condition is sufficient for A to have continuous trace. Theorem D.8.19. [57] Let A be a C ∗ -algebra of type I. Then A contains an essen-tial ideal which has continuous trace. Moreover, A has an essential composition series { I α | ≤ α ≤ β } such that I α + / I α has continuous trace for each α < β . Definition D.8.20. [35] A C ∗ -algebra is homogeneous of order n if every irreducible*-representation is of the same finite dimension n . Theorem D.8.21. [35] Every homogeneous C ∗ -algebra of order n is isomorphic withsome C ( B ) , where B is a fibre bundle with base space ˆ A, fibre space M n ( C ) , and groupPU ( n ) . Theorem D.8.22. [42] Let A be a C ∗ -algebra in which for every primitive ideal P, Pis finite-dimensional and of order independent of P. Then the structure space of A isHausdorff. efinition D.8.23. [61] If A is a C ∗ -algebra with Hausdorff spectrum ˆ A and U isan open subset of ˆ A then the set A | U = \ x ∈ ˆ A \U ker rep x (D.8.2)is said to be the open restriction of A on U . Remark D.8.24.
It is shown n [61] that the Jacobson topology on of the primitivespectrum (cf. Definition D.2.18) always determines the ideal structure of A : theopen sets U in ˇ A are in one-to-one correspondence with the ideals U ↔ A | U . (D.8.3)On the other it is proven in [61] that any C ∗ -algebra A with Hausdorff spectrumis CCR -algebra. From the Theorem D.8.9 it turns out that the spectrum of A coincides with its primitive spectrum. So the Equation D.8.3 establishes the one-to-one correspondence with the ideals of A and opens sets of the spectrum of A . Lemma D.8.25. [61] Suppose that A is a C ∗ -algebra with Hausdorff spectrum X andthat U is an open subset of ˆ A. If A | U is given by (D.2.7) then A | U is the closure of thespace { f · a | a ∈ A and f ∈ C ( X ) vanishes off U } , or, equivalently the closure of the space C ( U ) A . Lemma D.8.26. [61] Suppose A is a C ∗ -algebra with Hausdorff spectrum X .(a) If a , b ∈ A and rep x ( a ) = rep x ( b ) for every x ∈ X , then a = b.(b) For each a ∈ A the function x
7→ k rep x ( a ) k is continuous on X , vanishes atinfinity and has sup-norm equal to k a k . D.8.2 Fields of operators
Let X be a locally compact Hausdorff space called the base space; and for each x in X , let A x be a (complex) Banach space. A vector field (with values in the { A x } ) is a function a on X given by x a x such that a x ∈ A x for each x in X .Obviously the vector fields form a complex linear space. If each A x is a *-algebra,then the vector fields form a *-algebra under the pointwise operations; in that casethe vector fields will usually be referred to as operator fields . Here, either each A x will be a Hilbert space or each A x will be a C ∗ -algebra.482 efinition D.8.27. [35] A continuity structure for X and the { A x } is a linear space F of vector fields on X , with values in the { A x } , satisfying:(a) If a ∈ F , the real-valued function x
7→ k a x k is continuous on X .(b) For each x in X , { a x | a ∈ F } is dense in A x .(c) If each A x is a C *-algebra, we require also that F is closed under pointwisemultiplication and involution. Definition D.8.28. [35] A vector field a is continuous (with respect to F ) at x , iffor each ε > b of F and a neighborhood U of x such that k a x − b x k < ε for all x in U . We say that a is continuous on X if it is continuous atall points of X . Lemma D.8.29. [35] If a sequence of vector fields { a n } continuous (with respect to F )at x converges uniformly on X to a vector field a, then a is continuous at x (with respectto F ). Lemma D.8.30. [35] If a vector field a is continuous with respect to F at x , thenx
7→ k a x k is continuous at x . Lemma D.8.31. [35] The vector fields continuous (with respect to F ) at x form a linearspace, closed under multiplication by complex-valued functions on X which are continuousat x . If each A x is a C ∗ -algebra, the vector fields continuous at x are also closed underpointwise multiplication and involution. D.8.3 C ∗ -algebras as cross sections and their multiplies Here I follow to [1]. The following definition is a specialization of the DefinitionD.8.28.For any C ∗ -algebra A denote by M ( A ) β the algebra M ( A ) of multipliers withthe strict topology (cf. Definition D.1.12). Definition D.8.32.
A bounded section a of the fibered space {X , M ( A x ) } is saidto be strictly continuous (with respect to) F if for each ε > c ∈ F there is an element b ∈ F and an neighborhood U of x such that k c x ( a x − b x ) k + k ( a x − b x ) c x k < ε for every x in U .We denote by C b (cid:16) X , M ( A x ) β , F (cid:17) the set of all bounded strictly continuouscross sections in {X , M ( A x ) } heorem D.8.33. [1] There is the natural *-isomorphismM ( C ( X , A x , F )) ∼ = C b (cid:16) X , M ( A x ) β , F (cid:17) . Theorem D.8.34. [32] Let A be a liminal C ∗ -algebra whose spectrum X is Hausdorff.Let (cid:0) X , { A x } x ∈X . F (cid:1) be the continuous field of non-zero elementary C ∗ -algebras over X defined by A. Let A ′ be the C ∗ -algebra defined by (cid:0) X , { A x } x ∈X , F (cid:1) . For every a ∈ A,let a ′ = { a ( x ) ∈ A x } x ∈X be the element of C (cid:0) X , { A x } x ∈X , F (cid:1) defined by a. Thena ′ ∈ A ′ and a a ′ is an isomorphism of A onto A ′ . D.8.35. [1] Theorem D.8.33 allows us (in principle) to determine the multipliers ofany liminal algebra C ∗ -algebra A with Hausdorff spectrum. Because in this case A can be represented as an algebra of continuous cross sections A = C (cid:0) ˆ A , A ( t ) , A (cid:1) where A ( t ) = K ( H t ) . Since M ( K ( H t )) is equal to B ( H ) equipped with thestrong* topology, denoted by B ( H ) s ∗ , we can state the following. Corollary D.8.36. [1] If A is a liminal algebra C ∗ -algebra A with Hausdorff spectrum,so what A = C (cid:0) ˆ A , K ( H t ) , A (cid:1) , then M ( A ) = C b (cid:0) ˆ A , B ( H ) s ∗ , A (cid:1) . Remark D.8.37. [1] Even in case where A has only one and two dimensionalrepresentations it is not necessary true that M ( A ) has Hausdorff spectrum. Theorem D.8.38. [35] Let X be a locally compact Hausdorff space, and let { A x } x ∈X be a family of C ∗ -algebras. If F is a continuity structure for X and the { A x } . If A x ∼ = K (cid:0) ℓ ( N ) (cid:1) for all x ∈ X then C ( X , { A x } , F ) is a C ∗ -algebra with continuous trace. Remark D.8.39.
There are C ∗ -algebras with continuous trace which do not matchto the Theorem D.8.38, i.e. these algebras cannot be represented as C ( X , { A x } , F ) .484 ppendix E Spectral triples
This section contains citations of [39].
E.1 Definition of spectral triples
Definition E.1.1. [39] A (unital) spectral triple ( A , H , D ) consists of: • an unital pre- C ∗ -algebra A with an involution a a ∗ , equipped with afaithful representation on: • a Hilbert space H ; and also • a selfadjoint operator D on H , with dense domain Dom D ⊂ H , such that a ( Dom D ) ⊆ Dom D for all a ∈ A . Definition E.1.2. [39] A real spectral triple is a spectral triple ( A , H , D ) , togetherwith an antiunitary operator J : H → H such that J ( Dom D ) ⊂ Dom D , and [ a , Jb ∗ J − ] = a , b ∈ A . Definition E.1.3. [39] A spectral triple ( A , H , D ) is even if there is a selfadjointunitary grading operator Γ on H such that a Γ = Γ a for all a ∈ A , Γ ( Dom D ) = Dom D , and D Γ = − Γ D . If no such Z -grading operator Γ is given, we say thatthe spectral triple is odd .There is a set of axioms for spectral triples described in [39, 73]. In this articlethe following axioms are used only. Axiom E.1.4. [73](Regularity) For any a ∈ A , [ D , a ] is a bounded operator on H ,and both a and [ D , a ] belong to the domain of smoothness T ∞ k = Dom ( δ k ) of thederivation δ on B ( H ) given by δ ( T ) def = [ | D | , T ] .485 xiom E.1.5 (Real structure) . The antiunitary operator J : H → H satisfying J = ± JDJ − = ± D , and J Γ = ± Γ J in the even case, where the signs depend onlyon n mod 8 (and thus are given by the table of signs for the standard commutativeexamples). n mod 8 0 2 4 6 J = ± + − − + JD = ± DJ + + + + J Γ = ± Γ J + − + − n mod 8 1 3 5 7 J = ± + − − + JD = ± DJ − + − + Moreover, b Jb ∗ J − is an antirepresentation of A on H (that is, a representationof the opposite algebra A op ), which commutes with the given representation of A : [ a , Jb ∗ J − ] =
0, for all a , b ∈ A ,(cf. Definition E.1.2) Axiom E.1.6. [First order] For each a , b ∈ A , the following relation holds: [[ D , a ] , Jb ∗ J − ] =
0, for all a , b ∈ A . (E.1.1)This generalizes, to the noncommutative context, the condition that D be a first-order differential operator. Since [[ D , a ] , Jb ∗ J − ] = [[ D , Jb ∗ J − ] , a ] + [ D , [ a , Jb ∗ J − ] | {z } = ] ,this is equivalent to the condition that [ a , [ D , Jb ∗ J − ]] = Lemma E.1.7. [39] Let A be an unital Fréchet pre-C ∗ -algebra, whose C ∗ -completionis A. If ˜ q = ˜ q = ˜ q ∗ is a projection in A, then for any ε > , we can find a projectionq = q = q ∗ ∈ A such that k q − ˜ q k < ε . Theorem E.1.8. [37] If A is a Fréchet pre-C ∗ -algebra with C ∗ -completion A, then theinclusion j : A →
A induces an isomorphism K j : K ( A ) → K ( A ) . Remark E.1.9.
The K -symbol in the above Theorem is the K -functor of K -theory(cf. [5, 6]). The K -functor is related to projective finitely generated modules. Oth-erwise any projective finitely generated module corresponds to the idempotent ofthe matrix algebra. The idea of the proof of the Theorem E.1.8 contains followingingredients: 486 For any idempotent ˜ e ∈ M n ( A ) one constructs and idempotent e ∈ M n ( A ) such that there is the isomorphisms ˜ eA n ∼ = eA n of A -modules. • The inverse to K j homomorphism is roughly speaking given by [ e A n ] [ ˜ eA n ] where [ · ] means the K -theory class of projective finitely generated module.From the isomorphism ˜ eA n ∼ = eA n the above equation can be replaced with [ e A n ] [ eA n ] (E.1.2) E.2 Representations of smooth algebras
Let ( A , H , D ) be a spectral triple. Similarly to [52] we define a representation of π : A → B ( H ) given by π ( a ) = (cid:18) a [ D , a ] a (cid:19) . (E.2.1)We can inductively construct representations π s : A → B (cid:0) H s (cid:1) for any s ∈ N . If π s is already constructed then π s + : A → B (cid:16) H s + (cid:17) is given by π s + ( a ) = (cid:18) π s ( a ) [ D , π s ( a )] π s ( a ) (cid:19) (E.2.2)where we assume diagonal action of D on H s , i.e. D x ... x s = Dx ... Dx s ; x , ..., x s ∈ H .For any s ∈ N there is a seminorm k·k s on A given by k a k s = k π s ( a ) k . (E.2.3)The definition of spectral triple requires that A is a Fréchet algebra with respectto seminorms k·k s . 487 .3 Noncommutative differential forms Definition E.3.1. [18](a) A cycle of dimension n is a triple (cid:0) Ω , d , R (cid:1) where Ω = L nj = Ω j is a gradedalgebra over C , d is a graded derivation of degree 1 such that d =
0, and R : Ω n → C is a closed graded trace on Ω ,(b) Let A be an algebra over C . Then a cycle over A is given by a cycle (cid:0) Ω , d , R (cid:1) and a homomorphism A → Ω . E.3.2.
We let Ω ∗ A be the reduced universal differential graded algebra over A (cf. [18] Chapter III.1). It is by definition equal to A in degree 0 and is generatedby symbols da ( a ∈ A ) of degree 1 with the following presentation: ( α ) d ( ab ) = ( da ) b + adb ∀ a , b ∈ A , ( β ) d = Ω A is isomorphic as an A -bimodule to the kernel ker ( m ) of the multiplication mapping m : A ⊗ A → A , the isomorphism being given bythe mapping ∑ a j ⊗ b j ∈ ker ( m ) ∑ a j db j ∈ Ω A The involution * of A extends uniquely to an involution on * with the rule ( da ) ∗ def = − da ∗ .The differential d on A is defined unambiguously by d ( a da ... da n ) = da da ... da n ∀ a j ∈ A ,and it satisfies the relations d ω = ∀ ω ∈ Ω ∗ A , d ( ω ω ) = ( d ω ) ω + ( − ) ∂ω ω d ω . Proposition E.3.3. [18] Let ( A , H , D ) be a spectral triple.1. The following equality defines a *-representation π of the reduced universal algebra Ω ∗ A on H : π ( a da ... da n ) = a [ D , a ] ... [ D , a n ] ∀ a j ∈ A . (E.3.1)488 . Let J = ker π be the graded two-sided ideal of Ω ∗ A given byJ k = n ω ∈ Ω k (cid:12)(cid:12)(cid:12) π ( ω ) = o (E.3.2) then J = J + dJ (E.3.3) is a graded differential two-sided ideal of Ω ∗ A . Remark E.3.4.
Using 2) of Proposition E.3.3, we can now introduce the gradeddifferential algebra Ω D = Ω ∗ A / J . (E.3.4)Thus any spectral triple ( A , H , D ) naturally defines a cycle ρ : A → Ω D (cf.Definition E.3.1). In particular for any spectral triple there is an A -bimodule Ω D ⊂ B ( H ) of differential forms which is the C -linear span of operators given by a [ D , b ] ; a , b ∈ A . (E.3.5)There is the differential map d : A → Ω D , a [ D , a ] . (E.3.6) Definition E.3.5.
We say that that both the cycle ρ : A → Ω D and the differential(E.3.6) are associated with the triple ( A , H , D ) . We say that A -bimodule Ω D is the module of differential forms associated with the spectral triple ( A , H , D ) . E.3.1 Noncommutative connections and curvatures
Definition E.3.6. [18] Let A ρ −→ Ω be a cycle over A , and E a finite projectivemodule over A . Then a connection ∇ on E is a linear map ∇ : E → E ⊗ A Ω suchthat ∇ ( ξ x ) = ∇ ( ξ ) x = ξ ⊗ d ρ ( x ) ; ∀ ξ ∈ E , ∀ x ∈ A . (E.3.7)Here E is a right module over A and Ω is considered as a bimodule over A . Remark E.3.7.
In case of associated cycles (cf. Definition E.3.6) the connectionequation (E.3.7) has the following form ∇ ( ξ a ) = ∇ ( ξ ) a + ξ [ D , a ] . (E.3.8)489 emark E.3.8. The map ∇ : E → E ⊗ A Ω is an algebraic analog of the map ∇ : Γ ( E ) → Γ ( E ⊗ T ∗ ( M )) given by (H.2.2). Proposition E.3.9. [18] Following conditions hold:(a) Let e ∈ End A ( E ) be an idempotent and ∇ is a connection on E ; then ξ ( e ⊗ ) ∇ ξ (E.3.9) is a connection on e E ,(b) Any finitely generated projective module E admits a connection,(c) The space of connections is an affine space over the vector space Hom A (cid:0) E , E ⊗ A Ω (cid:1) ,(d) Any connection ∇ extends uniquely up to a linear map of e E = E ⊗ A Ω into itselfsuch that ∇ ( ξ ⊗ ω ) = ∇ ( ξ ) ω + ξ ⊗ d ω ; ∀ ξ ∈ E , ω ∈ Ω . (E.3.10) E.3.2 Connection and curvature
Definition E.3.10. A curvature of a connection ∇ is a (right A -linear) map F ∇ : E → E ⊗ A Ω (E.3.11)defined as a restriction of ∇ ◦ ∇ to E , that is, F ∇ = ∇ ◦ ∇| E where ∇ is given bythe Equation (E.3.10). A connection is said to be flat if its curvature is identicallyequal to 0 (cf. [13]). Remark E.3.11.
Above algebraic notions of curvature and flat connection are gen-eralizations of corresponding geometrical notions explained in [46] and the SectionH.2.For any projective A module E there is a trivial connection ∇ : E ⊗ A Ω → E ⊗ A Ω , ∇ = Id E ⊗ d .From d = d ◦ d = ( Id E ⊗ d ) ◦ ( Id E ⊗ d ) = 0, i.e. any trivialconnection is flat. 490 .4 Commutative spectral triples This section contains citation of [39, 73].
E.4.1 Spin c manifolds Let M be a compact n -dimensional orientable Riemannian manifold with a met-ric g on its tangent bundle T M . For any section X ∈ Γ ( M , T M ) of the tangentbundle there is the derivative (cf. [46]) X : C ∞ ( M ) → C ( M ) We say that X is smooth if X ( C ∞ ( M )) ⊂ C ∞ ( M ) . Denote by Γ ∞ ( M , T M ) the spaceof smooth vector fields. A tensor bundle corresponds to a tensor products Γ ( M , T M ) ⊗ C ( M ) ... ⊗ C ( M ) Γ ( M , T M ) Smooth sections of tensor bundle correspond to elements of Γ ∞ ( M , T M ) ⊗ C ∞ ( M ) ... ⊗ C ∞ ( M ) Γ ∞ ( M , T M ) ⊂⊂ Γ ( M , T M ) ⊗ C ( M ) ... ⊗ C ( M ) Γ ( M , T M ) .The metric tensor g is a section of the tensor bundle, or equivalently g ∈ Γ ( M , T M ) ⊗ C ( M ) Γ ( M , T M ) , we suppose that the section is smooth, i.e g ∈ Γ ∞ ( M , T M ) ⊗ C ∞ ( M ) Γ ∞ ( M , T M ) . On the other hand g yields the isomorphism Γ ( M , T M ) ∼ = Γ ( M , T ∗ M ) between sections of the tangent and the cotangent bundle. The section of cotangentbundle is said to be smooth if it is an image of the smooth section of the tangentbundle. Below we consider tensor bundles which correspond to tensor products ofexemplars of Γ ( M , T ∗ M ) and/or Γ ( M , T ∗ M ) . The metric tensor yields the givenby v = q(cid:2) g jk (cid:3) dx ... dx n (E.4.1) volume element (cf. [26]. The volume element can be regarded as an element of Γ ( M , T ∗ M ) ⊗ C ( M ) ... ⊗ C ( M ) Γ ( M , T ∗ M ) , i.e. v is a section of the tensor field. On theother hand element v defines the unique Riemannian measure µ on M . If E → M is acomplex bundle then from the Serre-Swan Theorem A.3.9 it turns out that there isan idempotent e ∈ M n ( C ( M )) such that there is the C ( M ) -module isomorphism Γ ( M , E ) ∼ = eC ( M ) n From the Theorem E.1.8 one can suppose that e ∈ M n ( C ∞ ( M )) (cf. Remark E.1.9).491 efinition E.4.1. The isomorphic to eC ∞ ( M ) n group with the induced by Γ ∞ ( M , E ) ∼ = eC ∞ ( M ) n ⊂ eC ( M ) n ∼ = Γ ( M , E ) e ∈ M n ( C ∞ ( M )) (E.4.2)inclusion Γ ∞ ( M , E ) ⊂ Γ ( M , E ) is said to be the subgroup of smooth sections .It is clear that C ∞ ( M ) Γ ∞ ( M , E ) ⊂ Γ ∞ ( M , E ) (E.4.3)and the map Γ ∞ ( M , E ) Γ ( M , E ) yields the given by the Theorem E.1.8 isomorphism K ( C ∞ ( M )) ∼ = K ( C ( M )) .It can be proved that the Definition E.4.1 complies with the above definition ofsmooth tensor fields. We build a Clifford algebra bundle C ℓ ( M ) → M whosefibres are full matrix algebras (over C ) as follows. If n is even, n = m , then C ℓ x ( M ) : = C ℓ ( T x M , g x ) ⊗ R C ≃ M m ( C ) is the complexified Clifford algebra overthe tangent space T x M . If n is odd, n = m +
1, the analogous fibre splits as M m ( C ) ⊕ M m ( C ) , so we take only the even part of the Clifford algebra: C ℓ x ( M ) def = C ℓ even ( T x M ) ⊗ R C ≃ M m ( C ) .What we gain is that in all cases, the bundle C ℓ ( M ) → M is a locally trivialfield of (finite-dimensional) elementary C ∗ -algebras, so B = Γ ( M , C ℓ ( M )) is a C ∗ -algebra.Locally, one finds trivial bundles with fibres S x such that C ℓ x ( M ) ≃ End ( S x ) ; the class δ ( C ℓ ( M )) is precisely the obstruction to patching them together(there is no obstruction to the existence of the algebra bundle C ℓ ( M ) ). It wasshown by Plymen [59] that δ ( C ℓ ( M )) = W ( T M ) ∈ H ( M , Z ) , the integral classthat is the obstruction to the existence of a spin c structure in the conventional senseof a lifting of the structure group of T M from SO ( n ) to Spin c ( n ) : see [50] for moreinformation on W ( T M ) .Thus M admits spin c structures if and only if δ ( C ℓ ( M )) =
0. But in theDixmier–Douady theory, δ ( C ℓ ( M )) is the obstruction to constructing (within the C ∗ -category) a B - A -bimodule S that implements a Morita equivalence between A = C ( M ) and B = C ( M , C ℓ ( M )) . Let us paraphrase Plymen’s redefinition of aspin c structure, in the spirit of noncommutative geometry: Definition E.4.2.
Let M be a Riemannian manifold, A = C ( M ) and B = C ( M , C ℓ ( M )) . We say that the tangent bundle T M admits a spin c structure if and only if it is orientable and δ ( C ℓ ( M )) =
0. In that case, a spin c structure on T M is a pair ( ε , S ) where ε is an orientation on T M and S is a B - A -equivalencebimodule. 492hat is this equivalence bimodule S ? By the Serre-Swan theorem A.3.9, itis of the form Γ ( M , S ) for some complex vector bundle S → M that also carriesan irreducible left action of the Clifford algebra bundle C ℓ ( M ) . This is the spinorbundle whose existence displays the spin c structure in the conventional picture.We call C ∞ ( M , S ) the spinor module ; it is an irreducible Clifford module in theterminology of [3], and has rank 2 m over C ( M ) if n = m or 2 m + Remark E.4.3. If B def = Γ ∞ ( M , C ℓ ( M )) then B is an be an unital Fréchet pre- C ∗ -algebra, such that B is dense in B . Similarly if A def = C ∞ ( M ) then A is dense in A def = C ( M ) . The Morita equivalence between B and A is given by the projective B - A bimodule Γ ( M , S ) . Similarly The Morita equivalence between B and A isgiven by the projective B - A bimodule Γ ∞ ( M , S ) . E.4.2 The Dirac operator
As soon as a spinor module makes its appearance, one can introduce the
Diracoperator . Let µ be the Riemannian measure given by the volume element (cf.Equation (E.4.1)). If H def = L ( M , S , µ ) is the space H def = L ( M , S , µ ) of square-integrable spinors then Γ ∞ ( M , S ) ⊂ H .This is a selfadjoint first-order differential operator D / defined on the space H of square-integrable spinors, whose domain includes the space of smooth spinors S = Γ ∞ ( M , S ) . The Riemannian metric g = [ g ij ] defines isomorphisms T x M ≃ T ∗ x M and induces a metric g − = [ g ij ] on the cotangent bundle T ∗ M . Via this iso-morphism, we can redefine the Clifford algebra as the bundle with fibres C ℓ x ( M ) : = C ℓ ( T ∗ x M , g − x ) ⊗ R C (replacing C ℓ by C ℓ even when dim M is odd). Let Γ ( M , T ∗ M ) be the C ( M ) -module of 1-forms on M . The spinor module S is then a B - A -bimodule on which the algebra B = Γ ( B , C ℓ ( M )) acts irreducibly. If γ : B ∼ = End A ( S ) the natural isomorphism then γ obeys the anticommutation rule { γ ( α ) , γ ( β ) } = − g − ( α , β ) = − g ij α i β j ∈ C ( M ) for α , β ∈ Γ ( M , T ∗ M ) .The action γ of Γ ( M , C ℓ ( M )) on the Hilbert-space completion H of S is called the spin representation .The metric g − on T ∗ M gives rise to a canonical Levi-Civita connection ∇ g : Γ ∞ ( M , T ∗ M ) → Γ ∞ ( M , T ∗ M ) ⊗ A Γ ∞ ( M , T ∗ M ) that, as well as obeying theLeibniz rule ∇ g ( ω a ) = ∇ g ( ω ) a + ω ⊗ da ,preserves the metric and is torsion-free. The spin connection is then a linear oper-ator ∇ S : Γ ∞ ( M , S ) → Γ ∞ ( M , S ) ⊗ A Γ ∞ ( M , T M ) satisfying two Leibniz rules, one493or the right action of A and the other, involving the Levi-Civita connection, forthe left action of the Clifford algebra: ∇ S ( ψ a ) def = ∇ S ( ψ ) a + ψ ⊗ da , ∇ S ( γ ( ω ) ψ ) = γ ( ∇ g ω ) ψ + γ ( ω ) ∇ S ψ , (E.4.4)for a ∈ A , ω ∈ Γ ∞ ( M , T ∗ M ) , ψ ∈ Γ ∞ ( M , S ) .Once the spin connection is found, we define the Dirac operator as the compo-sition γ ◦ ∇ S ; more precisely, the local expression D / def = n ∑ j = γ ( dx j ) ∇ S ∂ / ∂ x j (E.4.5)is independent of the local coordinates and defines D / on the domain S ⊂ H . Onecan check that this operator is symmetric; it extends to an unbounded selfadjointoperator on H , also called D / . In result one has the commutative spectral triple (cid:0) C ∞ ( M ) , L ( M , S ) , D / (cid:1) . (E.4.6)It is shown in [49, 73] that the first order condition (cf. Axiom E.1.6) is equivalentto the following equation [ / D , a ] b = b [ / D , a ] ∀ a , b ∈ C ∞ ( M ) . (E.4.7) E.5 Coverings of Riemannian manifolds
It is known that the covering manifold is a manifold. Following propositionstates this fact.
Proposition E.5.1. (Proposition 5.9 [46])(i) Given a connected manifold M there is a unique (unique up to isomorphism) uni-versal covering manifold, which will be denoted by e M.(ii) The universal covering manifold e M is a principal fibre bundle over M with group π ( M ) and projection p : e M → M, where π ( M ) is the first homotopy group ofM. iii) The isomorphism classes of covering spaces over M are in 1:1 correspondence withthe conjugate classes of subgroups of π ( M ) . The correspondence is given as follows.To each subgroup H of π ( M ) , we associate E = e M / H. Then the covering manifoldE corresponding to H is a fibre bundle over M with fibre π ( M ) / H associatedwith the principal bundle e M ( M , π ( M )) . If H is a normal subgroup of π ( M ) ,E = e M / H is a principal fibre bundle with group π ( M ) / H and is called a regularcovering manifold of M.
E.5.2. If e M is a covering space of Riemannian manifold M then it is possible togive e M a Riemannian structure such that π : e M → M is a local isometry (thismetric is called the covering metric ). cf. [26] for details.From the Theorem 1.1.1 it turns out any (noncommutative) C ∗ -algebra may beregarded as a generalized (noncommutative) locally compact Hausdorff topolog-ical space. The article [56] contain noncommutative analogs of coverings. Thespectral triple [39, 73] can be regarded as a noncommutative generalization of Rie-mannian manifold. Having analogs of both coverings and Riemannian manifoldsone can proof a noncommutative generalization of the Proposition E.5.1. E.6 Finite spectral triples
Here I follow to [49]. Finite spectral triples are particular cases of spectral triplesof dimension 0. The latter are rigorously defined within the axioms of noncommu-tative geometry and yield a general theory of discrete spaces. Among all discretespaces, we focus on finite ones, thus the algebra is finite dimensional. Futher-more, we will also assume that the Hilbert space is finite dimensional, an infinitedimensional one corresponding to a theory with an infinite number of elementaryfermions. Accordingly, a finite spectral triple ( A , H , D ) is defined as a spectraltriple of dimension 0 such that both A and H are finite dimensional. Using sucha triple, it is possible to a construct Yang-Mills theory with spontaneous symme-try breaking whose gauge group is the group of unitary elements of A , H is thefermionic Hilbert space and D is the mass matrix.It is known that finite dimensional real involutive algebras which admit a faith-ful representation on a finite dimensional Hilbert space are just direct sums ofmatrix algeras over the fields of real numbers, complex numbers and quaternions.495herefore, we write the algebra as a direct sum A = N M j = M n j ( K ) ,where M n ( K ) denotes the algebra of square matrices of order n with entries in thefield M = R , C or H . This remark simplifies considerably finite noncommutativegeometry and it becomes possible to give a detailed account of all finite spectraltriples. We also consider the particular case of complex finite spectral triples,whose algebra is a complex algebra and the representation π is assumed to belinear over complex numbers.Here we consider the case where A = L Kj = M n j ( C ) which acts on the Hilbertspace C N , i.e. there is a representation ρ : A → B (cid:0) C N (cid:1) = M N ( C ) . The Diracoperator D is represented by the mass matrix M ∈ M N ( C ) , i.e. D ξ = M ξ ; ξ ∈ C N . (E.6.1) E.7 Product of spectral triples
Here I follow to [20]. Let both ( A , H , D ) and ( A , H , D ) be spectral triples,we would like to find their direct product ( A , H , D ) = ( A , H , D ) × ( A , H , D ) .It is known that there are even and odd spectral triples, so there are followingcases:(i) Even-even case . In this case A def = A ⊗ A and H = H ⊗ H . Moreoverif ρ : A → B ( H ) and ρ : A → B ( H ) are representations of spectraltriples then the representation of the product is given by ρ def = ρ ⊗ ρ : A ⊗ A → B ( H ⊗ H ) . The Dirac operator is given by D def = D ⊗ id H + χ ⊗ D , D ′ def = D ⊗ χ + id H ⊗ D , (E.7.1)where χ (resp. χ ) is the chirality operator of ( A , H , D ) (resp. ( A , H , D ) )(cf. [20]) Operators D and D ′ are unitary equivalent.(ii) Even-odd case . The algebra, the *-representation, and Dirac operator are thesame as in the even-even case.(iii)
Odd-odd case . This case is not considered it this work, so it not describedhere. 496 ppendix F
Noncommutative torus and Moyalplane
F.1 Noncommutative torus T n Θ F.1.1 Definition of noncommutative torus T n Θ Definition F.1.1. [74] Denote by " · " the scalar product on R n . The matrix Θ iscalled quite irrational if, for all λ ∈ Z n , the condition exp ( π i λ · Θ µ ) = λ , µ ∈ Z n implies λ = Definition F.1.2.
Let Θ be an invertible, real skew-symmetric quite irrational n × n matrix. A noncommutative torus C (cid:0) T n Θ (cid:1) is the universal C ∗ -algebra generated bythe set { U k } k ∈ Z n of unitary elements which satisfy to the following relations. U k U p = e − π ik · Θ p U k + p ; (F.1.1)Following condition holds U k U p = e − π ik · Θ p U p U k . (F.1.2)An alternative description of C (cid:0) T n Θ (cid:1) is such that if Θ = θ . . . θ n θ θ n ... ... . . . ... θ n θ n . . . 0 (F.1.3)497hen C (cid:0) T n Θ (cid:1) is the universal C ∗ -algebra generated by unitary elements u , ..., u n ∈ U (cid:0) C (cid:0) T n Θ (cid:1)(cid:1) such that following condition holds u j u k = e − π i θ jk u k u j . (F.1.4)Unitary operators u , ..., u n correspond to the standard basis of Z n and they aregiven by u j = U k j , where k j =
0, ..., 1 |{z} j th − place , ..., 0 (F.1.5) Definition F.1.3.
The unitary elements u , ..., u n ∈ U (cid:0) C (cid:0) T n θ (cid:1)(cid:1) which satisfy therelations (F.1.4), (F.1.5) are said to be generators of C (cid:0) T n Θ (cid:1) . The set { U l } l ∈ Z n is saidto be the basis of C (cid:0) T n Θ (cid:1) .If a ∈ C (cid:0) T n Θ (cid:1) is presented by a series a = ∑ l ∈ Z n c l U l ; c l ∈ C and the series ∑ l ∈ Z n | c l | is convergent then from the triangle inequality it followsthat the series is C ∗ -norm convergent and the following condition holds. k a k ≤ ∑ l ∈ Z n | c l | . (F.1.6)In particular if sup l ∈ Z n ( + k k k ) s | c l | < ∞ , ∀ s ∈ N then ∑ l ∈ Z n | c l | < ∞ and takinginto account the Equation (C.2.2) there is the natural inclusion φ ∞ : S ( Z n ) ⊂ C (cid:0) T n Θ (cid:1) of vector spaces. If C ∞ ( T n Θ ) def = φ ∞ ( S ( Z n )) ⊂ C ( T n Θ ) (F.1.7)then C ∞ (cid:0) T n Θ (cid:1) is a pre- C ∗ -algebra, and the Fourier transformation C.2.3 yields the C -isomorphism C ∞ ( T n ) ∼ = C ∞ ( T n Θ ) . (F.1.8) F.1.4.
There is a state τ : C ( T n Θ ) → C ; ∑ k ∈ Z n a k U k a ( ) ; where a k ∈ C , (F.1.9)498hich induces the faithful GNS representation The C ∗ -norm completion C (cid:0) T n Θ (cid:1) of C ∞ (cid:0) T n Θ (cid:1) is a C ∗ -algebra and there is a faithful representation C ( T n Θ ) → B (cid:0) L ( C ( T n Θ ) , τ ) (cid:1) . (F.1.10)(cf. Definition D.2.4). Similarly to the equation (D.2.1) there is a C -linear map Ψ Θ : C ( T n Θ ) ֒ → L ( C ( T n Θ ) , τ ) . (F.1.11)If ξ k def = Ψ Θ ( U k ) (F.1.12)then from (F.1.1), (F.1.9) it turns out τ ( U ∗ k U l ) = ( ξ k , ξ l ) = δ kl , (F.1.13)i.e. the subset { ξ k } k ∈ Z n ⊂ L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) is an orthogonal basis of L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) .Hence the Hilbert space L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) is naturally isomorphic to the Hilbertspace ℓ ( Z n ) given by ℓ ( Z n ) = ( ξ = { ξ k ∈ C } k ∈ Z n ∈ C Z n | ∑ k ∈ Z n | ξ k | < ∞ ) and the C -valued scalar product on ℓ ( Z n ) is given by ( ξ , η ) ℓ ( Z n ) = ∑ k ∈ Z n ξ k η k . (F.1.14)From (F.1.1) and (F.1.12) it follows that the representation C (cid:0) T n Θ (cid:1) ֒ → B (cid:0) L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1)(cid:1) corresponds to the following action C ( T n Θ ) × L ( C ( T n Θ ) , τ ) → L ( C ( T n Θ ) , τ ) ; U k ξ l = e − π ik · Θ l ξ k + l . (F.1.15) F.1.2 Geometry of noncommutative tori
In the below text we imply that Θ is quite irrational. The restriction of given by(F.1.9) state on C ∞ (cid:0) T n Θ (cid:1) satisfies to the following equation τ ( f ) = b f ( ) (F.1.16)499here b f means the Fourier transformation. From C ∞ (cid:0) T n Θ (cid:1) ≈ S ( Z n ) it followsthat there is a C -linear isomorphism ϕ ∞ : C ∞ ( T n Θ ) ≈ −→ C ∞ ( T n ) . (F.1.17)such that following condition holds τ ( f ) = ( π ) n Z T n ϕ ∞ ( f ) dx . (F.1.18)From (F.1.18) it follows that for any a , b ∈ C ∞ (cid:0) T n Θ (cid:1) the scalar product on L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) is given by ( a , b ) = Z T n a ∗ comm b comm dx (F.1.19)where a comm ∈ C ∞ ( T n ) (resp. b comm ) is a commutative function which corre-sponds to a (resp. b ). Definition F.1.5. If Θ is non-degenerated, that is to say, σ ( s , t ) def = s · Θ t to be symplectic . This implies even dimension, n = N . One then selects Θ = θ J def = θ (cid:18) N − N (cid:19) (F.1.20)where θ > θ N def = det Θ . Denote by C ∞ (cid:0) T N θ (cid:1) def = C ∞ (cid:0) T N Θ (cid:1) and C (cid:0) T N θ (cid:1) def = C (cid:0) T N Θ (cid:1) .Denote by δ µ ( µ =
1, . . . , n ) the analogues of the partial derivatives i ∂∂ x µ on C ∞ ( T n ) which are derivations on the algebra C ∞ ( T n Θ ) given by δ µ ( U k ) = k µ U k .These derivations have the following property δ µ ( a ∗ ) = − ( δ µ a ) ∗ ,and also satisfy the integration by parts formula τ ( a δ µ b ) = − τ (( δ µ a ) b ) , a , b ∈ C ∞ ( T n Θ ) .The spectral triple describing the noncommutative geometry of noncommuta-tive n -torus consists of the algebra C ∞ ( T n Θ ) , the Hilbert space H = L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) ⊗ m , where m = [ n /2 ] with the representation π ⊗ B ( C m ) : C ∞ ( T n Θ ) → B ( H ) where π : C ∞ ( T n Θ ) → B (cid:0) L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1)(cid:1) is given by (F.1.10). The Dirac operator is givenby D = / ∂ def = n ∑ µ = ∂ µ ⊗ γ µ ∼ = n ∑ µ = δ µ ⊗ γ µ , (F.1.21)where ∂ µ = δ µ , seen as an unbounded self-adjoint operator on L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) and γ µ s are Clifford (Gamma) matrices in M m ( C ) satisfying the relation γ i γ j + γ j γ i = δ ij M m ( C ) . (F.1.22)There is a spectral triple (cid:0) C ∞ ( T n Θ ) , L ( C ( T n Θ ) , τ ) ⊗ C m , D (cid:1) . (F.1.23)There is an alternative description of D . The space C ∞ ( T n ) (resp. C ∞ (cid:0) T n Θ (cid:1) )is dense in L ( T n ) (resp. L (cid:0) C ∞ (cid:0) T n Θ (cid:1) , τ (cid:1) ), hence from the C -linear isomorphism ϕ ∞ : C ∞ (cid:0) T n Θ (cid:1) ≈ −→ C ∞ ( T n ) given by (F.1.17) it follows isomorphism of Hilbertspaces ϕ : L ( C ( T n Θ ) , τ ) ≈ −→ L ( T n ) .Otherwise T n admits a spin bundle S such that L (cid:0) T , S (cid:1) ≈ L ( T n ) ⊗ C m . It turnsout an isomorphism of Hilbert spaces Φ : L ( C ( T n Θ ) , τ ) ⊗ C m ≈ −→ L ( T n , S ) .There is a commutative spectral triple (cid:0) C ∞ ( T n ) , L ( T n , S ) , / D (cid:1) (F.1.24)such that D is given by D = Φ − ◦ / D ◦ Φ . (F.1.25)Noncommutative geometry replaces differentials with commutators such that thedifferential d f corresponds to i [ / D , f ] and the well known equation d f = n ∑ µ = ∂ f ∂ x µ dx µ is replaced with [ / D , f ] = n ∑ µ = ∂ f ∂ x µ (cid:2) / D , x µ (cid:3) (F.1.26)501n case of commutative torus we on has dx µ = iu ∗ µ du µ where u µ = e − ix µ , so what equation (F.1.26) can be written by the following way [ / D , f ] = n ∑ µ = ∂ f ∂ x µ u ∗ µ (cid:2) / D , u µ (cid:3) (F.1.27)We would like to prove a noncommutative analog of (F.1.27), i.e. for any a ∈ C ∞ (cid:0) T n Θ (cid:1) following condition holds [ D , a ] = n ∑ µ = ∂ a ∂ x µ u ∗ µ (cid:2) D , u µ (cid:3) (F.1.28)From (F.1.21) it follows that (F.1.28) is true if and only if (cid:2) δ µ , a (cid:3) = n ∑ µ = ∂ a ∂ x µ u ∗ µ (cid:2) δ µ , u µ (cid:3) ; µ =
1, . . . n . (F.1.29)In the above equation ∂ a ∂ x µ means that one considers a as element of C ∞ ( T n ) , takes ∂∂ x µ of it and then the result of derivation considers as element of C ∞ (cid:0) T n Θ (cid:1) . Sincethe linear span of elements U k is dense in both C ∞ (cid:0) T n Θ (cid:1) and L (cid:0) C (cid:0) T n Θ (cid:1) , τ (cid:1) theequation (F.1.29) is true if for any k , l ∈ Z n following condition holds (cid:2) δ µ , U k (cid:3) U l = ∂ U k ∂ x µ u ∗ µ (cid:2) δ µ , u µ (cid:3) U l .The above equation is a consequence of the following calculations: (cid:2) δ µ , U k (cid:3) U l = δ µ U k U l − U k δ k U l = ( k + l ) U k U l − lU k U l = kU k U l , ∂ U k ∂ x µ u ∗ µ (cid:2) δ µ , u µ (cid:3) U l = kU k u ∗ (cid:0) δ µ u µ U l − u µ δ U l (cid:1) = kU k u ∗ (cid:0) ( l + ) u µ U l − lu µ δ U l (cid:1) == kU k u ∗ µ u µ U l = kU k U l .For any k ∈ Z n following condition holds u ∗ µ (cid:2) δ µ , u µ (cid:3) U k = u ∗ µ δ µ u µ U k − u ∗ µ u µ δ µ U k = u ∗ µ (cid:0) ( k + ) u µ U k − ku µ U k (cid:1) = U k it turns out u ∗ µ (cid:2) δ µ , u µ (cid:3) = C ( T n Θ ) ; u ∗ µ (cid:2) D , u µ (cid:3) = γ µ . (F.1.30)502rom (F.1.21), (F.1.28) and (F.1.30) it turns out [ D , f ] = n ∑ µ = ∂ f ∂ x µ γ µ . (F.1.31) γ µ ∈ Ω D (F.1.32)where Ω D is the module of differential forms associated with the spectral triple(F.1.23) (cf. Definition E.3.5). F.2 Moyal plane
Definition F.2.1.
Denote the
Moyal plane product ⋆ θ on S (cid:0) R N (cid:1) given by ( f ⋆ θ h ) ( u ) = Z y ∈ R N f (cid:18) u − Θ y (cid:19) g ( u + v ) e π iy · v dydv (F.2.1)where Θ is given by (F.1.20).There is the tracial property [34] of the Moyal product Z R N ( f ⋆ θ g ) ( x ) dx = Z R N f ( x ) g ( x ) dx . (F.2.2)The Fourier transformation of the star product satisfies to the following condition. F ( f ⋆ θ g ) ( x ) = Z R N F f ( x − y ) F g ( y ) e π iy · Θ x dy . (F.2.3) Proposition F.2.2. [34] The algebra S (cid:0) R N , ⋆ θ (cid:1) has the (nonunique) factorization prop-erty: for all h ∈ S (cid:0) R N (cid:1) there exist f , g ∈ S (cid:0) R N (cid:1) that h = f ⋆ θ g. Definition F.2.3. [23] Denote by S ′ ( R n ) the vector space dual to S ( R n ) , i.e. thespace of continuous functionals on S ( R n ) . The Moyal product can be defined, byduality, on larger sets than S (cid:0) R N (cid:1) . For T ∈ S ′ (cid:0) R N (cid:1) , write the evaluation on g ∈ S (cid:0) R N (cid:1) as h T , g i ∈ C ; then, for f ∈ S we may define T ⋆ θ f and f ⋆ θ T aselements of S ′ (cid:0) R N (cid:1) by h T ⋆ θ f , g i def = h T , f ⋆ θ g ih f ⋆ θ T , g i def = h T , g ⋆ θ f i (F.2.4)503sing the continuity of the star product on S (cid:0) R N (cid:1) . Also, the involution is ex-tended to by h T ∗ , g i def = h T , g ∗ i . Consider the left and right multiplier algebras: M θ L def = { T ∈ S ′ ( R N ) : T ⋆ θ h ∈ S ( R N ) for all h ∈ S ( R N ) } , M θ R def = { T ∈ S ′ ( R N ) : h ⋆ θ T ∈ S ( R N ) for all h ∈ S ( R N ) } , M θ def = M θ L ∩ M θ R . (F.2.5)In [23] it is proven that M θ R ⋆ θ S ′ (cid:16) R N (cid:17) = S ′ (cid:16) R N (cid:17) and S ′ (cid:16) R N (cid:17) ⋆ θ M θ L = S ′ (cid:16) R N (cid:17) . (F.2.6)It is known [34] that the domain of the Moyal plane product can be extendedup to L (cid:0) R N (cid:1) . Lemma F.2.4. [34] If f , g ∈ L (cid:0) R N (cid:1) , then f ⋆ θ g ∈ L (cid:0) R N (cid:1) and k f k op < ( πθ ) − N k f k .where k·k be the L -norm given by k f k = (cid:12)(cid:12)(cid:12)(cid:12) Z R N | f | dx (cid:12)(cid:12)(cid:12)(cid:12) . (F.2.7) and the operator norm k T k op def = sup { k T ⋆ g k / k g k : 0 = g ∈ L (cid:16) R N ) (cid:17) } (F.2.8) Definition F.2.5.
Denote by S (cid:0) R N θ (cid:1) (resp. L (cid:0) R N θ (cid:1) ) the operator algebra whichis C -linearly isomorphic to S (cid:0) R N (cid:1) (resp. L (cid:0) R N (cid:1) ) and product coincides with ⋆ θ . Both S (cid:0) R N θ (cid:1) and L (cid:0) R N θ (cid:1) act on the Hilbert space L (cid:0) R N (cid:1) . Denote by Ψ θ : S (cid:16) R N (cid:17) ≈ −→ S (cid:16) R N θ (cid:17) (F.2.9)the natural C -linear isomorphism. Definition F.2.6. [34] Let S ′ (cid:0) R N (cid:1) be a vector space dual to S (cid:0) R N (cid:1) . Denote by C b (cid:0) R N θ (cid:1) def = { T ∈ S ′ (cid:0) R N (cid:1) : T ⋆ θ g ∈ L (cid:0) R N (cid:1) for all g ∈ L ( R N ) } , providedwith the given by (F.2.8) operator norm k · k op . Denote by C (cid:0) R N θ (cid:1) the operatornorm completion of S (cid:0) R N θ (cid:1) . Remark F.2.7.
Obviously S (cid:0) R N θ (cid:1) ֒ → C b (cid:0) R N θ (cid:1) . But S (cid:0) R N θ (cid:1) is not dense in C b (cid:0) R N θ (cid:1) , i.e. C (cid:0) R N θ (cid:1) ( C b (cid:0) R N θ (cid:1) (cf. [34]).504 emark F.2.8. L (cid:0) R N θ (cid:1) is the k · k norm completion of S (cid:0) R N θ (cid:1) hence from theLemma F.2.4 it follows that L (cid:16) R N θ (cid:17) ⊂ C (cid:16) R N θ (cid:17) . (F.2.10) Remark F.2.9.
Notation of the Definition F.2.6 differs from [34]. Here symbols A θ , A θ , A θ are replaced with C b (cid:0) R N θ (cid:1) , S (cid:0) R N θ (cid:1) , C (cid:0) R N θ (cid:1) respectively. Remark F.2.10.
The C -linear space C (cid:0) R N θ (cid:1) is not isomorphic to C (cid:0) R N (cid:1) .There are elements (cid:8) f nm ∈ S (cid:0) R (cid:1)(cid:9) m , n ∈ N , described in [23], which satisfy tothe following Lemma. Lemma F.2.11. [23]
Let m , n , k , l ∈ N . Then f mn ⋆ θ f kl = δ nk f ml and f ∗ mn = f nm . Thusf nn is an orthogonal projection and f mn is nilpotent for m = n. Moreover, h f mn , f kl i = δ mk δ nl . The family { f mn : m , n ∈ N } ⊂ S (cid:0) R (cid:1) ⊂ L ( R ) is an orthogonal basis. Remark F.2.12.
One has Z R f mn = δ mn . (F.2.11) Proposition F.2.13. [23, 34] Let N = . Then S (cid:0) R N θ (cid:1) = S (cid:0) R θ (cid:1) has a Fréchet algebraisomorphism with the matrix algebra of rapidly decreasing double sequences c = ( c mn ) ofcomplex numbers such that, for each k ∈ N ,r k ( c ) def = (cid:18) ∞ ∑ m , n = θ k (cid:0) m + (cid:1) k (cid:0) n + (cid:1) k | c mn | (cid:19) (F.2.12) is finite, topologized by all the seminorms ( r k ) ; via the decomposition f = ∑ ∞ m , n = c mn f mn of S ( R ) in the { f mn } basis. The twisted product f ⋆ θ g is the matrix product ab, where ( ab ) mn def = ∞ ∑ k = a µν b kn . (F.2.13) For N > , C ∞ (cid:0) R N θ (cid:1) is isomorphic to the (projective) tensor product of N matrixalgebras of this kind, i.e. S (cid:16) R N θ (cid:17) ∼ = S (cid:0) R θ (cid:1) ⊗ · · · ⊗ S (cid:0) R θ (cid:1)| {z } N − times (F.2.14) with the projective topology induced by seminorms r k given by (F.2.12) . emark F.2.14. If A is C ∗ -norm completion of the matrix algebra with the norm(F.2.12) then A ≈ K , i.e. C (cid:0) R θ (cid:1) ≈ K . (F.2.15)Form (F.2.14) and (F.2.15) it follows that C (cid:16) R N θ (cid:17) ∼ = C (cid:0) R θ (cid:1) ⊗ · · · ⊗ C (cid:0) R θ (cid:1)| {z } N − times ≈ K ⊗ · · · ⊗ K | {z } N − times ≈ K (F.2.16)where ⊗ means minimal or maximal tensor product ( K is nuclear hence bothproducts coincide). F.2.15. [34] By plane waves we understand all functions of the form x exp ( ik · x ) for k ∈ R N . One obtains for the Moyal product of plane waves:exp ( ik · ) ⋆ Θ exp ( ik · ) = exp ( ik · ) ⋆ θ exp ( ik · ) = exp ( i ( k + l ) · ) e − π ik · Θ l . (F.2.17)It is proven in [34] that plane waves lie in C b (cid:0) R N θ (cid:1) . F.2.16.
Let us consider the unitary dilation operators E a given by E a f ( x ) def = a N /2 f ( a x ) ,It is proven in [34] that f ⋆ θ g = ( θ /2 ) − N /2 E θ ( E θ /2 f ⋆ E θ /2 g ) . (F.2.18)We can simplify our construction by setting θ =
2. Thanks to the scaling rela-tion (F.2.18) any qualitative result can is true if it is true in case of θ =
2. We usethe following notation f × g def = f ⋆ g (F.2.19)Introduce the symplectic Fourier transform F by F f ( x ) def = ( π ) − N Z f ( t ) e ix · Jt d N t ; e F f ( x ) def = ( π ) − N Z f ( t ) e ix · Jt d N t ; (F.2.20)The twisted convolution [23] f ⋄ g is defined by f ⋄ g ( u ) def = Z f ( u − t ) g ( t ) e − iu · Jt dt . (F.2.21)506ollowing conditions hold [23]: F ( f × g ) = F f ⋄ F g ; F ( f ⋄ g ) = F f × F g ; (F.2.22) f × g = F f ⋄ g = f ⋄ e Fg ; f ⋄ g = F f × g = f × e Fg ; (F.2.23) ( f × g ) × h = f × ( g × h ) ; ( f ⋄ g ) ⋄ h = f ⋄ ( g ⋄ h ) ; (F.2.24) ( f × g ) ∗ = g ∗ × h ∗ ; ( f ⋄ g ) ∗ = g ∗ ⋄ h ∗ . (F.2.25) Definition F.2.17. [34] We may as well introduce more Hilbert spaces G st (for s , t ∈ R ) of those f ∈ S ′ ( R ) = ∞ ∑ m , n = c mn f mn for which the following sum is finite: k f k st def = ∞ ∑ m , n = ( m + ) s ( n + ) t | c mn | .for G st . Remark F.2.18.
It is proven in [23] f , g ∈ L (cid:0) R (cid:1) , then f × g ∈ L (cid:0) R (cid:1) and k f × g k ≤ k f k k g k . Moreover, f × g lies in C (cid:0) R (cid:1) : the continuity follows byadapting the analogous argument for (ordinary) convolution. Remark F.2.19.
It is shown in [23] that S (cid:0) R (cid:1) = \ s , t ∈ R G st . (F.2.26) F.2.20.
This part contains a useful equations proven in [23]. There are coordinatefunctions p , q on R such that for any f ∈ S (cid:0) R (cid:1) following conditions hold q × f = (cid:18) q + i ∂∂ p (cid:19) f ; p × f = (cid:18) p − i ∂∂ q (cid:19) f ; f × q = (cid:18) q − i ∂∂ p (cid:19) f ; f × p = (cid:18) p + i ∂∂ q (cid:19) f . (F.2.27)From q × f , f × q , p × f , f × p ∈ S (cid:0) R N (cid:1) it follows that p , q ∈ M (cf. (F.2.5)).From (F.2.6) it follows that q × S ′ (cid:16) R N (cid:17) ⊂ S ′ (cid:16) R N (cid:17) ; p × S ′ (cid:16) R N (cid:17) ⊂ S ′ (cid:16) R N (cid:17) ; S ′ (cid:16) R N (cid:17) × q ⊂ S ′ (cid:16) R N (cid:17) ; S ′ (cid:16) R N (cid:17) × p ⊂ S ′ (cid:16) R N (cid:17) . (F.2.28)507f f ∈ S ′ (cid:0) R (cid:1) then from (F.2.27) it follows that ∂∂ p f = − iq × f + i f × q , ∂∂ q f = ip × f − i f × p (F.2.29)If a def = q + ip √ a def = q − ip √ ∂∂ a def = ∂ q + i ∂ p √ ∂∂ a def = ∂ q − i ∂ p √ H def = aa = (cid:0) p + q (cid:1) , a × a = H − a × a = H + a × f = a f + ∂ f ∂ a , f × a = a f − ∂ f ∂ a , a × f = a f − ∂ f ∂ a , f × a = a f + ∂ f ∂ a , (F.2.31) H × f mn = ( m + ) f mn ; f mn × H = ( n + ) f mn (F.2.32) a × f mn = √ m f m − n ; f mn × a = √ n + f m , n + ; a × f mn = √ m + f m + n ; f m + n × a = √ n f m , n − . (F.2.33)It is proven in [23] that ∂ j ( f × g ) = ∂ j f × g + f × ∂ j g ; (F.2.34)where ∂ j = ∂∂ x j is the partial derivation in S (cid:0) R N (cid:1) .508 ppendix G Foliations and operator algebras
G.1 Foliations
Definition G.1.1. [18] Let M be a smooth manifold and T M its tangent bundle, sothat for each x ∈ M , T x M is the tangent space of M at x . A smooth subbundle F of T M is called integrable if and only if one of the following equivalent conditionsis satisfied:(a) Every x ∈ M is contained in a submanifold W of M such that T y ( W ) = F y ∀ y ∈ W ,(b) Every x ∈ M is in the domain U ⊂ M of a submersion p : U → R q ( q = codim F ) with F y = Ker ( p ∗ ) y ∀ y ∈ U ,(c) C ∞ ( F ) = { X ∈ C ∞ ( T M ) , X x ∈ F x ∀ x ∈ M } is a Lie algebra,(d) The ideal J ( F ) of smooth exterior differential forms which vanish on F isstable by exterior differentiation. G.1.2. [18] A foliation of M is given by an integrable subbundle F of T M . Theleaves of the foliation ( M , F ) are the maximal connected submanifolds L of M with T x ( L ) = F x , ∀ x ∈ L , and the partition of M in leaves M = ∪ L α , α ∈ X x ∈ M has aneighborhood U and a system of local coordinates ( x j ) j = V called foliationcharts , so that the partition of U in connected components of leaves correspondsto the partition of R dim M = R dim F × R codim F in the parallel affine subspaces R dim F × pt. The corresponding foliation will bedenoted by (cid:0) R n , F p (cid:1) (G.1.1)where p = dim F p . To each foliation ( M , F ) is canonically associated a C ∗ - algebra C ∗ r ( M , F ) which encodes the topology of the space of leaves. To take this intoaccount one first constructs a manifold G , dim G = dim M + dim F . Definition G.1.3. [10] Let N ⊂ M be a smooth submanifold. We say that F is transverse to N (and write F ⋔ N ) if, for each leaf L of F and each point x ∈ L ∩ N , T x ( L ) ans T x ( N ) together span T x ( M ) . At the other extreme At the other extreme,we say that F is tangent to N if, for each leaf L of F , either L ∩ N = ∅ or L ⊂ N .The symbol F p denotes either the full Euclidean space R p or Euclidean halfspace H p = { ( x , ..., x n ) ∈ R p | x ≤ } . Definition G.1.4. [10] A rectangular neighborhood in F n is an open subset of theform B = J × ... × J n , where each J j is a (possibly unbounded) relatively openinterval in the j th coordinate axis. If J is of the form ( a , 0 ] , we say that B hasboundary ∂ B { ( x , ..., x n ) } ⊂ B . Definition G.1.5. [10] Let M be an n -manifold. A foliated chart on M of codi-mension q is a pair ( U , ϕ )) , where U ⊂ M is open and ϕ : U ≈ −→ B τ × B ⋔ is adiffeomorphism, B ⋔ being a rectangular neighborhood in F q and B τ a rectangularneighborhood in F n − q . The set P y = ϕ − ( B τ × { y } ) , where y ∈ B ⋔ , is calleda plaque of this foliated chart. For each x ∈ B τ , the set S x = ϕ − ( { x } × B ⋔ ) iscalled a transversal of the foliated chart. The set ∂ τ U = ϕ − ( B τ × ( ∂ B ⋔ )) is calledthe tangential boundary of U and ∂ ⋔ U = ϕ − ( ∂ ( B τ ) × ∂ B ⋔ ) is called the transverseboundary of U . Definition G.1.6.
Let M be an n -manifold, possibly with boundary and corners,and let F = { L λ } λ ∈ Λ be a decomposition of M into connected, topologicallyimmersed submanifolds of dimension k = n − q . Suppose that M admits an atlas {U α } α ∈ A of foliated charts of codimension q such that, for each α ∈ A and each λ ∈ Λ , L λ ∩ U α is a union of plaques. Then F is said to be a foliation of M of510odimension q (and dimension k ) and {U α } α ∈ A l is called a foliated atlas associatedto F . Each L x is called a leaf of the foliation and the pair ( M , F ) is called a foliatedmanifold . If the foliated atlas is of class C r (0 ≤ r ≤ ∞ or r = ω ), then the foliation F and the foliated manifold ( M , F ) . is said to be of class C r . Definition G.1.7. If ( M , F ) is a foliation and U ⊂ M be an open subset F | U is therestriction of F on U then we say that ( U , F | U ) is the restriction of ( M , F ) to U .(cf. [18]) Definition G.1.8. A foliated atlas of codimension q and class C r on the n -manifold M is a C r -atlas A def = {U α } α ∈ A of foliated charts of codimension q which are coher-ently foliated in the sense that, whenever P and Q are plaques in distinct charts of A , then P ∩ Q is open both in P and Q . Definition G.1.9.
Two foliated atlases lt and A on A ′ of the same codimension andsmoothness class C r are coherent ( A ≈ A ′ ) if A ∪ A ′ is a foliated C ∗ -atlas. Lemma G.1.10.
Coherence of foliated atlases is an equivalence relation.
Lemma G.1.11.
Let A and A ′ be foliated atlases on M and suppose that A is associatedto a foliation F . Then A and A ′ are coherent if and only if A ′ is also associated to F . Definition G.1.12.
A foliated atlas A def = {U α } α ∈ A of class C r is said to be regular if(a) For each α ∈ A , the closure U α of U α is a compact subset of a foliated chart {V α } and ϕ α = ψ | U α .(b) The cover {U α } is locally finite.(c) if U α and U β are elements of A , then the interior of each closed plaque P ∈ U α meets at most one plaque in U β . Lemma G.1.13.
Every foliated atlas has a coherent refinement that is regular.
Theorem G.1.14.
The correspondence between foliations on M and their associated foli-ated atlases induces a one-to-one correspondence between the set of foliations on M.
We now have an alternative definition of the term "foliation".
Definition G.1.15. A foliation F of codimension q and class C r on M is a coherenceclass of foliated atlases of codimension q and class C r on M .By Zorn’s lemma, it is obvious that every coherence class of foliated atlasescontains a unique maximal foliated atlas.511 efinition G.1.16. A foliation of codimension q and class C r on M is a maximalfoliated C r -atlas of codimension q on M . G.1.17.
Let Π ( M , F ) be the space of paths on leaves, that is, maps α : [
0, 1 ] → M that are continuous with respect to the leaf topology on M . For such a path let s ( α ) = α ( ) be its source or initial point and let r ( α ) = α ( ) be its range orterminal point. The space Π ( M , F ) has a partially defined multiplication: theproduct α · β of two elements α and β is defined if the terminal point of β is theinitial point of α , and the result is the path β followed by the path α . (Note thatthis is the opposite to the usual composition of paths α β = β · α used in definingthe fundamental group of a space.) Definition G.1.18.
In the situation of G.1.17 we say that the topological space Π ( M , F ) is the space of path on leaves . Definition G.1.19. [10] A groupoid G on a set X is a category with inverses,having X as its set of objects. For y , z ∈ X the set of morphisms of G from y to z is denoted by G zy . Definition G.1.20. [11] The graph , or holonomy groupoid , of the foliated space ( M , F ) is the quotient space of Π ( M , F ) by the equivalence relation that iden-tifies two paths α and β if they have the same initial and terminal points, and theloop α · β has trivial germinal holonomy. The graph of ( M , F ) will be denoted by G ( M , F ) , or simply by G ( M ) or by G when all other variables are understood. Remark G.1.21.
There is the natural surjective continuous map Φ : Π ( M , F ) → G ( M , F ) (G.1.2)from the space of path on leaves to the foliation graph. Proposition G.1.22.
Let A = {U ι } be a regular foliated atlas of M. For each finitesequence of indices { α , ..., α k } , the product V α = G ( U ι ) ... G ( U ι k ) ∈ G ( M , F ) α = ( ι , ..., ι k ) (G.1.3) is either empty or a foliated chart for the graph G . The collection of all such finite productsis a covering of G by foliated charts. Theorem G.1.23.
The graph G of ( M , F ) is a groupoid with unit space G = M, andthis algebraic structure is compatible with a foliated structure on G and M. Furthermore,the following properties hold. i) The range and source maps r , s : G →
M are topological submersions.(ii) The inclusion of the unit space M → G is a smooth map.(iii) The product map
G × M G →
G, given by ( γ , γ ) γ · γ , is smooth.(iv) There is an involution j : G → G , given by j ( γ ) = γ − , which is a diffeomorphismof G , sends each leaf to itself, and exchanges the foliations given by the range Lemma G.1.24.
If f ∈ Γ c (cid:0) U α , Ω (cid:1) and f ∈ Γ c (cid:0) U α , Ω (cid:1) then their convolutionis a well-defined element f ∗ f ∈ Γ c (cid:0) U α · U α , Ω (cid:1) For each leaf L of ( M , F ) one has a natural representation of this ∗ -algebraon the L space of the holonomy covering ˜ L of L . Fixing a base point x ∈ L , oneidentifies ˜ L with G x = { γ , s ( γ ) = x } and defines ( ρ x ( f ) ξ ) ( γ ) = Z γ ◦ γ = γ f ( γ ) ξ ( γ ) ∀ ξ ∈ L ( G x ) , (G.1.4) Proposition G.1.25. If V ⊂ G is a foliated chart for the graph of ( M , F ) and f ∈ Γ c (cid:0) V , Ω (cid:1) , then ρ x ( f ) , given by (G.1.4) , is a bounded integral operator on L ( G x ) . G.1.26.
The space of compactly supported half-densities on G is taken as given bythe exact sequence M α α Γ c (cid:16) U α α , Ω (cid:17) → M α Γ c (cid:16) U α , Ω (cid:17) Γ ⊕ −→ Γ c (cid:16) G , Ω (cid:17) (G.1.5)associated to a regular cover for (( M , F )) as above. The first step for defining aconvolution is to do it at the level of L α Γ c (cid:0) U α Ω (cid:1) , as the following lemmaindicates. Definition G.1.27.
The reduced C ∗ -algebra of the foliated space ( M , F ) is the com-pletion of Γ c (cid:0) G , Ω (cid:1) with respect to the pseudonorm k f k = sup x ∈ M k ρ x ( f ) k (G.1.6)where ρ x is given by (G.1.4). This C ∗ -algebra is denoted by C ∗ r ( M , F ) .An obvious consequence of the construction of C ∗ r ( M , F ) is the following. Corollary G.1.28.
Let M be a foliated space and let A be a regular cover by foliatedcharts. Then the algebra generated by the convolution algebras Γ c (cid:0) G ( U ) , Ω (cid:1) , U ⊂ A ,is dense in C ∗ r ( M , F ) . .1.29. Let ( M , F ) be an arbitrary foliated space and let U ⊂ M be an opensubset. Then ( U , F | U ) is a foliated space and the inclusion U ֒ → M induces ahomomorphism of groupoids G ( U ) ֒ → G , hence a mapping j U : Γ c (cid:16) G ( U ) , Ω (cid:17) ֒ → Γ c (cid:16) G ( M ) , Ω (cid:17) (G.1.7)that is an injective homomorphism of involutive algebras. Proposition G.1.30.
Let U be an open subset of the foliated space M. Then the inclusion U ֒ → M induces an isometry of C ∗ r ( U , F | U ) into C ∗ r ( M , F ) . Lemma G.1.31.
Each element γ ∈ G induces a unitary operator ρ γ : L ( s ( γ )) ≈ −→ L ( r ( γ )) that conjugates the operators ρ s ( γ ) ( f ) and ρ r ( γ ) ( f ) . In particular, the norm of ρ x ( f ) is independent of the point in the leaf through x. Lemma G.1.32.
If f ∈ Γ c (cid:0) G , Ω (cid:1) does not evaluate to zero at each γ ∈ G , then thereexists a point x in M such that ρ x ( f ) = . Definition G.1.33. [10] A foliated space ( M , F ) is a fibration if for any x there is anopen transversal N such that x ∈ N and for every leaf L of ( M , F ) the intersection L ∩ N contains no more then one point. Proposition G.1.34.
The reduced C ∗ -algebra of a foliated space M consisting of exactlyone leaf is the algebra K (cid:0) L ( M ) (cid:1) of compact operators on L ( M ) . Proposition G.1.35. [11] The reduced C ∗ -algebra C ∗ r ( N × Z ) of the trivial foliated spaceN × Z is the tensor product
K ⊗ C ( Z ) , where K is the algebra of compact operators onL ( N ) and C ( Z ) is the space of continuous functions on Z that vanish at infinity. Theorem G.1.36. [11] Assume that ( M , F ) is given by the fibers of a fibration p : M → B with fiber F. Then ( M , F ) is isomorphic to C ( B ) ⊗ K (cid:0) L ( N ) (cid:1) . Remark G.1.37.
It is proven in [11] (See Claim 2, page 56) that for any x ∈ M the given by (G.1.4) representation ρ x : C ∗ r ( M , F ) → B (cid:0) L ( G x ) (cid:1) corresponds to astate τ x : C ∗ r ( M , F ) → C . Theorem G.1.38. [11] Let ( M , F ) be a foliated space and let x ∈ M. Then the represen-tation ρ x is irreducible if and only if the leaf through x has no holonomy. Proposition and Definition G.1.39. [10] Let A be a regular foliated atlas of class C r andlet γ = (cid:8) γ α , β (cid:9) α , β ∈ A be its holonomy cocycle. Then the set Γ A of holonomy transformationsis the C r pseudogroup on S generated by γ , called the holonomy pseudogroup of A . M in leaves M = ∪ L α . An element γ of G is given by two points x = s ( γ ) , y = r ( γ ) of M together with an equivalence class of smooth paths: γ ( t ) ∈ M , t ∈ [
0, 1 ] ; γ ( ) = x , γ ( ) = y , tangent to the bundle F ( i.e. with ˙ γ ( t ) ∈ F γ ( t ) , ∀ t ∈ R ) up to the following equivalence: γ and γ are equivalent if and only ifthe holonomy of the path γ ◦ γ − at the point x is the identity . The graph G hasan obvious composition law. For γ , γ ′ ∈ G , the composition γ ◦ γ ′ makes senseif s ( γ ) = r ( γ ′ ) . If the leaf L which contains both x and y has no holonomy, thenthe class in G of the path γ ( t ) only depends on the pair ( y , x ) . In general, if onefixes x = s ( γ ) , the map from G x = { γ , s ( γ ) = x } to the leaf L through x , givenby γ ∈ G x y = r ( γ ) , is the holonomy covering of L . Both maps r and s fromthe manifold G to M are smooth submersions and the map ( r , s ) to M × M is animmersion whose image in M × M is the (often singular) subset { ( y , x ) ∈ M × M : y and x are on the same leaf } .For x ∈ M one lets Ω x be the one dimensional complex vector space of mapsfrom the exterior power ∧ k F x , k = dim F , to C such that ρ ( λ v ) = | λ | ρ ( v ) ∀ v ∈ ∧ k F x , ∀ λ ∈ R .Then, for γ ∈ G , one can identify Ω γ with the one dimensional complex vectorspace Ω y ⊗ Ω x , where γ : x → y . In other words Ω G = r ∗ ( Ω M ) ⊗ s ∗ ( Ω M ) . (G.1.8) G.2 Operator algebras of foliations
Here I follow to [18] The bundle Ω M is trivial on M , and we could choose onceand for all a trivialisation ν turning elements of Γ c (cid:16) G , Ω G (cid:17) into functions. Let ushowever stress that the use of half densities makes all the construction completelycanonical. For f , g ∈ Γ c (cid:16) G , Ω G (cid:17) , the convolution product f ∗ g is defined by theequality ( f ∗ g )( γ ) = Z γ ◦ γ = γ f ( γ ) g ( γ ) . (G.2.1)This makes sense because, for fixed γ : x → y and fixing v x ∈ ∧ k F x and v y ∈∧ k F y , the product f ( γ ) g ( γ − γ ) defines a 1-density on G y = { γ ∈ G , r ( γ ) = y } , which is smooth with compact support (it vanishes if γ / ∈ supp f ), and hence515an be integrated over G y to give a scalar, namely ( f ∗ g )( γ ) evaluated on v x , v y .The ∗ operation is defined by f ∗ ( γ ) = f ( γ − ) , i.e. if γ : x → y and v x ∈ ∧ k F x , v y ∈ ∧ k F y then f ∗ ( γ ) evaluated on v x , v y is equal to f ( γ − ) evaluated on v y , v x .We thus get a ∗ -algebra Γ c (cid:16) G , Ω G (cid:17) . where ξ is a square integrable half densityon G x . Given γ : x → y one has a natural isometry of L ( G x ) on L ( G y ) whichtransforms the representation ρ x in ρ y . Definition G.2.1. [18] C ∗ r ( M , F ) is the C ∗ -algebra completion of Γ c (cid:16) G , Ω G (cid:17) with the norm k f k = sup x ∈ M k ρ x ( f ) k . (G.2.2) Example G.2.2.
Linear foliation on torus.
Consider a vector field ˜ X on R given by˜ X = α ∂∂ x + β ∂∂ y with constant α and β . Since ˜ X is invariant under all translations, it determinesa vector field X on the two-dimensional torus T = R / Z . The vector field X determines a foliation F on T . The leaves of F are the images of the parallellines ˜ L = { ( x + t α , y + t β ) : t ∈ R } with the slope θ = β / α under the projection R → T . In the case when θ is rational, all leaves of F are closed and are circles,and the foliation F is determined by the fibers of a fibration T → S . In thecase when θ is irrational, all leaves of F are everywhere dense in T . Denote by (cid:0) T , F θ (cid:1) this foliation. G.2.1 Restriction of foliation
Lemma G.2.3. [18] If
U ⊂
M is an open set and ( U , F | U ) is the restriction of ( M , F ) to U then the graph G ( U , F | U ) is an open set in the graph G ( M , F ) , and the inclusionC ∞ ( U , F | U ) ֒ → C ∞ ( M , F ) extends to an isometric *-homomorphism of C ∗ -algebrasC ∗ r ( U , F | U ) ֒ → C ∗ r ( M , F ) . Remark G.2.4. [18] This lemma, which is still valid in the non-Hausdorff case [17],allows one to reflect algebraically the local triviality of the foliation. Thus one cancover the manifold M by open sets U λ such that F restricted to U λ has a Hausdorff516pace of leaves, V λ = U λ / F . and hence such that the C*-algebras C ∗ r ( U λ , F | U λ ) are strongly Morita equivalent to the commutative C ∗ -algebras C ( B λ ) . Thesesubalgebras C ∗ -algebras C ∗ r ( U λ , F | U λ ) generate C ∗ r ( M , F ) . but of course they fittogether in a very complicated way which is related to the global properties of thefoliation. G.2.2 Lifts of foliations
Let M be a smooth manifold and let is an F ⊂
T M be an integrable subbundle.If p : e M → M is a covering and e F ⊂ T e M is the lift of F given by a followingdiagram e F T e M F T M ֒ → ֒ → then e F is integrable. Definition G.2.5.
In the above situation we say that a foliation (cid:16) e M , e F (cid:17) is the induced by p covering of ( M , F ) or the p - lift of ( M , F ) . Remark G.2.6.
The p -lift of a foliation is described in [27, 75]. G.2.7. If γ : [
0, 1 ] → M is a path which corresponds to an element of the holonomygroupoid then we denote by [ γ ] its equivalence class, i.e. element of groupoid.There is the space of half densities Ω e M on e M which is a lift the space of halfdensities Ω M on M . If L is a leaf of ( M , F ) , L ′ = π − ( L ) then a space e L ofholonomy covering of L coincides with the space of the holonomy covering of L ′ .It turns out that L (cid:16) e G e x (cid:17) ≈ L (cid:16) G π ( e x ) (cid:17) for any e x ∈ e M . If G (resp. e G ) is a holonomygroupoid of ( M , F ) (resp. (cid:16) e M , e F (cid:17) ) then there is the surjective map p G : e G → G given by [ e γ ] [ p ◦ e γ ] If the covering is finite-fold then the map p G : e G → G induces a natural involutivehomomorphism C ∞ c (cid:16) G , Ω M (cid:17) ֒ → C ∞ c (cid:16) e G , Ω e M (cid:17) C ∞ c (cid:16) G , Ω M (cid:17) and C ∞ c (cid:16) e G , Ω e M (cid:17) with respect to given by (G.2.2)norms gives an injective *- homomorphism π : C ∗ r ( M , F ) ֒ → C ∗ r (cid:16) e M , e F (cid:17) (G.2.3)of C ∗ -algebras. The action of the group G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) of covering transformationson e M naturally induces an action of G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) on (cid:16) e M , e F (cid:17) . It follows that thereis the natural action C ∗ r (cid:16) e M , e F (cid:17) such that C ∗ r ( M , F ) = C ∗ r (cid:16) e M , e F (cid:17) G ( e M | M ) (G.2.4)Let G ( M , F ) and G (cid:16) e M , e F (cid:17) be the holonomy groupoids of ( M , F ) and (cid:16) e M , e F (cid:17) respectively. The natural surjective map G (cid:16) e M , e F (cid:17) → G ( M , F ) in-duces the injective *-homomorphism C ∗ r (cid:16) e M , e F (cid:17) ֒ → C ∗ r (cid:16) e M , e F (cid:17) . Assume both G ( M , F ) and G (cid:16) e M , e F (cid:17) are Hausdorff. Let G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) be the covering group of p : e M → M . The G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) -action on e M can be naturally extended to G (cid:16) e M , e F (cid:17) by sending e γ to g e γ for a representative path e γ in e M and g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . Lemma G.2.8. [75] Let p : e M → M be a regular covering manifold with covering groupG (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) N ⊂ M a connected submanifold, and e N a connected component of p − ( N ) Then the restriction p e N of p to N is also regular, with the covering group G (cid:16) e N (cid:12)(cid:12)(cid:12) N (cid:17) being a subgroup G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . In particular, if L e x is the leave in (cid:16) e M , e F (cid:17) containing e x ∈ p − ( x ) , where x ∈ M , then L e x is a regular cover of L x . We denote the covering group by G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) x .On the other hand, for each x ∈ M , there is a holonomy group G xx , and we havethe holonomy group bundle {G xx } over M . If x , x are on the same leaf, then anypath γ connecting x and x induces an isomorphism γ ∗ : G x x ∼ = −→ G x x by mapping [ γ ] to (cid:2) γγ γ − (cid:3) . As a local homeomorphism, the covering map p induces anembedding p : G e x e x → G xx for each e x ∈ p − ( x ) . Lemma G.2.9. [75] The group p ∗ x (cid:0) G e x e x (cid:1) is a normal subgroup of G xx . Equivalently,p ∗ x (cid:16) G e x e x (cid:17) = p ∗ x (cid:16) G e x e x (cid:17) for e x , e x ∈ p − ( x ) if L e x = L e x , (cid:8) G e x e x (cid:9) over M . There isan obvious group homomorphism φ x : G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) x → G xx / G e x e x defined as follows.An element g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) x corresponds to a point x g ∈ p − ( x ) ∩ L e x if we fix e x corresponding to the unit e . A path γ g starting at e x and ending at g e x gives aloop π (cid:0) e γ g (cid:1) in M representing an element φ x ( g ) in G x , whose class in G xx / G e x e x isuniquely defined by g . Given any [ γ ] in G xx there is a preimage e γ in L e x starting at e x . The point r ( e γ ) ∈ p − ( x ) ∩ L e x corresponding to some g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) x . So φ x is onto. Definition G.2.10. [75] The covering map p : (cid:16) e M , e F (cid:17) → ( M , F ) of foliationsis said to be regular if the map φ is an isomorphism from the leaf covering groupbundle to the quotient holonomy group bundle. Remark G.2.11.
Every regular covering map p : (cid:16) e M , e F (cid:17) → ( M , F ) of foliationsinduces nontrivial action of G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) on C ∗ r (cid:16) e M , e F (cid:17) such that C ∗ r (cid:16) e M , e F (cid:17) G ( e M | M ) ∼ = C ∗ r (cid:16) e M , e F (cid:17) . Remark G.2.12.
If a map p : (cid:16) e M , e F (cid:17) → ( M , F ) is regular covering of foliationsand a leaf e L ∈ e M has no holonomy then g e L = e L for all nontrivial g ∈ G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) ,i.e. G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) transitively acts on leaves having no holonomy.51920 ppendix H Miscellany
H.1 Pre-order category
Definition H.1.1. [36] A category is said to be a pre-order category if there is atmost one morphism between different objects.
Definition H.1.2.
A binary relation R ⊂ Λ × Λ on the set Λ (writing pRq in placeof ( p , q ) ∈ R ) is said to be pre-ordering if it is(a) reflexive , i.e. for each p we have pRp and(b) transitive , whenever pRq and qRs , we have pRs . Remark H.1.3.
For any set Λ with pre-ordering R there is a pre-order categorysuch that(a) Objects of the category are elements of Λ .(b) Morphisms are pairs ( p , q ) ∈ R .(c) The composition of morphisms is given by ( q , s ) ◦ ( p , q ) = ( p , s ) . H.2 Flat connections in the differential geometry
Here I follow to [46]. Let M be a manifold and G a Lie group. A ( differentiable ) principal bundle over M with group G consists of a manifolfd P and an action of G on P satisfying the following conditions:(a) G acts freely on P on the right: ( u , a ) ∈ P × G ua = R a u ∈ P ;521b) M is the quotient space of P by the equivalence relation induced by G , i.e. M = P / G , and the canonical projection π : P → M is differentiable;(c) P is locally trivial, that is, every point x of M has an open neighborhood U such that π − ( U ) is isomophic to U × G in the sense that there is a diffeo-morphism ψ : π − ( U ) → U × G such that ψ ( u ) = ( π ( u ) , ϕ ( u )) where ϕ isa mapping of π − ( U ) into G satisfying ψ ( ua ) = ( ψ ( u )) a for all u ∈ π − ( U ) and a ∈ G .A principal fibre bundle will be denoted by P ( M , G , π ) , P ( M , G ) or simply P .Let P ( M , G ) be a principal fibre bundle over a manifold with group G . Foreach u ∈ P let T u ( P ) be a tangent space of P at u and G u the subspace of T u ( P ) consisting of vectors tangent to the fibre through u . A connection Γ in P is anassignment of a subspace Q u of T u ( P ) to each u ∈ P such that(a) T u ( P ) = G u ⊕ Q u (direct sum);(b) Q ua = ( R a ) ∗ Q u for every u ∈ P and a ∈ G , where R a is a transformation of P induced by a ∈ G , R a u = ua .Let P = M × G be a trivial principal bundle. For each a ∈ G , the set M × { a } is a submanifold of P . The canonical flat connection in P is defined by taking thetangent space to M × { a } at u = ( x , a ) as the horizontal tangent subspace at u .A connection in any principal bundle is called flat if every point has a neighbor-hood such that the induced connection in P | U = π − ( U ) is isomorphic with thecanonical flat connection. Corollary H.2.1. (Corollary II 9.2 [46]) Let Γ be a connection in P ( M , G ) such thatthe curvature vanishes identically. If M is paracompact and simply connected, then P isisomorphic to the trivial bundle and Γ is isomorphic to the canonical flat connection inM × G. If e π : e M → M is a covering then the e π - lift of P is a principal e P (cid:16) e M , G (cid:17) bundle,given by e P = n ( u , e x ) ∈ P × e M | π ( u ) = e π ( e x ) o .If Γ is a connection on P ( M , G ) and e M → M is a covering then is a canonicalconnection e Γ on e P (cid:16) e M , G (cid:17) which is the lift of Γ , that is, for any e u ∈ e P the horizontal522pace e Q e u is isomorphically mapped onto the horizontal space Q e π ( e u ) associatedwith the connection Γ . If Γ is flat then from the Proposition (II 9.3 [46]) it turns outthat there is a covering e M → M such that e P (cid:16) e M , G (cid:17) (which is the lift of P ( M , G ) )is a trivial bundle, so the lift e Γ of Γ is a canonical flat connection (cf. CorollaryH.2.1). From the the Proposition (II 9.3 [46]) it follows that for any flat connection Γ on P ( M , G ) there is a group homomorphism ϕ : G (cid:16) e M | M (cid:17) → G such that(a) There is an action G (cid:16) e M | M (cid:17) × e P → e P ≈ e M × G given by g ( e x , a ) = ( g e x , ϕ ( g ) a ) ; ∀ e x ∈ e M , a ∈ G ,(b) There is the canonical diffeomorphism P = e P / G (cid:16) e M | M (cid:17) ,(c) The lift ˜ Γ of Γ is a canonical flat connection. Definition H.2.2.
In the above situation we say that the flat connection Γ is induced by the covering e M → M and the homomorphism G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) → G , or we say that Γ comes from G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) → G . Remark H.2.3.
The Proposition (II 9.3 [46]) assumes that e M → M is the universalcovering however it is not always necessary requirement. Remark H.2.4. If π ( M , x ) is the fundamental group [70] then there is the canon-ical surjective homomorphism π ( M , x ) → G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) . So there exist the com-position π ( M , x ) → G (cid:16) e M (cid:12)(cid:12)(cid:12) M (cid:17) → G . It follows that any flat connection comesfrom the homomorphisms π ( M , x ) → G .Suppose that there is the right action of G on P and suppose that F is a manifoldwith the left action of G . There is an action of G on P × F given by a ( u , ξ ) = (cid:0) ua , a − ξ (cid:1) for any a ∈ G and ( u , ξ ) ∈ P × F . The quotient space P × G F =( P × F ) / G has the natural structure of a manifold and if E = P × G F then E ( M , F , G , P ) is said to be the fibre bundle over the base M, with (standard) fibre F, and (structure)group G which is associated with the principal bundle P (cf. [46]). If P = M × G isthe trivial bundle then E is also trivial, that is, E = M × F . If F = C n is a vectorspace and the action of G on C n is a linear representation of the group then E isthe linear bundle. Denote by T ( M ) (resp. T ∗ ( M ) ) the tangent (resp. contangent)bundle, and denote by Γ ( E ) , Γ ( T ( M )) , Γ ( T ∗ ( M )) the spaces of sections of E ,523 ( M ) , T ∗ ( M ) respectively. Any connection Γ on P gives a covariant derivativeon E , that is, for any section X ∈ Γ ( T ( M )) and any section ξ ∈ Γ ( E ) there is thederivative given by ∇ X ( ξ ) ∈ Γ ( E ) .If E = M × C n , Γ is the canonical flat connection and ξ is a trivial section, that is, ξ = M × { x } then ∇ X ξ = ∀ X ∈ T ( M ) . (H.2.1)For any connection there is the unique map ∇ : Γ ( E ) → Γ ( E ⊗ T ∗ ( M )) (H.2.2)such that ∇ X ξ = ( ∇ ξ , X ) where the pairing ( · , · ) : Γ ( E ⊗ T ∗ ( M )) × Γ ( T ( M )) → Γ ( E ) is induced by thepairing Γ ( T ∗ ( M )) × Γ ( T ( M )) → C ∞ ( M ) . H.3 Quantum SU ( ) and SO ( ) There is a quantum generalization of SU ( ) and we will introduce a quantumanalog of SO ( ) . Let q be a real number such that 0 < q <
1. A quantum group C (cid:0) SU q ( ) (cid:1) is the universal C ∗ -algebra algebra generated by two elements α and β satisfying the following relations: α ∗ α + β ∗ β = αα ∗ + q ββ ∗ = αβ − q βα = αβ ∗ − q β ∗ α = β ∗ β = ββ ∗ . (H.3.1)From C ( SU ( )) ≈ C ( SU ( )) it follows that C (cid:0) SU q ( ) (cid:1) can be regarded asa noncommutative deformation of the space SU ( ) . The dense pre- C ∗ -algebra C ∞ (cid:0) SU q ( ) (cid:1) ⊂ C (cid:0) SU q ( ) (cid:1) is defined in [28]. Let Q , S ∈ B (cid:0) ℓ (cid:0) N (cid:1)(cid:1) be given by Qe k = q k e k , Se k = (cid:26) e k − k > k = R ∈ B ( ℓ ( Z )) be given by e k e k + . There is a faithful representation [77] C (cid:0) SU q ( ) (cid:1) → B (cid:0) ℓ (cid:0) N (cid:1) ⊗ ℓ ( Z ) (cid:1) given by524 S p − Q ⊗ β Q ⊗ R . (H.3.2)There is a faithful state h : C (cid:0) SU q ( ) (cid:1) → C given by h ( a ) = ∞ ∑ n = q n ( e n ⊗ e , ae n ⊗ e ) (H.3.3)where a ∈ C (cid:0) SU q ( ) (cid:1) and e ⊗ e n ∈ ℓ (cid:0) N (cid:1) ⊗ ℓ ( Z ) (cf. [77]). Definition H.3.1.
The state h is said to be the Haar measure .Denote by L (cid:0) C (cid:0) SU q ( ) (cid:1) , h (cid:1) the GNS space associated with the state h . Therepresentation theory of SU q ( ) is strikingly similar to its classical counterpart. Inparticular, for each l ∈ { , 1, . . . } , there is a unique irreducible unitary represen-tation t ( l ) of dimension 2 n +
1. Denote by t ( l ) jk the jk th entry of t ( l ) . These are allelements of A f and they form an orthogonal basis for L (cid:0) C (cid:0) SU q ( ) (cid:1) , h (cid:1) . Denoteby e ( l ) jk the normalized t ( l ) jk ’s, so that { e ( l ) jk : n = , 1, . . . , i , j = − n , − n +
1, . . . , n } is an orthonormal basis. The definition of equivariant operators (with respect toaction of quantum groups) is described in [29]. It is proven in [28] that any un-bounded equivariant operator e D satisfies to the following condition e D : e ( l ) jk d ( l , j ) e ( l ) jk , (H.3.4)Moreover if d ( l , j ) = ( l + l = j , − ( l + ) l = j , (H.3.5)then there is a 3-summable spectral triple (cid:16) C ∞ (cid:0) SU q ( ) (cid:1) , L (cid:0) C (cid:0) SU q ( ) (cid:1) , h (cid:1) , e D (cid:17) (H.3.6)described in [28].According to [25] (equations (4.40)-(4.44) ) following condition holds t ( l ) jk = M ljk α − j − k β k − j p l + k (cid:16) ββ ∗ ; q − ( k − j ) q ( j + k ) | q (cid:17) ; j + k ≤ k ≥ j , t ( l ) jk = M ljk α − j − k β ∗ ( k − j ) p l + k (cid:16) ββ ∗ ; q − ( k − j ) q ( j + k ) | q (cid:17) ; j + k ≤ k ≤ j , t ( l ) jk = M lj , k p l − k (cid:16) ββ ∗ ; q − ( k − j ) q ( j + k ) | q (cid:17) β k − j α ∗ j + k ; j + k ≥ k ≥ j , t ( l ) jk = M lk , j p l − j (cid:16) ββ ∗ ; q − ( k − j ) q ( j + k ) | q (cid:17) β ∗ j − k α ∗ j + k ; j + k ≥ j ≥ k (H.3.7)525here M ljk ∈ R for any l , j , k and p l − k (cid:16) x ; q − ( k − j ) q ( j + k ) | q (cid:17) is little Jacobi poly-nomial (cf. [25]). Denote by g ∈ Z the unique nontrivial element. There is asurjective group homomorphism Φ : SU ( ) → SO ( ) , ker Φ = Z = {± } and the natural action of Z on SU ( ) such that SO ( ) ∼ = SU ( ) / Z , g (cid:18) α − ββ α (cid:19) = (cid:18) − α β − β − α (cid:19) ; ∀ (cid:18) α − ββ α (cid:19) ∈ SU ( ) . (H.3.8)This action induces an action of Z on a C ∗ -algebra C ( SU ( )) given by g α = − α , g β = − β where α , β are regarded as functions SU ( ) → C . Indeed SU ( ) is an orientedmanifold, SO ( ) is an unoriented one, and SU ( ) → SO ( ) is a two-fold cover-ing There is a quantum generalization of SU ( ) and we will introduce a quantumanalog of SO ( ) . Let q be a real number such that 0 < q <
1. There is a noncom-mutative analog of the action (H.3.8) described in [30, 47, 60] Z × C (cid:0) SU q ( ) (cid:1) → C (cid:0) SU q ( ) (cid:1) , g α = − α , g β = − β . (H.3.9)In [30, 47, 60] the quantum group SO q ( ) is defined, moreover in it is provenin [30, 60] following C (cid:0) SO q ( ) (cid:1) def = C (cid:0) SU q ( ) (cid:1) Z ∼ = (cid:8)e a ∈ C (cid:0) SU q ( ) (cid:1) , g e a = e a (cid:9) .Above equation can be used as the definition of SO q ( ) Definition H.3.2.
Denote by C (cid:0) SO q ( ) (cid:1) def = C (cid:0) SU q ( ) (cid:1) Z ∼ = (cid:8)e a ∈ C (cid:0) SU q ( ) (cid:1) , g e a = e a (cid:9) . (H.3.10)The C ∗ -algebra C (cid:0) SO q ( ) (cid:1) is said to be the quantum SO ( ) . Remark H.3.3.
Our definition of SO q ( ) differs from the Definition given in othersources. For example the quantum group defined in [66] is a quantization of O ( ) and not of SO ( ) (cf. [30]). 526 heorem H.3.4. [77] Let q = , and let A f the dense involutive subalgebra of C (cid:0) SU q ( ) (cid:1) generated by α , β . The set of elements of the form α k β n β ∗ m and α ∗ k ′ β n β ∗ m (H.3.11) where k , m , n =
0, . . . ; k ′ =
1, 2, . . . forms a basis in A f : any element of A f can bewritten in the unique way as a finite linear combination of elements of (H.3.11) . H.4 Presheaves and Sheaves
Definition H.4.1. [38] Let X be a topological space. A presheaf F of Abeliangroups on X consists of the data(a) for every open subset U ⊆ X , an Abelian group F ( U ) , and(b) for every inclusion V ⊆ U of open subsets of X , a morphism of Abeliangroups ρ U V : F ( U ) → F ( V ) ,subject to conditions(0) F ( V ) =
0, where ∅ is the empty set,(1) ρ U U is the identity map, and(2) if
W ⊆ V ⊆ U are three open sets, then ρ U W = ρ VW ◦ ρ U V . Definition H.4.2. [38] A presheaf F on a a topological space X is a sheaf if itsatisfies the following supplementary conditions:(3) If U is an open set, if {V α } is an open covering of U , and if s ∈ F ( U ) is anelement such that s | V α = α , then s = U is an open set, if {V α } is an open covering of U , and we have elements s α for each α , with property that for each α , β , s α | V α ∩V β = s α | V β ∩V β , then thereis an element s ∈ F ( V ) such that s | V α = s α for each α .(Note condition (3) implies that s is unique.) Definition H.4.3. [38] If F is a presheaf on X , and if x is a point of X we definethe stalk F x of F at x to be the direct limit of groups F ( U ) for all open sets U containing x , via restriction maps ρ . 527 roposition and Definition H.4.4. [38] Given a presheaf F , there is a sheaf F + anda morphism θ : F → F + , with the property that for any sheaf G , and any morphism ϕ : F → G , there is a unique morphism ψ : F + → G such that ϕ = ψ ◦ θ . Furthermorethe pair ( F + , θ ) is unique up to unique isomorphism. F + is called the sheaf associated to the preseaf F . H.4.5.
Following text is the citation of the proof of H.4.4 (cf. [38]). For any openset U , let F + ( U ) be set of functions s from U to the union S x ∈U F x of stalks of F over points of U , such that(1) for each x ∈ U , s ( x ) ∈ F x , and(2) for each x ∈ U , there is a neighborhood V of x contained in U and an element t ∈ F ( V ) , such that for all y ∈ V the germ t y of t at y is equal to s ( y ) . Definition H.4.6. [38] Let f : X → Y be a continuous map of topologicalspaces. For any sheaf F on X , we define the direct image sheaf f ∗ F on Y by ( f ∗ F ) ( V ) = F (cid:0) f − ( V ) (cid:1) for any open set V ⊆ Y . For any sheaf G on Y , wedefine the inverse image sheaf f − G on X be the sheaf associated to the presheaf U 7→ lim V⊇ f ( U ) G ( V ) , where U is any open set in X , and the limit is taken overall open sets V of V containing f ( U ) . Definition H.4.7. [38] Let F , G be sheaves of Abelian groups on X . For anyopen set U ⊆ X the set of morphisms Hom ( F | U , G | U ) has the natural structureof Abelian group. It is a sheaf (cf. [38]). It is called the sheaf of local morphisms of F → G , "sheaf hom" for short, and is denoted by H om ( F , G ) .Recall that for s ∈ F ( X ) , supp s = { x ∈ X | s ( x ) = } denotes the support ofthe section s . H.5 Isospectral deformations
A very general construction of isospectral deformations of noncommutative ge-ometries is described in [19]. The construction implies in particular that anycompact Riemannian manifold M which admits a spin c structure (cf. Defini-tion E.4.2), whose isometry group has rank ≥ M θ . We let ( C ∞ ( M ) , H = L ( M , S ) , / D ) be the canonical spectral triple associated with a compact spin-manifold M . We recall that A = C ∞ ( M ) is the algebra of smooth functions on528 , S is the spinor bundle and / D is the Dirac operator. Let us assume that thegroup Isom ( M ) of isometries of M has rank r ≥
2. Then, we have an inclusion T ⊂ Isom ( M ) , (H.5.1)with T = R /2 π Z the usual torus, and we let U ( s ) , s ∈ T , be the correspondingunitary operators in H = L ( M , S ) so that by construction U ( s ) / D = / D U ( s ) .Also, U ( s ) a U ( s ) − = α s ( a ) , ∀ a ∈ A , (H.5.2)where α s ∈ Aut ( A ) is the action by isometries on the algebra of functions on M .We let p = ( p , p ) be the generator of the two-parameters group U ( s ) so that U ( s ) = exp ( i ( s p + s p )) .The operators p and p commute with D . Both p and p have integral spectrum,Spec ( p j ) ⊂ Z , j =
1, 2 .One defines a bigrading of the algebra of bounded operators in H with the oper-ator T declared to be of bidegree ( n , n ) when, α s ( T ) = exp ( i ( s n + s n )) T , ∀ s ∈ T ,where α s ( T ) = U ( s ) T U ( s ) − as in (H.5.2).Any operator T of class C ∞ relative to α s (i. e. such that the map s → α s ( T ) isof class C ∞ for the norm topology) can be uniquely written as a doubly infinitenorm convergent sum of homogeneous elements, T = ∑ n , n b T n , n ,with b T n , n of bidegree ( n , n ) and where the sequence of norms || b T n , n || is ofrapid decay in ( n , n ) . Let λ = exp ( π i θ ) . For any operator T in H of class C ∞ we define its left twist l ( T ) by l ( T ) = ∑ n , n b T n , n λ n p , (H.5.3)529nd its right twist r ( T ) by r ( T ) = ∑ n , n b T n , n λ n p ,Since | λ | = p , p are self-adjoint, both series converge in norm. Denote by C ∞ ( M ) n , n ⊂ C ∞ ( M ) the C -linear subspace of elements of bidegree ( n , n ) .One has, Lemma H.5.1. [19] a) Let x be a homogeneous operator of bidegree ( n , n ) and y be a homogeneous oper-ator of bidegree ( n ′ , n ′ ) . Then,l ( x ) r ( y ) − r ( y ) l ( x ) = ( x y − y x ) λ n ′ n λ n p + n ′ p (H.5.4) In particular, [ l ( x ) , r ( y )] = if [ x , y ] = . b) Let x and y be homogeneous operators as before and definex ∗ y = λ n ′ n xy ; then l ( x ) l ( y ) = l ( x ∗ y ) . The product ∗ defined in (H.5.1) extends by linearity to an associative product onthe linear space of smooth operators and could be called a ∗ -product. One couldalso define a deformed ‘right product’. If x is homogeneous of bidegree ( n , n ) and y is homogeneous of bidegree ( n ′ , n ′ ) the product is defined by x ∗ r y = λ n n ′ xy .Then, along the lines of the previous lemma one shows that r ( x ) r ( y ) = r ( x ∗ r y ) .We can now define a new spectral triple where both H and the operator / D areunchanged while the algebra C ∞ ( M ) is modified to l ( C ∞ ( M )) . By Lemma H.5.1 b)one checks that l ( C ∞ ( M )) is still an algebra. Since / D is of bidegree (
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