A Mayer-Vietoris Spectral Sequence for C*-Algebras and Coarse Geometry
aa r X i v : . [ m a t h . K T ] M a y A Mayer-Vietoris Spectral Sequencefor C*-Algebras and Coarse Geometry
Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades„Doctor rerum naturalium“der Georg-August-Universität Göttingenim Promotionsprogramm „School of Mathematical Sciences“der Georg-August University School of Science (GAUSS)vorgelegt vonSimon Naarmannaus LippstadtGöttingen, 2018 etreuungsausschuss
Erstbetreuer:
Prof. Dr. Thomas Schick
Mathematisches InstitutGeorg-August-Universität GöttingenZweitbetreuer:
Prof. Dr. Ralf Meyer
Mathematisches InstitutGeorg-August-Universität Göttingen
Mitglieder der Prüfungskommission
Referent:
Prof. Dr. Thomas Schick
Mathematisches InstitutGeorg-August-Universität GöttingenKoreferent:
Prof. Dr. Ralf Meyer
Mathematisches InstitutGeorg-August-Universität Göttingen
Weitere Mitglieder der Prüfungskommission
Prof. Dr. Gert Lube
Institut für Numerische und Angewandte MathematikGeorg-August-Universität Göttingen
Prof. Dr. Thorsten Hohage
Institut für Numerische und Angewandte MathematikGeorg-August-Universität Göttingen
Prof. Dr. Ingo Witt
Mathematisches InstitutGeorg-August-Universität Göttingen
Prof. Dr. Chenchang Zhu
Mathematisches InstitutGeorg-August-Universität Göttingen
Tag der mündlichen Prüfung
10. September 2018 bstract
Let A be a C*-algebra that is the norm closure A = P β ∈ α I β of an arbitrary sumof C*-ideals I β ⊆ A . We construct a homological spectral sequence that takes asinput the K-theory of T j ∈ J I j for all finite nonempty index sets J ⊆ α and convergesstrongly to the K-theory of A .For a coarse space X , the Roe algebra C ∗ X encodes large-scale properties. Givena coarsely excisive cover { X β } β ∈ α of X , we reshape C ∗ X β as input for the spectralsequence. From the K-theory of C ∗ (cid:0) T j ∈ J X j (cid:1) for finite nonempty index sets J ⊆ α ,we compute the K-theory of C ∗ X if α is finite, or of a direct limit C*-ideal of C ∗ X if α is infinite.Analogous spectral sequences exist for the algebra D ∗ X of pseudocompact finite-propagation operators that contains the Roe algebra as a C*-ideal, and for Q ∗ X = D ∗ X/ C ∗ X . iii cknowledgements I sincerely thank my main advisor Professor Dr. Thomas Schick for sparking my in-terest in coarse geometry, for suggesting research goals that matched my backgroundin algebraic topology, and for his excellent supervision throughout my project. Hegranted me the liberty to expand my research in directions that interested me most.I have felt deep mutual trust both as his doctorate student and as his teachingassistant. I continue to respect his knowledge and intuition.I thank my secondary advisor Professor Dr. Ralf Meyer for his support and hisformidable strategic advice. I have known both Professor Dr. Ralf Meyer and Pro-fessor Dr. Thomas Schick since my Bachelor’s and Master’s studies; they laid thefoundation for my research.I thank my parents Elianne and Bernd Naarmann and my brother Julian for theirpatience and encouragement throughout this project. From my childhood on, theyhave conveyed the importance of good education.I thank my girlfriend Sylvia for her emotional support and her understanding.I thank my friends and fellow mathematicians Thorsten and Mehran for theirunconditional enthusiasm and their inspiration.My work was funded by the German Research Foundation (DFG) via the ResearchTraining Group 1493, Mathematical Structures in Modern Quantum Physics, fromJune 2013 through September 2013. Afterwards, I worked half-time as a scientificassistant at the Mathematical Institute in Göttingen from October 2013 through Au-gust 2017. Finally, I received a DFG stipend from November 2017 through December2017.I am grateful for all support and funding. v ontents
Contents K ∗ C ∗ R n . . . . . . . . . . . . . . . . . . . . . . . . . . . 55vii ontents Z ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.4 Wedge sum of rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Introduction
C*-algebras arise in mathematics and theoretical physics alike. K-theory is a funda-mental tool to study and classify these highly-structured algebras.As a covariant Z -graded functor of C*-algebras over C , K-theory is continuous,is additive, admits suspension isomorphisms K s SA ∼ = K s +1 A , and admits a cyclic6-term exact sequence induced by C*-ideal inclusions I ⊆ A via boundary maps andBott periodicity β : K s A ∼ = K s +2 A . Furthermore, for a sum A = I + I of twoC*-ideals, there is a Mayer-Vietoris exact sequence, K ( I ∩ I ) K I ⊕ K I K AK A K I ⊕ K I K ( I ∩ I ) . ∂ MV2 ◦ β∂ MV1
In algebraic topology, a similar Mayer-Vietoris sequence computes the homology orcohomology of a space X from a cover X = X ◦ ∪ X ◦ by relating the (co)homologyof X , X , and X ∩ X . This topological Mayer-Vietoris sequence generalizes to aspectral sequence: Given a suitable cover { X β } β ∈ α of a topological space, the spectralsequence takes as input the (co)homology of T j ∈ J X j for all finite nonempty J ⊆ α and converges to the (co)homology of X , the full space.It is natural to seek analogous spectral sequences for the K-theory Mayer-Vietorisexact sequence. This is our first main result: Theorem 6.7.1 (Spectral sequence for arbitrary sums) . Let α be an arbitrary indexset: finite, countable, or uncountable. Let A = P β ∈ α I β be the norm closure of asum of | α | -many C*-ideals I β ⊆ A . There is a spectral sequence { E rp,q , d r } r,p,q with E p,q ∼ = M | J | = p +1 K q (cid:16) \ j ∈ J I j (cid:17) for p ≥ , for p < ,where J ranges over all nonempty finite index subsets J ⊆ α . In general, this is ahalf-page spectral sequence, any term E p,q with p ≥ may be nonzero.This spectral sequence converges strongly to K ∗ A . It is functorial with respect to ∗ -homomorphisms that preserve α -indexed ideal decompositions. To prove this, we begin with the finite case A = I + I + · · · + I n and construct1 Introduction
C*-algebras of continuous functions from the standard simplex ∆ n to A , interlockingallowed ranges in the I j ⊆ A on different regions of the simplex.Sums of these function algebras become a chain of ideals Q ⊆ Q ⊆ · · · ⊆ Q n .This chain of ideals fits into an already-known spectral sequence that convergesstrongly to K ∗ Q n ∼ = K ∗ ( S n A ) , the K-theory of the n -fold suspension of A . Thisspectral sequence for ideal inclusions has been described by C. Schochet in [Sch81];we reprove it to highlight its inner workings, its differentials, and its filtration thatguarantees strong convergence. Compared to that spectral sequence, our Theorem6.7.1 relaxes the input conditions: We do not require that the I j form a chain ofinclusions.For countable index sets α , we link the spectral sequences for n ideals and n + 1 ideals – this is only possible on the level of K-theory, not on the level of C*-algebras– and construct a filtration on the spectral sequence via a suitable direct limit. Foruncountable sets α , we adapt our direct limit construction to the directed system offinite subsets of α . Coarse geometry studies the large-scale structure of metric spaces. If two spacesdiffer only within a compact set, coarse invariants will not detect any difference.For a coarse space X , i.e., a metric space ( X, d ) , the Roe algebra C ∗ X encodes suchlarge-scale properties. This C*-algebra and the larger algebra D ∗ X are introduced,e.g., by N. Higson and J. Roe in [HR00], or by J. Roe in [Roe96]. Via these algebras,the K-homology of X and further invariants of contemporary research are defined,such as the coarse index when X is a Riemannian manifold. For our work, it sufficesto define Q ∗ X = D ∗ X/ C ∗ X ; we will not formulate our results in the language ofK-homology.For certain sets X and X with X ∪ X = X , there is a coarse Mayer-Vietorisexact sequence: It relates the K-theory of C ∗ X , C ∗ X , and C ∗ ( X ∩ X ) to theK-theory of C ∗ X . Its proof, e.g., in [Roe96], relies on the Mayer-Vietoris sequencefor two abstract C*-ideals.We generalize to arbitrarily many regions. A decomposition X = S β ∈ α X β iscalled coarsely excisive if, for all nonempty finite subcollections J ⊆ α and R > ,there exists S > such that the intersection of the R -neighborhoods is contained inthe S -neighborhood of the intersection according to the metric d on X : \ j ∈ J N d ( X j , R ) ⊆ N d (cid:16) \ j ∈ J X j , S (cid:17) . This leads to our second main result:2 .3 Relations to other research
Theorem 7.2.1 (Spectral sequence for coarsely excisive covers) . Let ( X, d ) be acoarse space and let { X β } β ∈ α be a coarsely excisive cover of ( X, d ) . Let F ∗ be eitherthe functor C ∗ from the coarse category to C ∗ A or one of the functors D ∗ or Q ∗ fromthe coarse-continuous category to C ∗ A . There is a spectral sequence { E rp,q , d r } r,p,q with E p,q ∼ = M | J | = p +1 K q F ∗ (cid:16) \ j ∈ J X j (cid:17) for p ≥ , for p < ,where J ranges over all nonempty finite subcollections of indices in α . For finite α , this spectral sequence converges strongly to K ∗ F ∗ X . In general, the spectral se-quence converges strongly to the K-theory of S J P j ∈ J F ∗ ( X j ⊆ X ) , a C*-ideal of F ∗ X , where J ranges over all finite subcollections of indices in α . The spectral se-quence is functorial with respect to morphisms (coarse maps for C ∗ , or coarse andcontinuous maps for D ∗ and Q ∗ ) to other coarse spaces with compatible coarselyexcisive covers (Definition 5.1.4). As an example, we recompute the known K-theory of C ∗ R n . Also, we find aninfinite coarsely excisive cover of Z ∞ under many metrics, then show that the K-theory of the direct limit ideal of Roe algebras for this cover in C ∗ Z ∞ vanishes.Furthermore, we compute the K-theory of the direct limit of Roe algebras for acountable wedge sum W N [0 , ∞ [ in a single application of the spectral sequence; thisobviates inductive proofs with the Mayer-Vietoris exact sequence. Early motivation for this project was the Partitioned Manifold Index Theorem [Sie12,Proposition 4.9]: Given certain Riemannian manifolds N ⊆ M with G -equivariantcovers, the classes of their Dirac operators – elements in K-homology – map tothe same element in K ∗ C ∗ r G , the K-theory of the group C*-algebra for G , via thecoarse index maps. P. Siegel proves this by induction with the coarse Mayer-Vietorisprinciple for two regions. Our idea was to reprove this theorem by a single applicationof our Theorem 7.2.1.Spectral sequences, however, do not construct specific maps on their targets; evenstrong convergence only leads to isomorphism theorems. Still, if the spectral sequencecannot compute the equality for the Partitioned Manifold Index Theorem, it canclassify related C*-algebras for coarse spaces in this setting and decide about thestructure of possible morphisms between them.Sums or inclusion chains of abstract C*-ideals also arise in other settings. In[MM18], D. Mukherjee and R. Meyer construct a gauge-invariant C*-algebra T of3 Introduction the Toeplitz algebra T for partial product systems. By [MM18, Theorem 4.5], T isa direct limit along N ∈ N of images of maps from L n We will rehearse well-known constructions to establish notation and conventions,beginning with C*-algebras and their K-theory. We introduce the basics of coarsegeometry and spectral sequences.The set N of natural numbers includes . Definition 2.1.1 (Complex Banach algebra) . Let ( A, k−k ) be a normed associativealgebra over C that is topologically complete according to its norm. For all x , y ∈ A ,the following inequality shall hold: k xy k ≤ k x k k y k . Then we call A a complexBanach algebra . Definition 2.1.2 (C*-algebra, C*-ideal) . Let A be a complex Banach algebra. Let A carry an involution ∗ , i.e., a map A → A , x x ∗ , satisfying x ∗∗ = x , ( λx + y ) ∗ = λx ∗ + y ∗ , and ( xy ) ∗ = y ∗ x ∗ for all x , y ∈ A and λ ∈ C . Furthermore, ∗ shall satisfythe C*-identity k xx ∗ k = k x k . Then ( A, ∗ ) is a C*-algebra .A closed two-sided ideal in a C*-algebra is called a C*-ideal . Remark 2.1.3. The zero algebra is a C*-algebra. The complex numbers C them-selves are a C*-algebra with z 7→ | z | as the norm and z z as the ∗ -operation.For a C*-algebra A with a C*-ideal I ⊆ A , the quotient A/I is a well-definedC*-algebra.Let α be an arbitrary index set and let A β be a C*-algebra for each β ∈ α . Thenorm completion L β ∈ α A β of the algebraic direct sum is again a C*-algebra with thenorm k ( a β ) β ∈ α k = sup {k a β k : β ∈ α } and component-wise addition, multiplication,and involution. Definition 2.1.4 (Category C ∗ A of C*-algebras) . A bounded algebra homomor-phism f : A → B between two C*-algebras is called a ∗ -homomorphism if it preservesthe involution: f ( x ∗ ) = f ( x ) ∗ shall hold for all x ∈ A .The category C ∗ A encompasses all C*-algebras as objects, together with all ∗ -ho-momorphisms as arrows. Definition 2.1.5 (Homotopy of C*-algebras) . Let f , g : A → B be ∗ -homomor-phisms between C*-algebras. A homotopy in the category of C*-algebras between f and g is a map H : ( A × [0 , → B that satisfies: • For all a ∈ A , H ( a, 0) = f ( a ) and H ( a, 1) = g ( a ) . • For all a ∈ A , the map [0 , → B , t H ( a, t ) is continuous. 5 Fundamentals • For all t ∈ [0 , , the map A → B , a H ( a, t ) is a ∗ -homomorphism.If such a homotopy exists, then f and g are called homotopic , denoted by f ∼ g .A ∗ -homomorphism f : A → B is a homotopy equivalence if there exists a ∗ -ho-momorphism g : B → A such that g ◦ f ∼ id ( A ) and f ◦ g ∼ id ( B ) . Existence ofa homotopy equivalence A → B is denoted by A ≃ B ; then A is called homotopyequivalent to B .A C*-algebra A is called contractible if A ≃ .Homotopy equivalence as C*-algebras is a stronger condition than topological ho-motopy equivalence. Lemma 2.1.6. Let A be a contractible C*-algebra and p = p ∗ = p a projection in A . Then p = 0 .Proof. Let H : A × [0 , → A be the contracting homotopy. For all t ∈ [0 , , wehave k H ( p, t ) k = k H ( pp ∗ , t ) k = k H ( p, t ) k ∈ R ≥ because x H ( x, t ) is a ∗ -ho-momorphism. This implies k H ( p, t ) k ∈ { , } and, because of continuity, this normstays constant across all t ∈ [0 , . By construction, H ( p, 1) = 0 , thus k H ( p, k = 0 and p = 0 . Corollary 2.1.7. The C*-algebra C is contractible as a topological space, but notcontractible as a C*-algebra.Proof. One possible topological homotopy is ( z, t ) tz . Lemma 2.1.6 precludes aC*-contraction because · is a nonzero projection in C . Definition 2.1.8 (Cone) . Let A be a C*-algebra. The cone of A is the C*-algebra CA = { f : [0 , → A : f is continuous, f (0) = 0 } . It carries the uniform norm k f k = sup {k f ( x ) k : x ∈ [0 , } ; this is well-defined be-cause [0 , is compact. Algebra multiplication on CA is given by pointwise multi-plication of functions. The involution on the cone is defined by f ∗ ( x ) = f ( x ) ∗ .Let g : A → B be a ∗ -homomorphism. The cone map Cg : CA → CB is given by ( Cg )( f ) = g ◦ f : [0 , → B . This construction turns C : C ∗ A → C ∗ A into a covariantfunctor.This is still basic theory of C*-algebras, but it is reasonable to agree on whetherthe functions f in the cone must vanish at 0 or vanish at 1. In later sections, we willconstruct a spectral sequence for C*-algebras; some technical lemmas require conesof algebras.6 .2 C*-algebras for spaces Proposition 2.1.9. Let A be a C*-algebra. Then the cone CA is contractible; thezero algebra is a strong deformation retract of CA .Proof. Define a homotopy H : CA × [0 , → CA by H ( f, t )( x ) = f ( tx ) . This iscontinuous because f is continuous. It is the desired deformation retraction because H ( f, 0) = 0 and H ( f, 1) = f for all f ∈ CA . Since H (0 , t )( x ) = 0 for all t and x ∈ [0 , , it is even a strong deformation retraction. Definition 2.1.10 (Suspension) . Let A be a C*-algebra and CA its cone. The suspension of A is the subalgebra SA = { f ∈ CA : f (1) = 0 } . The suspension SA inherits its C*-algebra structure from CA . Likewise, ∗ -homomor-phisms ϕ : A → B induce Sϕ : SA → SB via Sϕ = ( Cϕ ) ↾ SA , making S : C ∗ A → C ∗ A another covariant functor. Definition 2.2.1 ( C ( X, A ) , C X ) . Let X be a compact Hausdorff space and A aC*-algebra. Then C ( X, A ) denotes the C*-algebra of A -valued continuous functionson X with the sup-norm k f k C ( X,A ) = sup {k f ( x ) k A : x ∈ X } , pointwise additionand multiplication, and f ∗ ( x ) = f ( x ) ∗ . If A is unital, C ( X, A ) contains the constantfunction that maps all points in X to ∈ A ; this function is then a unit.Often, A = C ; we abbreviate by setting C X = C ( X, C ) .When X fails to be compact, C X is not a normed algebra because some A -valued functions on X are unbounded. More interesting function algebras imposeboundedness: Definition 2.2.2 ( C ( X, A ) , C X ) . Let X be a locally compact Hausdorff space, A a C*-algebra. A continuous function f : X → A vanishes at infinity if, for all ε > ,there exists a compact set K ⊆ X with { x ∈ X : k f ( x ) k > ε } ⊆ K. The set of all such functions f is denoted C ( X, A ) . In the common case A = C , weshall write C X = C ( X, C ) .Again, C ( X, A ) carries a C*-algebra structure under pointwise multiplication andthe sup-norm k f k = sup {k f ( x ) k : x ∈ X } . 7 Fundamentals Remark 2.2.3. The sup-norm is well-defined because functions in C ( X, A ) arenecessarily bounded. The C*-algebra C ( X, A ) has a unit if and only if X is compactand A is unital. For compact X , the algebra C ( X, A ) coincides with C ( X, A ) .Equivalent definitions of C ( X, A ) embed X into an arbitrary compactification Y ,then define C ( X, A ) as the subset of all continuous functions f : Y → A such that f ↾ ( Y − X ) = 0 , then restrict these functions to X .Taking A -valued functions that vanish at infinity is a contravariant functor fromHausdorff spaces with proper continuous maps into C ∗ A : Let X and Y be Hausdorffspaces and let f : X → Y be a proper continuous map. Then ( − ◦ f ) : C ( Y, A ) → C ( X, A ) maps g : Y → A to g ◦ f : X → A . The composition g ◦ f vanishes atinfinity because f is proper. The exact constructions of the K-theory K ∗ A for a C*-algebra A are lengthy andshall be omitted; several textbooks, e.g., [WO93] or [RLL00], cover all technicaldetails. The zeroth K-theory group K A is the Grothendieck group of equivalenceclasses of projections in a ring of matrices over A modulo homotopy equivalence. Thefirst K-theory group K A results from a similar construction with unitary elementsof the matrix ring instead of projections.Both K and K become continuous covariant functors from C ∗ A to abelian groups:For a morphism f : A → A ′ , the resulting morphism K ∗ f : K ∗ A → K ∗ A ′ applies f to all matrix entries before taking equivalence classes. Theorem 2.3.1 (Suspension isomorphism) . For a C*-algebra A , there is an iso-morphism σ : K SA → K A . This allows N -graded K-theory by defining K s A as K s − SA inductively; some authors even define K A this way instead of via unitarymatrix elements. Theorem 2.3.2 (Bott isomorphism) . For all s ∈ N , there are Bott isomorphisms β : K s A → K s +2 A . This allows Z -graded K-theory by defining K s A = K s +2 A induc-tively for all s < . Theorem 2.3.3 (Six-term exact sequence) . Let I ⊆ A be a C*-ideal. For all s ∈ Z , K-theory admits boundary maps ∂ s : K s ( A/I ) → K s − I that make the followingsix-term sequence exact; the horizontal arrows are induced by ideal inclusion and .4 Roe algebras projection: K I K A K ( A/I ) K ( A/I ) K A K I. ∂ ◦ β∂ Theorem 2.3.4 (Abstract Mayer-Vietoris exact sequence) . Let A be a C*-algebrasuch that I , I ⊆ A are two C*-ideals with I + I = A . There is an exact sequencewith Mayer-Vietoris boundary morphisms: K ( I ∩ I ) K I ⊕ K I K AK A K I ⊕ K I K ( I ∩ I ) . ∂ MV2 ◦ β∂ MV1 Definition 2.4.1 (Ample representation) . Let A be a separable C*-algebra. Let ̺ : A → BH be a representation of C*-algebras on a separable Hilbert space H ,where BH denotes the C*-algebra of all bounded linear operators H → H . Then ̺ is called ample if • ̺ is nondegenerate, and • ̺ (0) = 0 is the only compact operator in im ( ̺ ) ⊆ BH . Definition 2.4.2 (Very ample representation) . Let A be a separable C*-algebra. Arepresentation ̺ : A → BH of C*-algebras is called very ample if it is a countablyinfinite sum of ample representations. Remark 2.4.3. To admit ample representations, the Hilbert space H must be bothseparable and infinite-dimensional. Then suitable ample representations ̺ alwaysexist. According to [HR00], because H is separable, the constructions in Section 2.4do not depend on the particular choice of H or ̺ up to isomorphy.Every very ample representation is ample. Most constructions require ample rep-resentations. Some isomorphism theorems call for very ample representations, but,because H ∼ = L N H , requiring very ample representations is merely a technical con-venience, not a fundamental restriction. Definition 2.4.4 (Pseudolocal operator) . Let X be a locally compact Hausdorffspace. For the C*-algebra C X , let ̺ : C X → BH be an ample representation. Let9 Fundamentals T ∈ BH be an operator such that ̺ ( f ) T − T ̺ ( f ) is a compact operator in BH forall f ∈ C X . Then T is called pseudolocal . Definition 2.4.5 (Finite propagation) . Let ( X, d ) be a locally compact metric spaceand ̺ : C X → BH an ample representation. An operator T ∈ BH has finitepropagation if there exists a constant R > such that for all f , g ∈ C X with d ( supp f, supp g ) ≥ R , the product ̺ ( f ) T ̺ ( g ) ∈ BH is zero. Definition 2.4.6 ( D ∗ A ) . Let ( X, d ) be a locally compact metric space. Fix anample representation ̺ : C X → BH . The norm closure of the set of all pseudolocaloperators in BH with finite propagation forms a C*-algebra, denoted by D ∗ X . Remark 2.4.7. The norm closure turns D ∗ X into a sub-C*-algebra of BH . Withoutthe norm closure, the operators with finite propagation do not form a closed subset.Pseudolocal operators by themselves already form a sub-C*-algebra in BH withoutadditional closure. Definition 2.4.8 (Locally compact operator) . Let ( X, d ) be a locally compact metricspace and ̺ : C X → BH an ample representation. Let T ∈ BH be an operatorsuch that, for all f ∈ C X , both ̺ ( f ) T and T ̺ ( f ) are compact operators in BH .Then T is called locally compact . Remark 2.4.9. Given ̺ : C X → BH ample for a locally compact metric space ( X, d ) , the locally compact operators form a C*-ideal in the algebra of pseudolocaloperators. Definition 2.4.10 ( C ∗ X , Roe algebra) . For a locally compact metric space ( X, d ) and an ample representation ̺ : C X → BH , the translation algebra or Roe algebra C ∗ X is the norm closure of the operators T ∈ BH that are both locally compact andhave finite propagation. Remark 2.4.11. The Roe algebra C ∗ X is a C*-ideal in D ∗ X . Remark 2.4.12 (K-homology) . Let A be a C*-algebra and A + the C*-algebra with aunit adjoined. It is possible to define an abstract dual algebra D ∗ A + by representing A amply and taking all pseudolocal operators, without defining finite propagation.For s ∈ Z , we may define the s -th K-homology group of A as K s A = K − s D ∗ A + . K-theory of C*-algebras is a covariant functor; K-homology becomes a contravariantfunctor of C*-algebras. Were we concerned only with abstract C*-algebras, we could10 .5 Coarse spaces consider “K-homology” a bad name for a contravariant functor and to rename it to“K-cotheory”. But K-homology becomes a covariant functor for topological spaces:Let X be a locally compact metric space and s ∈ Z . The abelian group K s X = K − s C X is the K-homology of the space X ; this defines a covariant functor from locally com-pact metric spaces to abelian groups. By [HR00, Lemma 12.3.2], there is an isomor-phism K s X = K s +1 ( D ∗ X/ C ∗ X ) .We will not need K-homology and will instead formulate all results in the languageof K-theory and Roe algebras. Thus we introduce a notation similar to [Sie12]: Notation 2.4.13 ( Q ∗ X ) . Let X be a locally compact metric space. We write Q ∗ X = D ∗ X/ C ∗ X. The most general definition of a coarse space X uses entourages or controlled sets – collections of subsets of X × X with axioms to capture a notion of closeness.Following [Roe96, Chapter 2], we will instead work with proper metric spaces, amodest restriction. If our spaces are manifolds, both methods bring the same results. Definition 2.5.1 (Coarse space) . A coarse space X = ( X, d ) is a proper metricspace; i.e., a metric space where closed d -bounded sets are compact. Definition 2.5.2 (Coarse map) . Let f : ( X, d X ) → ( Y, d Y ) be a map between coarsespaces. f is called coarse if • f is uniformly expansive: For R > , there exists S > such that for all x , x ′ ∈ X with d X ( x, x ′ ) ≤ R , we have d Y ( f x, f x ′ ) ≤ S . • f is proper as a map between the metric spaces X and Y : For each boundedset B ⊆ Y , the preimage f − ( B ) is bounded in X .Coarse maps are not required to be continuous. Remark 2.5.3. The identity id ( X ) : ( X, d ) → ( X, d ) is coarse. Compositions ofcoarse maps are coarse. Definition 2.5.4 (Coarse category, coarse-continuous category) . The coarse cate-gory has as objects all coarse spaces and as morphisms all coarse maps. 11 Fundamentals The coarse-continuous category is the subcategory of the coarse category that stillcomprises all coarse spaces, but that has as morphisms only the coarse maps thatare also continuous. Definition 2.5.5 (Closeness) . Let f , f ′ : ( X, d X ) → ( Y, d Y ) be two maps betweencoarse spaces. We call f close to f ′ , or coarsely equivalent to f ′ , if there exists S > such that for all x ∈ X , we have d Y ( f x, f ′ x ) ≤ S . Definition 2.5.6 (Coarse equivalence) . Let X and Y be coarse spaces with coarsemaps f : X → Y and g : Y → X such that g ◦ f is close to id ( X ) and f ◦ g is closeto id ( Y ) . We call X , Y coarsely equivalent and f , g coarse equivalences . Example 2.5.7. Fix n ∈ N . The lattice Z n is coarsely equivalent to Euclideanspace R n under the metric d ∞ with d ∞ ( x, x ′ ) = sup j Similarly, the constructions D ∗ and Q ∗ are functorial, but thesefunctors are merely well-defined on the coarse-continuous category. Only C ∗ is well-defined for coarse non-continuous maps.All three of C ∗ , D ∗ , and Q ∗ are covariant functors to C ∗ A : Passing from spaces X and Y to function algebras C X and C Y is contravariant, and passing from functionalgebras to locally compact or pseudocompact operators with finite propagation isagain contravariant. Corollary 2.5.10. Let f : X → Y and g : Y → X be coarse equivalences. Then K p C ∗ X ∼ = K p C ∗ Y for all p ∈ Z .Proof. The compositions g ◦ f and f ◦ g are close to the identities on X and Y . Theyinduce identities in K-theory, thus both K p C ∗ f and K p C ∗ g are isomorphisms.12 .6 Coarsely excisive pairs We will recapitulate coarse excision as defined by J. Roe in [Roe96]. Later, we willdefine coarsely excisive covers to generalize this idea. Definition 2.6.1 ( R -neighborhood) . Let ( X, d ) be a metric space and Y ⊆ X asubspace. For a real number R > , define the R -neighborhood of Y as N d ( Y, R ) = { x ∈ X : inf { d ( x, y ) : y ∈ Y } ≤ R } . When d is a standard metric such as the -metric d , the Euclidean metric d , or thesup-metric d ∞ on R n or Z n , we will also write N = N d or, similarly, N or N ∞ . Definition 2.6.2 (Coarsely excisive pair) . Let ( X, d ) be a metric space. Let U and V be subspaces of X with U ∪ V = X .The pair ( U, V ) is called a coarsely excisive pair for X if, for every distance R > ,there exists a distance S > such that the intersection of the R -neighborhoods iscontained in the S -neighborhood of the intersection: N d ( U, R ) ∩ N d ( V, R ) ⊆ N d ( U ∩ V, S ) . Example 2.6.3. For the metric space R with its standard metric d , the pair ofsubspaces ( R ≤ , R ≥ ) is coarsely excisive: The R -neighborhoods are N d ( R ≤ , R ) =] ∞ , R ] and N d ( R ≥ , R ) = [ − R, ∞ [ . Their intersection is [ − R, R ] , which, for S = R ,is the S -neighborhood of R ≤ ∩ R ≥ = { } .In the same vain, R n +1 admits the coarsely excisive pair ( R n × R ≤ , R n × R ≥ ) under d , d , or d ∞ . Example 2.6.4. For all S > , the S -neighborhood of ∅ is again ∅ . This imposesrestrictions on eligible coarsely excisive pairs: In any metric space ( X, d ) , disjointnonempty sets U and V cannot form a coarsely excisive pair. Choose R larger than inf { d ( x, y ) : x ∈ U , y ∈ V } , then N ( U, R ) ∩ N ( V, R ) contains points. This is nevera subset of N ( U ∩ V, S ) = N ( ∅ , S ) = ∅ . Theorem 2.6.5 ([HRY93, Section 5]) . For a coarsely excisive pair ( U, V ) of ( X, d ) ,there is an exact Mayer-Vietoris sequence: K C ∗ ( U ∩ V ) K C ∗ U ⊕ K C ∗ V K C ∗ XK C ∗ X K C ∗ U ⊕ K C ∗ V K C ∗ ( U ∩ V ) . Fundamentals Boths proofs in [HRY93] and [Roe96] reduce this to the Mayer-Vietoris principlefor abstract C*-ideals I , I ⊆ A , Theorem 2.3.4.The goal of this thesis is to construct a spectral sequence extending that abstractMayer-Vietoris principle and Theorem 2.6.5 alongside. A good introduction to spectral sequences is [McC01]. We will give the basic defini-tions to establish notation. We require no cohomological spectral sequences or ringstructures on the pages. Definition 2.7.1 (Spectral sequence) . A spectral sequence (of homological type,of abelian groups) is a system of bigraded differential abelian groups E rp,q for all = r ∈ N and p , q ∈ Z with differentials d r : E rp,q → E rp − r,q + r − for all r , p , q suchthat each E r +1 p,q is the homology of d r at E rp,q .We define convergence of spectral seuqences with notation similar to [Boa99, Sec-tion 5]; that exposition does not assert any common origin of the target group andthe E r ∗ , ∗ -terms of the spectral sequence. We re-index to match our spectral sequencesof homological type and make explicit the grading of the Z -graded target group. Definition 2.7.2. Let G be an abelian group with an increasing filtration · · · ⊆ F p G ⊆ F p +1 ⊆ · · · ⊆ G for p ∈ Z . We call the filtration { F p G } p ∈ Z • Hausdorff if T p ∈ Z F p G = 0 , • exhaustive if S p ∈ Z F p G = G , and • complete if the right-derived functor of taking the inverse limit yields the zerogroup for the inverse system F p G for p → −∞ . Definition 2.7.3 (Strong convergence) . Let { E rp,q , d r } r,p,q be a spectral sequence.For r ≥ and p , q ∈ Z , write Z rp,q = ker d r : E rp,q → E rp − r,q + r − ,B rp,q = im d r : E rp + r,q − r +1 → E rp,q . Because E r ∗ , ∗ for r ≥ is the homology of E r − ∗ , ∗ under d r − , an element in E rp,q maybe written as x + B r − p,q with x ∈ E r − p,q . Recursively, this allows us to treat Z rp,q and14 .7 Spectral sequences B rp,q as subgroups of E p,q and define E ∞ p,q = (cid:16) \ r ≥ Z rp,q (cid:17) . (cid:16) [ r ≥ B rp,q (cid:17) . Let G = L s ∈ Z G s be a Z -graded abelian group. The spectral sequence { E rp,q , d r } r,p,q converges strongly to G if there exist increasing filtrations { F p G s } p ∈ Z of each sum-mand G s such that these filtrations are Hausdorff, exhaustive, complete, and allowisomorphisms E ∞ p,q ∼ = F p G p + q /F p − G p + q . Definition 2.7.4 (Morphism of spectral sequences) . Given two spectral sequences { E rp,q , d r } r,p,q and { ¯ E rp,q , ¯ d r } r,p,q and ( p ′ , q ′ ) ∈ Z , a morphism of spectral sequencesof bidegree ( p ′ , q ′ ) is a system of morphisms of abelian groups, f = (cid:8) f rp,q : E rp,q → ¯ E rp + p ′ ,q + q ′ (cid:9) r,p,q , such that • the group morphisms commute with the differentials; i.e., f r ∗ , ∗ ◦ d r = ¯ d r ◦ f r ∗ , ∗ for all pages r , and • each map f r ∗ , ∗ induces f r +1 ∗ , ∗ by passing to homology on { E r ∗ , ∗ , d r } r,p,q and { ¯ E r ∗ , ∗ , ¯ d r } r,p,q . Remark 2.7.5. Spectral sequences with these morphisms form a category.By describing a morphism of spectral sequences on the R -th page, all subsequent f r ∗ , ∗ for r > R and r = ∞ are implicitly defined because the E r ∗ , ∗ -terms are iterativehomologies of the earlier E R ∗ , ∗ -term. In our setting, we will construct morphisms ofspectral sequences only for the E ∗ , ∗ -terms.In particular, if f R ∗ , ∗ is an isomorphism between the differential graded abeliangroups E R ∗ , ∗ and ¯ E R ∗ , ∗ , then all f r ∗ , ∗ for r > R and r = ∞ become isomorphisms. 15 Ideal inclusions Theorem 3.1.1 (Spectral sequence for ideal inclusions) . Let A = S p ∈ N I p be a C*-algebra, where the I ⊆ I ⊆ I ⊆ · · · ⊆ I p ⊆ · · · form a chain of closed two-sidedideals. There is a spectral sequence { E rp,q , d r } r,p,q with E p,q = K p + q ( I p /I p − ) . This spectral sequence converges strongly to K ∗ A ; i.e., given s ∈ Z , the groups E ∞ p,q along the diagonal s = p + q pose an extension problem to reconstruct K s A . In [Sch81], C. Schochet gave a proof of Theorem 3.1.1.Nonetheless, we will reprove Theorem 3.1.1 based on very general theory from[CE73]. This extra work reveals the inner mechanisms of the spectral sequence,shows that the convergenge is strong, and highlights naturality of all constructions:The morphisms in K-theory arise from natural inclusions of C*-ideals, quotients ofC*-ideals, and boundary maps.This spectral sequence serves as groundwork for the Mayer-Vietoris results in latersections. In [CE73], H. Cartan and S. Eilenberg construct an abstract spectral sequence froma bigraded system of groups, but they omit some details during their proof of con-vergence. Their construction uses cohomological differentials: On the E ∗ , ∗ r -page, thedifferential has the degree ( r, − r ) . For homological spectral sequences, they suggestthe renumbering E rp,q = E − p, − qr . We will state the main theorem of [CE73] in thisrenumbered notation, then prove it with all details. Definition 3.2.1 (Ungraded H-system) . Let H ( p, p ′ ) be abelian groups for p ′ ≤ p from the range Z ∪ {±∞} . We introduce the shorthand notations H ( p ) = H ( p, −∞ ) ,H = H ( ∞ ) = H ( ∞ , −∞ ) . For each ( p, p ′ ) and ( q, q ′ ) with −∞ ≤ p ≤ q ≤ ∞ and p ′ ≤ p ≤ ∞ and q ′ ≤ q ≤ ∞ ,let there be a morphism i : H ( p, p ′ ) → H ( q, q ′ ) . Ideal inclusions For each −∞ ≤ p ′′ ≤ p ′ ≤ p ≤ ∞ , let ∂ : H ( p, p ′ ) → H ( p ′ , p ′′ ) be a connecting homomorphism . We call this collection of groups together with theabove morphisms an ungraded H-system if the following axioms are satisfied:1. i : H ( p, p ′ ) → H ( p, p ′ ) is the identity.2. All triangle and square diagrams built with the morphisms i commute.3. For all p ′′ ≤ p ′ ≤ p , there is an exact sequence · · · ∂ −→ H ( p ′ , p ′′ ) i −→ H ( p, p ′′ ) i −→ H ( p, p ′ ) ∂ −→ H ( p ′ , p ′′ ) → · · · . (3.2.1.1)4. For each index p ′ ∈ Z ∪ {−∞} , the group H ( ∞ , p ′ ) is the direct limit of themorphisms i : H ( p, p ′ ) → H ( p + 1 , p ′ ) along p ′ ≤ p .In [CE73], the indices of these H-systems are denoted by ( p, q ) instead of ( p, p ′ ) .To avoid confusion with the bigrading ( p, q ) of the pages E rp,q later, we shall use H ( p, p ′ ) . Definition 3.2.2 (Graded H-system) . Let { H ( p, p ′ ) , i, ∂ } p,p ′ be an ungraded H-system as in Definition 3.2.1. We call this a graded H-system if it satisfies thefollowing extra axioms:5. All H ( p, p ′ ) carry a Z -grading: H ( p, p ′ ) = L s ∈ Z H s ( p, p ′ ) .6. All morphisms i : H ( p, p ′ ) → H ( q, q ′ ) are degree-preserving.7. All morphisms ∂ : H ( p, p ′ ) → H ( p ′ , p ′′ ) have degree − ; i.e.,im (cid:0) ∂ ↾ H s ( p, p ′ ) (cid:1) ⊆ H s − ( p ′ , p ′′ ) . Notation 3.2.3. Let H ( p, p ′ ) for −∞ ≤ p ′ ≤ p ≤ ∞ form a graded H-system. For r ≥ and q ∈ Z , write Z rp,q = im i : H p + q ( p, p − r − → H p + q ( p, p − ,B rp,q = im ∂ : H p + q +1 ( p + r, p ) → H p + q ( p, p − ,E r +1 p,q = Z rp,q /B rp,q . In this way, we define E rp,q only for r ≥ , not for r ≥ . Compared to [CE73], wehave shifted the index r in Z r ∗ , ∗ and B r ∗ , ∗ by to match our Definition 2.7.3 of these18 .3 Application: K-theory groups as closely as possible; e.g., we write Z p,q for what would be denoted by Z p,q in [CE73]. Lemma 3.2.4. We have B p,q = 0 and E p,q ∼ = Z p,q = H p + q ( p, p − . Proof. In the long exact sequence · · · → H p + q +1 ( p, p ) ∂ −→ H p + q ( p, p − i −→ H p + q ( p, p − i −→ H p + q ( p, p ) → · · · , the central map i : H ( p, p − → H ( p, p − is the identity by Definiton 3.2.1. Itsimage is Z p,q , which is all of H p + q ( p, p − . The exactness of the sequence forcesthe preceding map ∂ to vanish. The group B p,q is the image of ∂ , therefore it is thetrivial group.The main statement in [CE73] becomes: Theorem 3.2.5. With Notation 3.2.3, there is a spectral sequence { E rp,q , d rp,q } r,p,q ofhomological type. Its differentials d rp,q : E rp,q → E rp − r,q + r − are defined as the compo-sition E rp,q E rp − r,q + r − Z r − p,q /B r − p,q Z r − p,q /Z rp,q ∼ = B rp − r,q + r − /B r − p − r,q + r − Z r − p − r,q + r − /B r − p − r,q + r − , d rp,q where the three maps at the bottom are all constructed in [CE73, Chapter XV, Para-graph 1]: The bottom-left map arises from factoring out the larger group Z rp,q ⊇ B r − p,q ,the central map is an isomorphism, and the last map arises from the inclusion B rp − r,q + r − → Z r − p − r,q + r − . The homology of E r ∗ , ∗ under d r at ( p, q ) is isomorphicto E r +1 p,q . We will first look at our application to K-theory of C*-algebras, then return to thefull proof of convergence. We construct a graded H-system to compute the K-theory of a C*-algebra, takinga chain of ideals as data. This is a Z -graded theory. Bott periodicity forces K s A = Ideal inclusions K s +2 A for all C*-algebras A over the complex numbers, leaving only two differentK-groups to be computed.Throughout the remainder of Section 3, let ⊆ I ⊆ I ⊆ · · · ⊆ I p ⊆ · · · be an increasing chain of C*-ideals for p ∈ N and set A = [ p ∈ N I p . For convenience, set I p = 0 for p < , obtaining a Z -graded chain of ideals ( I p ) p ∈ Z .Commonly, the chain of ideals will stabilize after finitely many steps; i.e., thereexists n ∈ N with I n = I n +1 = I n +2 = · · · . Still, we develop the spectral sequencefor the general case without assuming stabilization. The extra work is marginal andthe final results of this thesis will rely on that general case. Definition 3.3.1. For all Z -indices p ′ ≤ p and K-theory degrees s ∈ Z , define H s ( p, p ′ ) = K s ( I p /I p ′ ) ,H s ( p ) = K s I p ,H s = K s A. The morphisms i : H s ( p, p ′ ) → H s ( p + 1 , p ′ ) in K-theory are induced by inclusionsof ideals. The morphisms of the form i : H s ( p, p ′ ) → H s ( p, p ′ + 1) are induced bythe natural projection I p /I p ′ → I p /I p ′ +1 ; this projection is well-defined because I p ′ ⊆ I p ′ +1 . All these morphisms commute with each other and preserve the degreein K-theory.The assignment H s ( p, p ′ ) = K s ( I p /I p ′ ) satisfies the direct limit axiom from Defi-nition 3.2.1 regarding H ( p ) and H : K s ( A/I q ) = colim p →∞ H s ( p, p ′ ) = H s ( ∞ , p ′ ) ,K s A = colim p →∞ H s ( p, −∞ ) = H s . For i : H s ( p ) → H s ( p, p ′ ) and i : H s ( p ) → H s , we use the respective limit maps.Because K-theory is a continuous functor, these are induced by inclusions and pro-jections of A . Again, these limit maps preserve the degree s as desired. Notation 3.3.2. We will deal with two kinds of boundary maps: The K-theoreticboundary map and the connecting homomorphism of the resulting H-system. To20 .3 Application: K-theory distinguish these, throughout Section 3, we shall denote the K-theoretic map by ∂ K and the connecting homomorpism by ∂ . Definition 3.3.3. Let p ′′ ≤ p ′ ≤ p be indices in Z . We implement the connectinghomomorphisms ∂ in the diagram · · · → H s ( p, p ′′ ) i −→ H s ( p, p ′ ) ∂ −→ H s − ( p ′ , p ′′ ) i −→ H s ( p, p ′′ ) → · · · (3.3.3.1)by a composition of maps in K-theory, making this diagram commutative: H s ( p, p ′ ) H s − ( p ′ , p ′′ ) K s ( I p /I p ′ ) K s − I p ′ K s − ( I p ′ /I p ′′ ) . ∂∂ K K s ( pr : I p ′ → I p ′ /I p ′′ ) Lemma 3.3.4. This choice of connecting homomorphism ∂ in Definition 3.3.3 makesthe sequence 3.3.3.1 exact.Proof. Let p ′′ ≤ p ′ ≤ p be indices in Z and I p ′′ ⊆ I p ′ ⊆ I p be a chain of ideals in A . According to Definition 3.3.1, we can rewrite the long exact sequence 3.3.3.1 intothe top row of the following commutative diagram: K s (cid:0) I p /I p ′′ (cid:1) K s (cid:0) I p /I p ′ (cid:1) K s − (cid:0) I p ′ /I p ′′ (cid:1) K s − (cid:0) I p /I p ′′ (cid:1) K s I p K s (cid:0) I p /I p ′ (cid:1) K s − I p ′ K s − I p . i ∂ if ∗ g ∗ h ∗ pr ∗ ∂ K incl ∗ The vertical arrows f ∗ , g ∗ , h ∗ arise from natural projections: They are induced inK-theory from factoring out I p ′′ . The identity arrow is also of this type because I p ′′ ⊆ I p ′ = ⇒ I p /I p ′′ I p ′ /I p ′′ ∼ = I p /I p ′ . The top row – except possibly at ∂ – matches the long exact sequence in K-theorythat corresponds to the short exact sequence → I p ′ /I p ′′ → I p /I p ′′ → I p /I p ′ → .We wish to prove that ∂ turns the upper row into a long exact sequence.We recognize ∂ K as the boundary map in K-theory for the short exact sequence → I p ′ → I p → I p /I p ′ → . By naturality of the exact sequence with respect tofactoring out I p ′′ , the composition g ∗ ◦ ∂ K is the K-theoretic connecting homomor-phism to make the upper row exact. By Definition 3.3.3, we have ∂ = g ∗ ◦ ∂ K . Thus21 Ideal inclusions ∂ is the correct arrow to construct the H-system. Definition 3.4.1. For s ∈ Z , the chain of ideals leads to a Z -indexed increasingfiltration { F p K s A } p ∈ Z of K s A , F p H s = F p K s A = im ( i : K s I p → K s A ) ⊆ K s A. To discuss this filtration, we will continue to write s ∈ Z for the index in K-theory.Afterwards, the K-theory groups will be indexed by ( p + q ) to show convergence ofthe spectral sequence. Proposition 3.4.2. For all s ∈ Z , the filtration { F p K s A } p ∈ Z in Definition 3.4.1 isHausdorff, exhaustive, and complete according to Definition 2.7.2.Proof. The Hausdorff property is immediate because I p ′ = 0 for p ′ < , therefore F p ′ H s = im ( i : 0 → K s A ) = 0 ⊇ T p ∈ Z F p H s .For exhaustion, consider the input of the spectral sequence: An inclusion chain ofC*-ideals I ⊆ I ⊆ I ⊆ · · · ⊆ I p ⊆ · · · with S p ∈ N I p = A . This closure is the directlimit object of the system of ideal inclusions ( I p → I p +1 ) p ∈ N in C ∗ A . K-theory is acontinuous functor, rendering the K-theory of the limit object A isomorphic to thelimit of the system of K-theory groups along the morphisms i : K s I p → K s I p +1 . ByDefinition 3.3.1, these are exactly the morphisms that appear in Definition 3.4.1 ofthe filtration { F p H s } p ∈ Z . Finally, universality of the limit object K s A guaranteesthat the above system of morphisms i exhausts K s A .Completeness is trivially satisfied because F p H s = 0 for all p < . Both the inverselimit and its right derivative vanish for this system. Remark 3.4.3. Even when one C*-ideal I ⊆ I is included in another, the K-theory groups need not be connected by a system of injections K ∗ I ֒ → K ∗ I ֒ → · · · ;for example, let H be an infinite-dimensional Hilbert space, then KH , the compactoperators of H , have K KH = Z , but KH ⊆ BH is an inclusion of ideals and K BH = 0 .Nonetheless, the filtration { F p H s } p ∈ Z = { F p K s A } p ∈ Z satisfies F p H s ⊆ F p +1 H s for all p ∈ Z : The group im ( i : K s I p → K s A ) is a subgroup of im ( i : K s I p +1 → K s A ) because, by definition of an H-system, the morphisms i factor through each other,22 .5 Convergence making this diagram commutative: K s I p K s A = colim p K s I p = K s colim p I p .K s I p +1 i ii We would like to show that this filtration makes the K-theory spectral sequenceconverge strongly to K ∗ A . Here we replace the K-theoretic degree s by p + q . Notation 3.5.1 ( Z ∞ p,q , B ∞ p,q ) . With Z rp,q and B rp,q as in Notation 3.2.5, write Z ∞ p,q = \ r ≥ Z rp,q , B ∞ p,q = [ r ≥ B rp,q . For each q ∈ Z , the filtration { F p K p + q A } p ∈ Z from Definition 3.4.1 leads to suc-cessive quotients F p K p + q A/F p − K p + q A across all p ∈ Z . We have to show that this ( p, q ) -indexed collection of quotients coincides with E ∞ p,q = Z ∞ p,q /B ∞ p,q = (cid:16) \ r ≥ Z rp,q (cid:17) . (cid:16) [ r ≥ B rp,q (cid:17) . Because the filtration has already been proven Hausdorff, exhaustive, and complete,the convergence will then be strong.We have shaped our input according to the most general axioms in [CE73]. Eventhough convergence for more specialized input is proven in that source, convergenceis merely claimed for our input. Henceforth, we shall give a full proof. Lemma 3.5.2. If there exists n ∈ N with A = I n = I n +1 = I n +2 = · · · , then thespectral sequence collapses at page n + 1 : We have E r ∗ , ∗ = E r +1 ∗ , ∗ for all r ≥ n + 1 ,thus E ∞ p,q = E n +1 p,q .Proof. Fix p and q ∈ Z . If p < , then B rp,q and Z rp,q vanish by definition for all r ≥ because I p = 0 and we have E r +1 p,q = 0 here.Consider the case p ≥ . For all r ≥ n , we have B rp,q = im ∂ : K p + q +1 ( I p + r /I p ) → K p + q ( I p /I p − ) , where I p + r = I n = A for all r ≥ n + 1 . Thus all B rp,q coincide for such high pagenumbers r . In similar fashion, all Z rp,q become im i : K p + q I p → K p + q ( I p /I p − ) . The23 Ideal inclusions collapse follows from the definition E r +1 p,q = Z rp,q /B rp,q for all r ∈ Z .This collapsing lemma reveals some structure of the pages E r ∗ , ∗ when the chain ofideals stabilizes. The following convergence theorem holds with or without stabiliza-tion of the ideals. Theorem 3.5.3. The E ∞ -term admits the desired filtration; i.e., E ∞ p,q ∼ = F p H p + q /F p − H p + q . Substituting the definitions lets us rewrite the claim like this, denoting the yet-undefined isomorphism in the middle by f : E ∞ p,q = Z ∞ p,q B ∞ p,q = im i : K p + q I p → K p + q ( I p /I p − ) im ∂ : K p + q +1 ( A/I p ) → K p + q ( I p /I p − ) f ∼ = im i : K p + q I p → K p + q A im i : K p + q I p − → K p + q A = F p H p + q F p − H p + q . (3.5.3.1)Our strategy is to construct the central isomorphism f in 3.5.3.1 explicitly. Aftergiving its construction, we show that f is well-defined, injective, and surjective. Thatwill constitute the proof of Theorem 3.5.3. Definition 3.5.4. For x + B ∞ p,q , an element in Z ∞ p,q /B ∞ p,q , we must define f ( x + B ∞ p,q ) .Find y ∈ K p + q I p with i ( y ) = x . Define f ( x + B ∞ p,q ) = i pA ( y ) + F p − K p + q A, where i pA : K p + q I p → K p + q A denotes the standard map from our H-system, inducedby the inclusion of algebras. Lemma 3.5.5. The morphism f is well-defined: The construction in Definition 3.5.4is independent of the choice of y ∈ K p + q I p with i ( y ) = x .Proof. Let y and y ′ ∈ K p + q I p with i ( y − y ′ ) ∈ B ∞ p,q . We have to show that f ( y ) = f ( y ′ ) , equivalently, that f ( y − y ′ ) ∈ F p − K p + q A = im i : K p + q I p − → K p + q A .Because i ( y − y ′ ) ∈ B ∞ p,q = im ∂ : K p + q +1 ( A/I p ) → K p + q ( I p /I p − ) , we find z ∈ K p + q +1 ( A/I p ) with ∂ ( z ) = i ( y − y ′ ) . This ∂ belongs to the exact sequence · · · → K p + q +1 ( A/I p − ) i ′ −→ K p + q +1 ( A/I p ) ∂ −→ K p + q ( I p /I p − ) i ′ −→ K p + q ( A/I p − ) → · · · . .5 Convergence Onto this exact sequence, we draw a commutative square, and then extend the right-hand side of the square to a vertical exact sequence in K-theory. K p + q I p − K p + q I p K p + q A · · · K p + q +1 ( A/I p ) K p + q ( I p /I p − ) K p + q ( A/I p − ) · · · . i p − A i pA i ii ′ ∂ i ′ i ′ (3.5.5.1)Chasing y − y ′ ∈ K p + q I p through this diagram, we obtain ( i ′ ◦ i )( y − y ′ ) = ( i ′ ◦ ∂ )( z ) = 0 ∈ K p + q ( A/I p − ) because the bottom row is exact. Then ( i ◦ i pA )( y − y ′ ) = 0 due to the commutativityof the square. With i pA ( y − y ′ ) ∈ ker i : K p + q A → K p + q ( A/I p − ) , we conclude fromthe exactness of the vertical sequence that i pA ( y − y ′ ) ∈ im i p − A = F p − K p + q A .This shows that f : E ∞ p,q → F p K p + q A/F p − K p + q A is well-defined. Lemma 3.5.6. The morphism f is injective.Proof. Let x + B ∞ p,q be a class in E ∞ p,q that vanishes under f . We have to show that x ∈ B ∞ p,q . This proof looks like the proof of Lemma 3.5.5 in reverse.Select y ∈ K p + q ( I p ) with i ( y ) = x ∈ im i : K p + q I p → K p + q ( I p /I p − ) . From f (cid:0) i ( y ) + B ∞ p,q (cid:1) = 0 and the definition of f , we know i pA ( y ) ∈ im i p − A : K p + q I p − → K p + q A . Consider diagram 3.5.5.1 again: By exactness of the vertical sequence, ( i ◦ i pA )( y ) = 0 . Commutativity of the square shows that ( i ′ ◦ i )( y ) = 0 . Exactness ofthe bottom row gives i ( y ) ∈ im ∂ : K p + q +1 ( A/I p ) → K p + q ( I p /I p − ) .After substituting i ( y ) = x and im ∂ = B ∞ p,q , we have shown x ∈ B ∞ p,q and thereforethe injectivity of f . Lemma 3.5.7. The morphism f is surjective.Proof. Let z + im i p − A be in F p K p + q A/F p − K p + q A . For this z ∈ F p K p + q A = im i pA ,we may find a lift y ∈ K p + q I p with i pA ( y ) = z . Then i ( y ) ∈ Z ∞ p,q already satisfies f (cid:0) i ( y ) + B ∞ p,q (cid:1) = z + im i p − A by definition of f . Thus f is surjective.These lemmas conclude the proof of Theorem 3.5.3. 25 Ideal inclusions We recall the main theorem of Section 3: Theorem 3.1.1 (Spectral sequence for ideal inclusions) . Let A = S p ∈ N I p be a C*-algebra, where the I ⊆ I ⊆ I ⊆ · · · ⊆ I p ⊆ · · · form a chain of closed two-sidedideals. There is a spectral sequence { E rp,q , d r } r,p,q with E p,q = K p + q ( I p /I p − ) . This spectral sequence converges strongly to K ∗ A ; i.e., given s ∈ Z , the groups E ∞ p,q along the diagonal s = p + q pose an extension problem to reconstruct K s A . The theorem is plausible: Fix n ∈ N , choose I p = 0 for p < n and I n = A . Thespectral sequence begins with E p,q = 0 for p = n and E n,q ∼ = K n + q A . In the onlynonzero column E n, ∗ ∼ = E ∞ n, ∗ , we see the expected K s A ∼ = E ∞ n,q for n + q = s . Proof of Theorem 3.1.1. The computation of E p,q is straightforward: E p,q = Z p,q B p,q = im id : K p + q ( I p /I p − ) → K p + q ( I p /I p − ) im ∂ : K p + q +1 ( I p /I p ) | {z } =0 → K p + q ( I p /I p − ) ∼ = K p + q ( I p /I p − ) . The differentials d r : E r ∗ , ∗ → E r ∗ , ∗ were defined in Theorem 3.2.5 and have the correctbidegrees ( − r, r − on page r . The strong convergence of this spectral sequencefollows from Proposition 3.4.2 and Theorem 3.5.3. Remark 3.6.1. Even though this is a half-plane spectral sequence, its convergenceis provable like the convergence of a single-quadrant spectral sequence because wehave exiting differentials : For each a bidegree ( p, q ) ∈ Z , all except finitely manydifferentials d rp,q : E rp,q → E rp − r,q + r − exit the half-plane E rp ′ , ∗ for p ′ ≥ of nonzerogroups.There are more intricate results for half-plane spectral sequences with enteringdifferentials or for whole-plane spectral sequences; these will not arise in our setting.Besides the classic reference [McC01], a good resource for convergence theorems is[Boa99].26 Finite sums of ideals Let A be a C*-algebra. There is a K-theory spectral sequence for ideals I ⊆ I ⊆ I ⊆ · · · ⊆ I n = A . We will postulate a new spectral sequence that weakens the “ ⊆ ”to a mere “ + ”: For ideals I , I , I , . . . , I n with P nj =0 I j = A , there is a spectralsequence that relates the K-theory of their intersections to the K-theory of A . Even though Section 4 deals only with the finite case, we will define ∗ -homomor-phisms that preserve arbitrarily-sized ideal decompositions in light of later sections. Definition 4.1.1 (Preservation of ideal decompositions) . Let α and α ′ be arbitraryindex sets with α ⊆ α ′ . Let A be the norm closure A = P β ∈ α I β of the sum of | α | -many C*-ideals I β . Let A ′ = P β ∈ α ′ I ′ β be another C*-algebra written as the sumof | α ′ | -many C*-ideals I ′ β .A ∗ -homomorphism f : A → A ′ preserves the ideal decomposition if f ( I β ) ⊆ I ′ β forevery β ∈ α . Both f and the specific decompositions { I β } β ∈ α and { I ′ β } β ∈ α ′ are partof the input data. Remark 4.1.2 (Naturality w.r.t. ideal decompositions) . It is conceivable to definea category of C*-ideal decompositions and decomposition-preserving ∗ -homomor-phisms, e.g., with cardinal numbers α as index sets to ensure α ⊆ α ′ wherever | α | ≤ | α ′ | . But it will be enough to work in C ∗ A , the standard category of C*-algebras, because all natural constructions here will already be natural will w.r.t.ideal decompositions of C*-algebras:Let C be any category. Let F , G : C ∗ A → C be functors of C*-algebras and let η : F → G be a natural transformation. Let f : A → A ′ be a ∗ -homomorphism thatpreserves an | α | -fold ideal decomposition as in Definition 4.1.1. Since f ( I β ) ⊆ I ′ β forall β ∈ α and ∗ -homomorphisms are compatible with sums, the following diagramcommutes in C : F ( A ) = F (cid:0) P β ∈ α I β (cid:1) G ( A ) = G (cid:0) P β ∈ α I β (cid:1) F ( A ′ ) = F (cid:0) P β ∈ α I ′ β (cid:1) G ( A ′ ) = G (cid:0) P β ∈ α I ′ β (cid:1) . η ( A ) η ( A ′ ) F ( f ) = F (cid:0) P β ∈ α f ↾ I β (cid:1) G ( f ) = G (cid:0) P β ∈ α f ↾ I β (cid:1) Finite sums of ideals To construct the spectral sequence for finite ideal decompositions, we will definefunction algebras over certain subsets of the standard simplex. Definition 4.2.1 (Cake piece) . Fix n ∈ N . The standard n -simplex is the topologicalspace ∆ n = ( ( x , x , . . . , x n ) ∈ [0 , n +1 : n X i =0 x i = 1 ) . Its boundary ∂ ∆ n shall be the subset of points with at least one zero entry. Let j ∈ N be an index with j ≤ n . This index defines the j -th cake piece ∆ nj = { ( x , x , . . . , x n ) ∈ ∆ n : x j ≤ x i for all i ≤ n } . Let J ⊆ { , , . . . , n } be a nonempty subset of the n + 1 indices. This determines anintersection of cake pieces: ∆ nJ = \ j ∈ J ∆ nj . We will see how ∆ nJ behaves very much like a j -th cake piece, and therefore also callit a cake piece . (1 , , 0) (0 , , , , ∆ { } Figure 4.2.2: The simplex ∆ with the cake piece ∆ { } = ∆ ⊆ ∆ markedCake pieces are closed subsets of ∆ n . The central point (cid:0) n +1 , n +1 , . . . , n +1 (cid:1) of the n -simplex is part of every cake piece; this point is the only element of ∆ nJ for the fullset J = { , , . . . , n } .Arbitrary index subsets J = ∅ make ∆ nJ look like ∆ n +1 −| J | : Proposition 4.2.3. For a nonempty J ⊆ { , , . . . , n } , the subset ∆ nJ is the imageof ∆ n +1 −| J | × { } | J |− under a nondegenerate affine transformation in R n +1 .Proof. We will analyze several cases explicitly by cardinality of J .28 .2 Cake pieces Full set. For J = { , , . . . , n } , the full set of ( n + 1) elements, we have alreadyargued how ∆ nJ contains only a single point. This is an image of ∆ × { } n ⊆ R n +1 . One element. For n ≥ and J = { j } , we have ∆ nJ = ∆ nj . Without loss ofgenerality, choose J = { } . The affine transformation of R n +1 to get ∆ n from thestandard n -simplex is f = ( f , f , . . . , f n ) : R n +1 → R n +1 ,f ( x ) = x n + 1 , f i ( x ) = x i + x n + 1 for i = 0 . The purpose of f i ( x ) is to equally distribute among the n other coordinates the value nx n +1 that has been taken away from x .The f i are nontrivial linear maps, and their direct product f = ( f , . . . , f n ) is anaffine automorphism of R n +1 . Its inverse is f − = g = ( g , g , . . . , g n ) : R n +1 → R n +1 ,g ( x ) = ( n + 1) · x , g i ( x ) = x i − x for i = 0 . Even though these maps are defined in R n +1 , they restrict well to maps on the sim-plex: For x = ( x , . . . , x n ) ∈ ∆ n , we have P ni =0 x i = 1 = P ni =0 f ( x i ) . Furthermore, f ↾ ∆ n maps into ∆ n because all points x ∈ ∆ n satisfy f i ( x ) ≥ f ( x ) . The restrictedinverse f − ↾ ∆ n maps into ∆ n : Positive coordinates stay positive because x i ≥ x for all ≤ i ≤ n . Several elements. For < | J | ≤ n , observe how | J | coordinates in ∆ nJ remain equalto each other at all times, and are always the smallest. Without loss of generality,let J be { , , . . . , | J | − } , the | J | first coordinates. We will construct an affineisomorphism h : ∆ n +1 −| J |{ } → ∆ nJ by defining h on the ( n + 2 − | J | ) corners of ∆ n +1 −| J |{ } , then extending h to the entirecake piece by preserving convex combinations. Thereby, h reduces the case of ∆ nJ tothe already-proven case | J | = 1 .The central point of ∆ n +1 −| J | is a corner of ∆ n +1 −| J |{ } . Have h map this point tothe center of ∆ n , this is extremal in ∆ nJ . Biject the remaining ( n + 1 − | J | ) cornersof ∆ n +1 −| J |{ } to the corners in ∆ n that belong to the coordinates in { , , . . . , n } − J ;these points remain extremal in ∆ nJ . This bijection can even be chosen to preservethe order of coordinates. 29 Finite sums of ideals Corollary 4.2.4. Let J ⊆ { , , . . . , n } be nonempty. Then ∆ n +1 −| J | ∼ = ∆ nJ ∼ = D n +1 −| J | , where D n +1 −| J | denotes the ( n + 1 − | J | ) -dimensional unit disk.In particular, if J = { j } , then ∆ nj ∼ = D n .Proof. This follows from ∆ n ∼ = D n and Proposition 4.2.3.These technical constructions relate various subspaces of simplices to disks. Theboundaries of disks are spheres. This will become useful once we consider C*-algebrasof functions on these subspaces of simplices: When we force functions to vanish onthe boundaries, the C*-algebras can be viewed as suspensions of other algebras. Definition 4.3.1 (Cake algebra) . Fix n ∈ N . Let A be a C*-algebra and I j ⊆ A beclosed two-sided ideals for j ∈ { , , . . . , n } with A = P nj =0 I j . This gives rise to asuspension-like C*-algebra, the cake algebra B = B ( I , I , . . . , I n )= f : ∆ n → A = n X j =0 I j : f continuous, f ↾ ∂ ∆ n = 0 , f (∆ nj ) ⊆ I j for all j . For J ⊆ { , , . . . , n } , define the sub-C*-algebra B J ⊆ B , again called a cake algebra ,by B J = (cid:8) f ∈ B : for each j ′ / ∈ J , f (∆ nj ′ ) = 0 (cid:9) . Remark 4.3.2. We observe B { , ,...,n } = B and B ∅ = 0 . Larger index sets meanlarger function algebras because fewer restrictions apply. Whenever J ′ ⊆ J is asubset, then B J ′ ⊆ B J is a subalgebra. For J = { j } , we can characterize B { j } : Proposition 4.3.3. B { j } is isomorphic to the n -fold C*-algebra suspension of I j .Proof. By ∂ ∆ nj , we denote the topological boundary of ∆ nj as a subset of R n +1 . Apoint x ∈ ∂ ∆ nj lies in ∂ ∆ n if x j = 0 . Otherwise, we have x j = x j ′ for an index j ′ = j and x then lies in ∆ nj ′ .Understanding this, we simplify the above definition for B J = B { j } : B { j } = (cid:8) f : ∆ n → A : f ↾ ∂ ∆ n = 0 , f (∆ nj ) ⊆ I j , f (∆ nj ′ ) = 0 for all j ′ = j (cid:9) = (cid:8) f : ∆ n → A : f ↾ ∂ ∆ n = 0 , f (∆ nj ) ⊆ I j , f (∆ n − ∆ nj ) = 0 (cid:9) ∼ = (cid:8) f : ∆ nj → I j : f ↾ ∂ ∆ nj = 0 (cid:9) . Because ∆ n { j } ∼ = D n , the algebra B { j } is isomorphic to the n -fold C*-algebra suspen-sion of I j ; i.e., the A -valued functions on D n that vanish on the boundary ∂D n .30 .3 Cake algebras Remark 4.3.4. As subsets of functions that vanish on a given set, the B J are closedtwo-sided ideals in B . For J ′ ⊆ J , B J ′ is a closed two-sided ideal in B J . Lemma 4.3.5. For subsets J and J ′ of { , , . . . , n } , we have B J ∩ B J ′ = B J ∩ J ′ .Proof. This is immediate from the definition of B J : B J ∩ B J ′ = (cid:8) f ∈ B : f (∆ nj ′ ) = 0 for j ′ / ∈ J (cid:9) ∩ (cid:8) f ∈ B : f (∆ nj ′ ) = 0 for j ′ / ∈ J ′ (cid:9) = (cid:8) f ∈ B : f (∆ nj ′ ) = 0 for j ′ with j ′ / ∈ J or j ′ / ∈ J ′ (cid:9) = B J ∩ J ′ . Definition 4.3.6 (Cake sums Q p for p ∈ Z ) . Let A and B ( I , I , . . . , I n ) be as inDefinition 4.3.1. For p ∈ Z , define the C*-algebra Q p , called a cake sum , by Q p = X | J |≤ p +1 B J , where J ranges over all subsets of { , , . . . , n } that have cardinality ( p + 1) or less. Remark 4.3.7 (Cake sums for p < or p ≥ n ) . For p < , the sum Q p is either B ∅ or an empty sum; both of these are the zero algebra Q p = 0 .For p ≥ n , the sum Q p is taken over all B J for all possible subsets J ⊆ { , , . . . , n } including { , , . . . , n } itself. By Remark 4.3.2, for all J ⊆ { , , . . . , n } , the cakealgebra B J is already a subalgebra of B { , ,...,n } . Thus Q p is identical to B { , ,...,n } = B for p ≥ n . Remark 4.3.8 (Inclusions Q p − ⊆ Q p ) . We have well-defined inclusions Q p − ⊆ Q p for all p ∈ Z because Q p collects at least the cake algebras from Q p − , possibly more.If p ≤ n , the relation B J ′ ⊆ B J for J ′ ⊆ J allows another characterization of Q p by summing over fewer sets: Q p = X | J |≤ p +1 B J = X | J | = p +1 B J . (4.3.8.1)For p > n , this characterization would be false because there are no subsets ofcardinality ( n + 2) in { , , . . . , n } . Lemma 4.3.9. For all p ′ ≤ p ∈ Z , the cake sum Q p ′ is a C*-ideal in Q p . Thus wehave well-defined quotients Q p /Q p ′ , in particular Q p /Q p − .Proof. For p ′ ≥ n , we trivially have Q p ′ = Q p = B .We will prove the case p ′ < n . Q p ′ and Q p are sub-C*-algebras of the samecommutative C*-algebra B and we have Q p ′ ⊆ Q p . It remains to show that Q p ′ isan algebraic ideal. 31 Finite sums of ideals For f ∈ Q p ′ and g ∈ Q p , find J ⊆ { , , . . . , n } such that f ∈ B J and | J | = p ′ + 1 ;such a J exists according to the characterization 4.3.8.1.By Definition 4.3.1 of B J , the function f must vanish on at least ( n − p ′ ) differentcake pieces. For any g ∈ Q p , the pointwise product f g must vanish on the same cakepieces, therefore f g ∈ Q p ′ . Remark 4.3.10. The algebra Q p is defined by summing over all B J with | J | ≤ p +1 ,not merely over those with | J | ≤ p . This index shift is deliberate: We are going todefine a spectral sequence with the K-theory of quotients of Q p /Q p − as input. Theindex shift will affect the layout of the first page { E p,q } p,q ∈ Z .Consider the trivial input n = 0 and A = I . Here Q p = B { } for p ≥ and Q p = 0 for p < . This leads to a spectral sequence with E ,q ∼ = K q A and E p,q = 0 for p = 0 . This is the most desirable layout; the K-theory of the lone ideal A = I isnot shifted in any way: K A K A K A − K − A − p.q We started with a sum of ideals I , I , . . . , I n and have developed a chain of ideals · · · ⊆ Q p ⊆ Q p +1 ⊆ · · · . The main theorem of Section 4.4 relates the K-theory ofthis chain to the K-theory of the original ideals: Theorem 4.4.1. For A = I + I + · · · + I n and the cake sums Q p defined as before,given p ∈ { , , . . . , n } and q ∈ Z , the K-theory of Q p /Q p − decomposes as K p + q ( Q p /Q p − ) ∼ = M | J | = p +1 K q + n (cid:16) \ j ∈ J I j (cid:17) . Remark 4.4.2. For p / ∈ { , , . . . , n } , the K-theory K ∗ ( Q p /Q p − ) vanishes because Q p = Q p − . Example 4.4.3. Before we prove Theorem 4.4.1 for all p ∈ { , , . . . , n } , we lookat the simplest case, p = 0 . One-fold intersections of ideals are merely the ideals32 .4 K-theory of cake algebras themselves. The above formula reduces to K q ( Q /Q − ) ∼ = M j ≤ n K q + n I j . Inserting the definitions Q − = 0 and Q = P j ≤ n B { j } , we rewrite our claim to K q (cid:16) X j ≤ n B { j } (cid:17) ∼ = M j ≤ n K q + n I j . Lemma 4.3.5 implies that B { j } and B { j ′ } overlap trivially as algebras for j = j ′ ; i.e., B { j } ∩ B { j ′ } contains only the zero function. The sum P j ≤ n B { j } on the left-handside is therefore isomorphic to a direct sum L j ≤ n B { j } of the function spaces B { j } .In Proposition 4.3.3, we have shown that B { j } is isomorphic to the n -fold suspen-sion of I j , providing the desired shift by n degrees, K q B { j } ∼ = K q S n I j ∼ = K q + n I j .Because taking K-theory commutes with taking direct sums, we have shown thetheorem for p = 0 .The main ingredient B { j } ∩ B { j ′ } = B ∅ = 0 must now be generalized to proveTheorem 4.4.1 for p > . Lemma 4.4.4. Let J ′ = J ′′ be nonempty ( p + 1) -element subsets of { , , . . . , n } andlet f ∈ Q p lie in the intersection B J ′ ∩ B J ′′ . Then f lies already in the next-smallerideal, f ∈ Q p − = X | J | = p B J . Proof. By Lemma 4.3.5, B J ′ ∩ B J ′′ = B J ′ ∩ J ′′ . The algebra B J ′ ∩ J ′′ is a summand of Q | J ′ ∩ J ′′ | This algebra is equal to or a subset of Q p − because | J ′ ∩ J ′′ | ≤ p . Lemma 4.4.5. Fix an index subset J ⊆ { , , . . . , n } . Let p ∈ N be a cardinality.Let f ∈ B J vanish on all ( p + 1) -fold intersections of cake pieces: f ↾ ∆ nL = 0 when | L | = p + 1 .Then f is a finite sum of functions f L with each f L ∈ B L for L ⊆ J and eachoccurring set L has cardinality | L | ≤ p . Remark 4.4.6. It follows that f is in Q p − , but the claim is stronger: Only sum-mands B L with L ⊆ J are required to construct f in Q p − . Proof of Lemma 4.4.5. We prove this by induction along p . The base case is p = 0 :One-fold intersections of cake pieces – where f vanishes by assumption – are thecake pieces themselves, thus f = 0 , the only function in the zero algebra B ∅ . Thisconcludes the base case. 33 Finite sums of ideals For the induction hypothesis, assume that all functions that vanish on p -fold in-tersections are sums of functions from B L with L ⊆ J and | L | ≤ p − . We will showthe claim for p : Let f ∈ B J vanish on ( p + 1) -fold intersections of cake pieces.Consider all subspaces ∆ nL for L ⊆ J with | L | = p . We may treat each as atopological submanifold of R n +1 on its own and consider its boundary ∂ ∆ nL . Eachpoint x ∈ ∂ ∆ nL lies on the boundary ∂ ∆ n of the entire simplex ∆ n or in a cake piece ∆ nj with j = L , see Figure 4.4.7. If x ∈ ∂ ∆ n , then f ( x ) = 0 by definition of B .If x ∈ ∆ nj with j = J , then also f ( x ) = 0 because now x ∈ ∆ nL ∪{ j } , a ( p + 1) -foldintersection of cake pieces. Thus f ↾ ∂ ∆ nL = 0 .On the interior ∆ nL − ∂ ∆ nL , f assumes values in T j ∈ L I j by definition of B J . Fromthis and the restriction f ↾ ∂ ∆ nL = 0 , we can find a function g L ∈ B L such that f ↾ ∆ nL = g L ↾ ∆ nL . After defining g L for each L ⊆ J of cardinality p , consider thefunction f ′ = f − X L ⊆ J | L | = p g L . This f ′ still lies in the C*-algebra B J because B L ⊆ B J for each L . Furthermore, f ′ vanishes on all p -fold intersections of cake pieces, not merely on the ( p + 1) -foldintersections.By our induction hypothesis, f ′ is a finite sum of functions from B L for L ⊆ J of cardinality | L | ≤ p − . Each g L is in B L with L ⊆ J and | L | = p . Since f = f ′ + P L g L , we have shown the induction case for cardinality p . (1 , , 0) (0 , , , , ∆ { , } Figure 4.4.7: The two-fold intersection ∆ { , } and its two-point boundary: one pointin ∂ ∆ , one in the three-fold intersection ∆ { , , } Lemma 4.4.8. Let J ⊆ { , , . . . , n } be a nonempty index set. • Let f be a function in P L $ J B L . Then f ↾ ∆ nJ = 0 . • Conversely, let g be a function in B J with g ↾ ∆ nJ = 0 . Then g ∈ P L $ J B L . .4 K-theory of cake algebras Proof. We have f ∈ P L $ J B L . Each L $ J lacks at least one index j ∈ J , therefore f ↾ ∆ nj = 0 by definition of B L . Since ∆ nJ is contained in the boundary ∂ ∆ nj , weconclude f ↾ ∆ nJ = 0 .Conversely, let g ∈ B J vanish on ∆ nJ . All functions in B J vanish on ( | J | + 1) -foldintersections of cake pieces. Furthermore, ∆ nJ is the only | J | -fold intersection thattouches the interior of the support of g . Thus g vanishes on all | J | -fold intersections.By Lemma 4.4.5, g is a sum of functions g L from B L with | L | < | J | and L ⊆ J .In the following technical proposition, A and B are general C*-algebras; they neednot coincide with B ( I , I , . . . , I n ) that we defined before. Nonetheless, we choosethe names A and B here because we will later apply this result to the B J fromTheorem 4.4.1. Proposition 4.4.9. Let X ⊆ R n be a compact set and let D ⊆ X be a compactsubspace of X . Let A be a C*-algebra, B ⊆ C ( X, A ) a C*-ideal of functions from X to A , and Van D ⊆ B the vanishing ideal of D ; i.e., the ideal of functions f ∈ B with f ↾ D = 0 .Then B/ Van D is isomorphic as a C*-algebra to B ′ = { f ↾ D : f ∈ B } . This is plausible: When we enlarge D , then functions in Van D are allowed lessvariation, thus Van D becomes smaller and the quotient space B/ Van D becomeslarger. Proof. Define the operator T : B/ Van D → B ′ by T [ f ] = f ↾ D . This is a well-definedlinear map because for f and f ′ ∈ [ f ] ∈ B/ Van D , we have f − f ′ ∈ Van D , therefore f ↾ D = f ′ ↾ D . T is a continuous operator with norm k T k = sup {k f ↾ D k : f ∈ B with k [ f ] k ≤ } , where k [ f ] k = inf {k f − g k : g ∈ Van D } . From k f ↾ D k ≤ k [ f ] k ≤ k f k , we see that T is continuous with norm k T k ≤ . T is bijective: If T [ f ] = 0 , then f ↾ D = 0 , therefore f ∈ Van D and [ f ] = 0 ∈ B/ Van D . On the other hand, given f ↾ D ∈ B ′ with f ∈ B , surely [ f ] is a preimageof f ↾ D under T .As a restriction of functions, T preserves products and the C*-involution. Togetherwith bijectivity of T , we conclude that k T k = 1 and that T is an isometric *-isomorphism by [Dav96, Theorem I.5.5].We could have obtained k T k = 1 from an analytical argument, too: Given f ↾ D ,force f : X → A to decay rapidly outside D ⊆ X by multiplying with bump functions.35 Finite sums of ideals B is an ideal in C ( X, A ) , and A admits an approximate unit.We shall now return to the setting where A = I + I + · · · + I n is a sum of idealsand B = B ( I , I , . . . , I n ) is the function algebra constructed over the n -simplex. Lemma 4.4.10. Let J ⊆ { , , . . . , n } be a nonempty index set of cardinality | J | = p + 1 . Then B J / ( B J ∩ Q p − ) ∼ = S n − p (cid:16) \ j ∈ J I j (cid:17) , where S n − p denotes the ( n − p ) -fold suspension of C*-algebras. Recall that Q p was a sum over all B L with index sets L of cardinality | L | = p + 1 ,thus B J ⊆ Q p and dividing by the intersection ( B J ∩ Q p − ) ⊆ Q p is meaningful. Proof. All functions in B J are supported in S j ∈ J ∆ nj . Write D = ∆ nJ = T j ∈ J ∆ nj forthe subset of S j ∈ J ∆ nj where functions in B J may take nonzero values in all I j for j ∈ J simultaneously. Now B J ∩ Q p − = X L $ J B L , and, by Lemma 4.4.8, functions in B J ∩ Q p − are exactly those functions in B J thatvanish on D . We can apply Proposition 4.4.9 to the function algebra B J on the basespace X = S j ∈ J ∆ nj and its subset D to get B J / ( B J ∩ Q p − ) ∼ = { f ↾ D : f ∈ B J } . (4.4.10.1)Considering D = ∆ nJ an ( n + 1 − | J | ) -dimensional topological manifold on its own, ∆ nJ itself has a boundary ∂ ∆ nJ and a nontrivial interior (∆ nJ − ∂ ∆ nJ ) . In the edgecase where J = { , , . . . , n } is the full set, ∆ n { , ,...,n } is a single point, which is azero-dimensional manifold with empty boundary ∂ ∆ n { , ,...,n } = ∅ .The boundary ∂ ∆ nJ is contained in the boundary of the original domain S j ∈ J ∆ nj .Therefore functions in B J , even when restricted to D as in 4.4.10.1, must still vanishon this new boundary ∂ ∆ nJ .On the interior of D = ∆ nJ , functions in B J must take values in T j ∈ J I j bydefinition of B J , but no further restrictions apply. We can rewrite 4.4.10.1 as B J / ( B J ∩ Q p − ) ∼ = f : ∆ nJ → \ j ∈ J I j : f is continuous and f ↾ ∂ ∆ nJ = 0 . Finally, by Lemma 4.2.4, ∆ nJ ∼ = ∆ n +1 −| J | = ∆ n − p is homeomorphic to the ( n − p ) -dimensional unit disk. This allows us to further rewrite the algebra B J / ( B J ∩ Q p − ) .4 K-theory of cake algebras as the ( n − p ) -fold suspension in the claim.We are now ready to prove the main theorem about the K-theory of the chain ofideals Q p . Proof of Theorem 4.4.1. Fix p ∈ { , , . . . , n } and q ∈ Z . With Q p = P | J | = p +1 B J according to the characterization 4.3.8.1, we have to show: K p + q ( Q p /Q p − ) ∼ = M | J | = p +1 K q + n (cid:16) \ j ∈ J I j (cid:17) . First, we will show that the quotient Q p /Q p − decomposes as a direct sum. Let f ∈ Q p lie in the images of different inclusions B J → Q p and B J ′ → Q p for | J | = | J ′ | = p + 1 . By Lemma 4.4.4, we have f ∈ Q p − and therefore [ f ] = 0 ∈ Q p /Q p − .This shows that Q p /Q p − is a direct sum. Each summand corresponds to one B J with | J | = p + 1 : Q p /Q p − = M | J | = p +1 B J / ( B J ∩ Q p − ) . For each J , we computed B J / ( B J ∩ Q p − ) ∼ = S n − p (cid:0) T j ∈ J (cid:1) in Lemma 4.4.10. Passingto K-theory, we can replace the ( n − p ) -fold suspension with a degree shift by ( n − p ) : K p + q (cid:0) B J / ( B J ∩ Q p − ) (cid:1) ∼ = K p + q (cid:18) S n − p (cid:16) \ j ∈ J I j (cid:17)(cid:19) ∼ = K q + n (cid:16) \ j ∈ J I j (cid:17) . The claim follows because taking K-theory commutes with taking direct sums. Theorem 4.4.11. The inclusion of algebras B → { f : ∆ n → A : f ↾ ∂ ∆ n = 0 } in-duces an isomorphism in K-theory. The following Lemmas 4.4.12 to 4.4.15 will prove this theorem. Define the followingintermediate algebras: R = { f : ∆ n → A : f ↾ ∂ ∆ n = 0 } ,R = R ∩ { f : f (∆ n ) ⊆ I } ,R = R ∩ { f : f (∆ n ) ⊆ I } , ... B = R n +1 = R n ∩ { f : f (∆ nn ) ⊆ I n } . To show that B → R is an isomorphism in K-theory, we show that each inclusionincl : R k → R k − Finite sums of ideals induces an isomorphism for k ∈ { n + 1 , n, . . . , , } . Lemma 4.4.12. For k > k ′ , the algebra R k is a C*-ideal in R k ′ .Proof. The additional restrictions to the set of functions in R k over R k ′ forces thefunctions to map points into the given C*-ideals of A instead of anywhere in A . Be-cause all I , . . . , I n are C*-ideals and the multiplication of functions happens point-wise, R k becomes a C*-ideal in R k ′ . Lemma 4.4.13. The pair of topological spaces (∆ nk , ∂ ∆ n ∩ ∆ nk ) is homeomorphic to ( D n − × [0 , , D n − × { } ) .Proof. ∂ ∆ n ∩ ∆ nk is exactly the k -th face of the n -simplex. In Proposition 4.2.3, wehave seen how ∆ nn ∼ = ∆ n . In particular, for one-element sets J = { n } , the proof showshow ∆ nJ and ∆ n are diffeomorphic via a stretch by the factor ( n + 1) . This stretchhas ∂ ∆ n ∩ ∆ nn as a set of fixed points. ∆ nn and ∆ nk are certainly homeomorphic.The cake piece ∆ nk is a compact, convex n -dimensional manifold within R n +1 and ∂ ∆ n ∩ ∆ nk is a convex ( n − -dimensional hypersurface within ∂ ∆ nk . Corollary 4.2.4relates the simplices to the desired disks. Lemma 4.4.14. For all k ∈ { , , . . . , n + 1 } , the quotient R k − /R k has trivialK-theory.Proof. The subset ∆ n − ∆ nk − is open in the entire space ∆ n . With the conventionthat { , , . . . , − } denotes the empty set and that { , , . . . , } = { } , we compute: R k − /R k = n f : ∆ n → A : f ( ∂ ∆ n ) = 0 and f (∆ nj ) ⊆ I j for j ∈ { , , . . . , k − } on f : ∆ n → A : f ( ∂ ∆ n ) = 0 and f (∆ nj ) ⊆ I j for j ∈ { , , . . . , k − } o ∼ = (cid:8) f : ∆ nk − → A : f ( ∂ ∆ n ∩ ∆ nk − ) = 0 (cid:9)(cid:8) f : ∆ nk − → A : f ( ∂ ∆ n ∩ ∆ nk − ) = 0 and f (∆ nk − ) ⊆ I k − (cid:9) ∼ = (cid:8) f : ∆ nk − → A/I k − : f ( ∂ ∆ n ∩ ∆ nk − ) = 0 (cid:9) . Because of Lemma 4.4.13, the quotient R k − /R k is isomorphic to the algebra R ′ = (cid:8) f : D n − × [0 , → A/I k − : f (cid:0) D n − × { } (cid:1) = 0 (cid:9) . This is a contractible algebra: The homotopy h : R ′ × I → R ′ , h ( f, t )( x, t ′ ) = f ( x, t ′ · t ) , defines a ∗ -homomorphism for each fixed t . This construction is analogous to theproof of Proposition 2.1.9 for the contractibility of cone algebras. Since K-theory38 .5 Review of developed theory is homotopy invariant and h ( f, 0) = 0 for all f , we conclude that K ∗ ( R k − /R k ) = K ∗ R ′ = 0 . Lemma 4.4.15. incl : R k → R k − induces an isomorphism in K-theory.Proof. We examine the six-term exact sequence associated to the inclusion of theideal. K R k K R k − K ( R k − /R k ) K ( R k − /R k ) K R k − K R k . incl pr ∂ ◦ β incl pr ∂ Since K p ( R k − /R k ) vanishes for both even and odd p as shown in Lemma 4.4.14,incl p is an isomorphism for all p . This also concludes the proof of Theorem 4.4.11,which is an n -fold application of these lemmas. Let A be a C*-algebra that can be written as a finite sum I + I + · · · + I n = A ofclosed two-sided ideals I j ⊆ A . For the cake pieces ∆ nj ⊆ ∆ n , we have constructedin Definition 4.3.1 a new C*-algebra B of functions into A , B = B ( I , I , . . . , I n )= f : ∆ n → A = n X j =0 I j : f continuous, f ↾ ∂ ∆ n = 0 , f (∆ nj ) ⊆ I j for all j . We worked with arbitrary index subsets J ⊆ { , , . . . , n } . For such a J , we havedefined B J = (cid:8) f ∈ B : for each j ′ / ∈ J , f (∆ nj ′ ) = 0 (cid:9) . These cake algebras B J are closed two-sided ideals in B = B { , ,...,n } . For p ∈ Z , wedefined the cake sums Q p = X | J |≤ p +1 B J . There are inclusions Q p − → Q p and quotients of C*-ideals, Q p /Q p − . This inclusionchain of C*-ideals is the decisive structure: We can later feed these algebras into thespectral sequence for ideal inclusions.For p ∈ { , , . . . , n } , Theorem 4.4.1 computes K p + q ( Q p /Q p − ) ∼ = M | J | = p +1 K q + n (cid:16) \ j ∈ J I j (cid:17) . Finite sums of ideals This expression continues to hold for p > n where K p + q ( Q p /Q p − ) = 0 : There areno subsets J with | J | = n + 2 . But the expression fails for p = − : To avoidintersections over the empty set, we must explicitly mention K p + q ( Q p /Q p − ) = 0 for p = − whenever we extend Theorem 4.4.1 to all p ∈ Z .Theorem 4.4.11 shows that the inclusion B → { f : ∆ n → A : f ↾ ∂ ∆ n = 0 } in-duces an isomorphism in K-theory. The C*-algebra on the right-hand side is iso-morphic to the n -fold suspension S n A , therefore K p B ∼ = K p + n A . With this, we cango back to A , the algebra of original interest, even though the quotients Q p encodeinformation about B . Theorem 4.6.1 (Spectral sequence for finite sums of C*-ideals) . Let A be a C*-algebra and I , I , . . . , I n be ( n + 1) C*-ideals in A with I + I + · · · + I n = A .There is a spectral sequence { E rp,q , d r } r,p,q with E p,q ∼ = M | J | = p +1 K q (cid:16) \ j ∈ J I j (cid:17) for ≤ p ≤ n , for p < or p > n . (4.6.1.1) This spectral sequence converges strongly to K ∗ A . This spectral sequence is functorial for ∗ -homomorphisms that preserve ideal de-compositions; this will be the next theorem. Proof of Theorem 4.6.1. For the C*-ideals I , I , . . . , I n , define cake algebras B = B ( I , I , . . . , I n ) and cake sums Q p ⊆ B for p ∈ Z as reviewed in Section 4.5. Wehave Q p = B for p ≥ n by Remark 4.3.7. For the series of inclusions · · · = 0 = 0 ⊆ Q ⊆ Q ⊆ · · · ⊆ Q n = B = Q n +1 = Q n +2 = · · · , Theorem 3.1.1 gives a spectral sequence { ¯ E rp,q , ¯ d r } r,p,q with ¯ E p,q ∼ = K p + q ( Q p /Q p − ) , (4.6.1.2)converging to K ∗ ( B ) . By Theorem 4.4.1, we can replace the K-theory of thesequotients by the K-theory of a more immediate intersection, ¯ E p,q ∼ = K p + q ( Q p /Q p − ) ∼ = M | J | = p +1 K q + n (cid:16) \ j ∈ J I j (cid:17) for ≤ p ≤ n , for p < or p > n . (4.6.1.3)40 .6 Main theorem This spectral sequence converges strongly to K ∗ (cid:0)S p ∈ Z Q p (cid:1) = K ∗ Q n = K ∗ B . ByTheorem 4.4.11, K q B ∼ = K q + n A . To simplify, we will shift down by n all degrees inK-theory, both in 4.6.1.3 and in the expression for the convergence. As a result, weobtain a new spectral sequence { E rp,q , d r } r,p,q with E p,q ∼ = M | J | = p +1 K q (cid:16) \ j ∈ J I j (cid:17) for ≤ p ≤ n , for p < or p > n .This spectral sequence converges strongly to K ∗ A . Theorem 4.6.2 (Functoriality of the spectral sequence from Theorem 4.6.1) . Thespectral sequence { E rp,q , d r } r,p,q from Theorem 4.6.1 is functorial with respect to ∗ -homomorphisms that preserve ( n + 1) -fold ideal decompositions (Definition 4.1.1):For n ′ ≥ n , let A ′ = I ′ + I ′ + · · · + I ′ n ′ be a C*-algebra and let f : A → A ′ be a ∗ -homomorphism such that f ( I j ) ⊆ I ′ j for all j ≤ n . Let { E rp,q , d r } r,p,q be the spectralsequence from Theorem 4.6.1 that converges to A and let { ¯ E rp,q , ¯ d r } r,p,q be the spectralsequence that converges to A ′ .Then f induces a morphism { f rp,q } r,p,q of spectral sequences (Definition 2.7.4) ofbidegree (0 , with f rp,q : E rp,q → ¯ E rp,q for all r ≥ and p , q ∈ Z that commutes with the differentials and, in turn, induces K ∗ f : K ∗ A → K ∗ A ′ on the convergence targets.Proof. All constructions since Section 4.3 on the level of C*-algebras have beenfunctorial with respect to ideal decompositions: Cake algebras, sums of cake algebras,cones, suspensions.Likewise, taking K-theory and constructing direct sums of K-theory groups foreach nonempty J ⊆ { , , . . . , n } are functorial in the same way. Thus { f rp,q } r,p,q exists and induces the correct morphism { f ∞ p,q } p,q : { E ∞ p,q } → { ¯ E ∞ p,q } , which inducesthe desired K ∗ f : K ∗ A → K ∗ A ′ on the convergence targets. 41 Finite coarse excision We generalize coarsely excisive pairs from Definition 2.6.2 to coarsely excisive coversof arbitrary cardinality. Later in this section, we will apply the spectral sequencefrom Theorem 4.6.1 for finitely many C*-ideals to C*-algebras obtained from finitecoarsely excisive covers. Definition 5.1.1 (Coarsely excisive cover) . Let ( X, d ) be a coarse space. Let { X β } β ∈ α be a cover of X of arbitrary cardinality | α | such that each X β is a closedsubset in X .The cover { X β } β ∈ α is called coarsely excisive if, for all nonempty finite sets J ⊆ α and for all distances R > , there exists a distance S > such that the intersection ofthe | J | -many R -neighborhoods lies in the S -neighborhood of the | J | -fold intersection: \ j ∈ J N d ( X j , R ) ⊆ N d (cid:16) \ j ∈ J X j , S (cid:17) . (5.1.1.1) Remark 5.1.2. The distance S may be chosen depending both on R and the par-ticular finite subcollection { X j } j ∈ J at hand. It is not required that, given R , a single S > satisfies 5.1.1.1 uniformly for all subcollections of the cover, or even only forall subcollections of a given cardinality | J | . Remark 5.1.3. This is a straightforward generalization of coarsely excisive pairs.These covers will yield C*-ideals in the Roe algebra of X suitable for our spectralsequence. They behave as expected: • A coarsely excisive pair of closed sets is a two-set coarsely excisive cover. • The extra requirement that each X β be closed in X does not affect any coarseproperties. An arbitrary subset Y ⊆ X and its closure Y have the sameneighborhoods N d ( Y, R ) = N d ( Y , R ) ⊆ X for any given R > because d is ametric. • Provided ∅ is not a member of a coarsely excisive cover, all intersections offinitely many sets in the cover must contain at least one point. Assume { X j } j ∈ J are | J | sets in the cover with empty intersection. Then { X j } j ∈ J does not satisfy5.1.1.1. This can be seen in a similar way as Example 2.6.4: In a metric space,any two nonempty sets have finite distance from each other, and thus R canbe chosen as the maximum of the pairwise distances among the X j , producinga nonempty T j ∈ J N d ( X j , R ) that is not a subset of N d ( ∅ , S ) = ∅ . 43 Finite coarse excision Definition 5.1.4 (Compatible coarsely excisive covers) . Let ( X, d ) and ( X ′ , d ′ ) becoarse spaces and f : X → X ′ a coarse map. Let { X β } β ∈ α be a finite or infinitecoarsely excisive cover of X and { X ′ β } β ∈ α ′ a coarsely excisive cover of X ′ with α ⊆ α ′ such that f ( X β ) ⊆ X ′ β for every β ∈ α .Then the two covers are called compatible with f , or, when f is clear from thecontext, simply compatible . Before we can use our spectral sequence for abstract C*-ideals on coarsely excisivecovers, we have to shape our data accordingly – we don’t have sums and intersectionsof C*-ideals, but rather unions and intersections of subspaces X j . We connect thecoarse world to the world of abstract C*-ideal intersections via relative Roe algebrasand relative D ∗ -algebras. Notation 5.2.1. In Sections 5.2 through 5.4, let ( X, d ) be a metric space. Fix avery ample representation ̺ : C X → BH for a separable Hilbert space H to define D ∗ X and C ∗ X .Let { X β } β ∈ α be a coarsely excisive cover of ( X, d ) for an arbitrary index set α .Let J ⊆ α be a nonempty finite subset of indices. Definition 5.2.2 (Support near a subset) . Let Y ⊆ X be a subspace. An operator T ∈ C ∗ X is supported near Y if there exists a constant R > (that may depend on T ) such that all f ∈ C X with d ( supp f, Y ) > R satisfy ̺ ( f ) T = T ̺ ( f ) = 0 ∈ BH . Definition 5.2.3 (Relative Roe algebra) . For Y ⊆ X closed, the relative Roe algebraof Y in X , denoted C ∗ ( Y ⊆ X ) , is defined as the norm closure of all operators in C ∗ X that are supported near Y . Definition 5.2.4 (Relative D ∗ algebra) . Let Y ⊆ X be a subspace. The C*-algebra C X contains C ( X − Y ) as a C*-ideal: The inclusion morphism C ( X − Y ) → C X extends functions by zero on Y .Let T ∈ D ∗ X be an operator such that, for all f ∈ C ( X − Y ) ⊆ C X , both ̺ ( f ) T and T ̺ ( f ) are compact operators. Then T is called locally compact outside Y .For Y ⊆ X closed, the relative D ∗ algebra D ∗ ( Y ⊆ X ) is the norm closure of alloperators in D ∗ X that are supported near Y and locally compact outside Y . Remark 5.2.5. The algebra C ∗ ( Y ⊆ X ) is a C*-ideal in C ∗ X ; the algebra D ∗ ( Y ⊆ X ) is a C*-ideal in D ∗ X . As subalgebras of C ∗ X and D ∗ X , all operators in theseideals are locally compact or pseudocompact, respectively. Each operator has finitepropagation or is a norm limit of operators with finite propagation.44 .2 Relative Roe algebras For Y = X , we have C ∗ ( X ⊆ X ) = C ∗ X and D ∗ ( X ⊆ X ) = D ∗ X . Operators in D ∗ ( Y ⊆ X ) act with three strengths on represented functions: pseudocompactly on Y , in a locally compact way near Y , and trivially far away from Y .Our guideline is [Roe96, Theorem 9.2]: If the coarsely excisive cover { X , X } hasonly two sets, then there are isomorphisms C ∗ ( X ⊆ X ) + C ∗ ( X ⊆ X ) ∼ = C ∗ X and C ∗ ( X ⊆ X ) ∩ C ∗ ( X ⊆ X ) ∼ = C ∗ ( X ∩ X ) .Besides for C ∗ , we prove a similar result for D ∗ and Q ∗ = D ∗ / C ∗ ; especially D ∗ requires extra work over the proofs for C ∗ . Also, finite subsets { X j } j ∈ J of arbitarycoarsely excisive covers { X β } β ∈ α require more care than an inductive application ofthe result for two regions. Even when S j ∈ J X j happens to be a small subset of X ,our C*-algebras still arise from representations of C X in its entirety. Our proofsmust safely ignore regions of X far away from S j ∈ J X j . Theorem 5.2.6. For all closed subsets Y ⊆ X and K-theory degrees s ∈ Z , thereare natural isomorphisms of K-theory groups: K s C ∗ ( Y ⊆ X ) ∼ = K s C ∗ Y,K s D ∗ ( Y ⊆ X ) ∼ = K s D ∗ Y. Remark 5.2.7. The proof for C ∗ in [Roe96, Theorem 9.2] passes from Y ⊆ X tothe coarsely equivalent N d ( Y, n ) ⊆ X for n ∈ N and takes the direct limit in K-theory along n → ∞ . The isomorphism is induced by the inclusion Y → X . Theconstruction for C ∗ is natural with respect to coarse maps. The construction for D ∗ is natural with respect to maps that are both coarse and continuous.A proof for D ∗ is in [Sie12, Proposition 3.8]. This construction calls for a veryample representation ̺ : C X → BH as we have required in Notation 5.2.1, notmerely for an ample representation. Lemma 5.2.8. The representation ̺ : C X → BH can be extended to all Borelfunctions on X .Proof. This follows from [Dav96, Theorem II.1.1] and [Dav96, Proposition II.1.2]:As a nondegenerate representation of C X , the given ̺ is equivalent to a direct sum L γ ∈ Γ ̺ γ of cyclic representations ̺ γ , each unitarily equivalent to pointwise multipli-cation with a continuous function f γ that depends only on the cyclic representation ̺ γ , on the Hilbert space L ( X ) of H -valued functions using a regular Borel proba-bility measure. Now extend by pointwise multiplication.The topological space in this construction is compact, but continuous functions ona compact space differ, as an algebra, from C X of a noncompact space X merelyby the value at the extra point of the one-point compactification of X . 45 Finite coarse excision Lemma 5.2.9. Let ψ : X → [0 , be a Borel function. Extend ̺ to all Borel functionsas in Lemma 5.2.8 such that ̺ ( ψ ) makes sense. Let Y ⊆ X be a closed subset. Let T be an operator in D ∗ ( Y ⊆ X ) . Then the following statments hold. • We have ̺ ( ψ ) T ∈ D ∗ ( Y ⊆ X ) . • Let R supp be a distance constant for the support of T near Y ; i.e., for functions f ∈ C X with d ( supp f, Y ) > R supp , the operators ̺ ( f ) T and T ̺ ( f ) vanish.Then ̺ ( ψ ) T is supported near Y with the same distance constant R supp . • If T has finite propagation with distance constant R prop , then ̺ ( ψ ) T has finitepropagation with distance constant R prop . (If T does not admit such an R prop ,then T is in the norm completion of operators that do.) • If T ∈ C ∗ ( Y ⊆ X ) , then also ̺ ( ψ ) T ∈ C ∗ ( Y ⊆ X ) . The main idea is that the support of a pointwise product of functions f ψ for f ∈ C X must be a subset of supp f within X . Later in this section, ψ will be afunction from a Borel partition of unity and supp ( f ψ ) may be much smaller thansupp f . Proof of Lemma 5.2.9. We show the claim for all finite-propagation operators. Theabsolute algebras D ∗ X and C ∗ Y are norm completions of such operators; the generalclaim follows because the relative algebras D ∗ ( Y ⊆ X ) and C ∗ ( Y ⊆ X ) carry thesame norm as subalgebras of the absolute algebras.Let R prop > be a constant of finite propagation for T ; i.e., for f , g ∈ C X with d ( supp f, supp g ) ≥ R prop , the product ̺ ( f ) T ̺ ( g ) ∈ BH is zero. For such f , g ∈ C X with d ( supp f, supp g ) ≥ R prop , we have supp ( f ψ ) ⊆ supp f , therefore d (cid:0) supp ( f ψ ) , supp g (cid:1) ≥ d ( supp f, supp g ) ≥ R prop and ̺ ( f ) ̺ ( ψ ) T ̺ ( g ) = ̺ ( f ψ ) T ̺ ( g ) = 0 . Thus ̺ ( ψ ) T has finite propagation with thesame constant R prop .Fix R supp > such that all f ∈ C X with d ( supp f, Y ) > R satisfy ̺ ( f ) T = T ̺ ( f ) = 0 ∈ BH ; such an R supp exists because T is supported near Y . Given f ∈ C X with d ( supp f, Y ) > R , we have supp ( f ψ ) ⊆ supp f , thus ̺ ( f ) ̺ ( ψ ) T = ̺ ( f ψ ) T = 0 . Furthermore, ̺ ( ψ ) T ̺ ( f ) = 0 because T ̺ ( f ) = 0 . Thus ̺ ( ψ ) T issupported near Y with the same distance constant R supp .For pseudocompactness, given f ∈ C X , we must show that the following operator46 .3 Intersections of relative algebras is compact: ̺ ( f ) ̺ ( ψ ) T − ̺ ( ψ ) T ̺ ( f ) = ̺ ( f ψ ) T − ̺ ( ψ ) T ̺ ( f )= ̺ ( ψf ) T − ̺ ( ψ ) T ̺ ( f )= ̺ ( ψ ) (cid:0) ̺ ( f ) T − T ̺ ( f ) (cid:1)| {z } compact since T ∈ D ∗ X , which is compact as a product with a compact operator. Thus ̺ ( ψ ) T is pseudolocal.For local compactness of ̺ ( ψ ) T outside Y , let f be a function in C ( X − Y ) Because T is already locally compact outside Y , ̺ ( f ) T and T ̺ ( f ) are compactoperators. Then ̺ ( f ) ̺ ( ψ ) T = ̺ ( ψ ) ̺ ( f ) T and ̺ ( ψ ) T ̺ ( f ) are also compact. Thus ̺ ( ψ ) T is locally compact outside Y .Additionally, if T ∈ C ∗ ( Y ⊆ X ) , the same argument applied to arbitrary f ∈ C X shows that ̺ ( f ) ̺ ( ψ ) T and ̺ ( ψ ) T ̺ ( f ) are compact for all f ∈ C X . Thus ̺ ( ψ ) T islocally compact if T is. Notation 5.3.1. Throughout Section 5.3, in addition to Notation 5.2.1 that definesthe finite nonempty subset { X j } j ∈ J of the coarsely excisive cover { X β } β ∈ α , write Z = \ j ∈ J X j . Lemma 5.3.2. Let F ∗ denote either the functor C ∗ or D ∗ . Then F ∗ ( Z ⊆ X ) ⊆ \ j ∈ J F ∗ ( X j ⊆ X ) . Proof. For T ∈ F ∗ ( Z ⊆ X ) , there exists R > such that ̺ ( f ) T = T ̺ ( f ) = 0 for all f ∈ C X with d ( supp f, Z ) > R by definition of T being supported near Z .Given j ∈ J , let f j ∈ C X be a function with d ( supp f j , X j ) > R . Then d ( supp f j , Z ) > R because Z ⊆ X j , therefore T is supported near X j . This holds forall j ∈ J . For F ∗ = C ∗ , this finishes the proof: T is in C ∗ ( X j ⊆ X ) for all j ∈ J .For F ∗ = D ∗ , we must show, in addition, that T is locally compact outside X j forthe given j ∈ J ; this holds because Z ⊆ X j and T ∈ D ∗ ( Z ⊆ X ) is locally compactoutside Z . Notation 5.3.3. For a subset Y ⊆ X , let χ ( Y ) : X → { , } Finite coarse excision denote the charateristic function of Y on X ; i.e., χ ( Y )( x ) = 1 if and only if x ∈ Y . Lemma 5.3.4. For F ∗ = C ∗ or F ∗ = D ∗ , we have F ∗ ( Z ⊆ X ) ⊇ \ j ∈ J F ∗ ( X j ⊆ X ); i.e., the inclusion from Lemma 5.3.2 is an equality of sets.Proof. Fix T ∈ T j ∈ J F ∗ ( X j ⊆ X ) . We will show that T ∈ F ∗ ( Z ⊆ X ) . Support of T near Z . Since T is supported near X j for all j ∈ J , there are constants R j > such that whenever f ∈ C X satisfies d ( supp f, X j ) > R j for at least one j ∈ J , then ̺ ( f ) T = T ̺ ( f ) = 0 . (It is not necessary that d ( supp f, X j ) > R j holdsfor all j ∈ J . It is enough if this holds for one j ∈ J because “support near a subset”states what happens on the complement of the subset, not on the subset itself.)The cover { X β } β ∈ α is coarsely excisive. In particular, for the constant R =(max j ∈ J R j ) + 1 and for the chosen finite index set J , there exists S > with T j ∈ J N d ( X j , R ) ⊆ N d ( Z, S ) or, reformulating with complement sets, (cid:0) X − N d ( Z, S ) (cid:1) ⊆ [ j ∈ J (cid:0) X − N d ( X j , R ) (cid:1) . This constant S depends on T and the X j , but not on any function in C X . Tofinish the proof, choose f ∈ C X with d ( supp f, Z ) > S + 1 . We must show that ̺ ( f ) T = T ̺ ( f ) = 0 .The support of this f lies within X − N d ( Z, S ) , thus there exists a j ∈ J withsupp f ⊆ (cid:0) X − N d ( X j , R ) (cid:1) and therefore d ( supp f, X j ) ≥ R > R j . Because T issupported near X j , we conclude that ̺ ( f ) T = T ̺ ( f ) = 0 as desired. Thus T issupported near Z . Local compactness of T outside Z . If F ∗ = C ∗ , the proof is finished because T islocally compact everywhere in X .If F ∗ = D ∗ , we must show that T is locally compact outside Z . Fix a function g ∈ C ( X − Z ) ⊆ C X . We must show that ̺ ( g ) T and T ̺ ( g ) are compact.The support of g may still overlap each X j . To remedy this, decompose X into | J | regions: Given L ⊆ J , write Y L = (cid:16) \ j ∈ L X j (cid:17) − [ j / ∈ L X j = { x ∈ X : x ∈ X j if and only if j ∈ L } . .3 Intersections of relative algebras For all L ⊆ J , the set Y L is a Borel set as a finite union, intersection, and setdifference of closed sets X j . For L = L ′ ⊆ J , the regions Y L and Y L ′ are disjoint.Furthermore, X = [ L ⊆ J Y L , Z = Y J = \ j ∈ J X j . Decompose g into | J | Borel functions on X by multiplying with indicator functions: g = X L ⊆ J χ ( Y L ) g. Extend ̺ from C X to all Borel functions on X . Lemma 5.2.9 shows that theoperators ̺ (cid:0) χ ( Y L ) (cid:1) T and T ̺ (cid:0) χ ( Y L ) (cid:1) remain in T j ∈ J D ∗ ( X j ⊆ X ) because the Y L are Borel sets.For each L ⊆ J , examine χ ( Y L ) g : For L = J , we have χ ( Y J ) g = χ ( Z ) g = 0 since g ∈ C ( X − Z ) . Both ̺ (cid:0) χ ( Y J ) g (cid:1) T and T ̺ (cid:0) χ ( Y J ) g (cid:1) are the zero operator, thuscompact. For L = J , fix an index j ∈ J − L . Then χ ( Y L ) g vanishes on X j because Y L contains no points from X j by definition. We have T ∈ D ∗ ( X j ⊆ X ) , therefore T is locally compact outside X j , making both ̺ (cid:0) χ ( Y L ) g (cid:1) T and T ̺ (cid:0) χ ( Y L ) g (cid:1) compact.This shows that the decomposition g = P L ⊆ J χ ( Y L ) g allows ̺ ( g ) T and T ̺ ( g ) tobe written as finite sums of | J | compact operators each. Such sums are compact.Thus T is locally compact outside Z . Proposition 5.3.5. Let s ∈ Z be a degree for K-theory. Then for the nonemptyfinite index set J ⊆ α , there are natural isomorphisms K s C ∗ (cid:16) \ j ∈ J X j (cid:17) ∼ = K s (cid:16) \ j ∈ J C ∗ ( X j ⊆ X ) (cid:17) ,K s D ∗ (cid:16) \ j ∈ J X j (cid:17) ∼ = K s (cid:16) \ j ∈ J D ∗ ( X j ⊆ X ) (cid:17) . Proof. Combine Lemma 5.3.2 with Lemma 5.3.4 for Z = T j ∈ J X j : C ∗ ( Z ⊆ X ) = \ j ∈ J C ∗ ( X j ⊆ X ) , D ∗ ( Z ⊆ X ) = \ j ∈ J D ∗ ( X j ⊆ X ) . Theorem 5.2.6 relates the K-theory of relative algebras with the K-theory of absolutealgebras. This yields the claimed isomorphisms. Proposition 5.3.6. Write Q ∗ X = D ∗ X/ C ∗ X as in Notation 2.4.13. Let s ∈ Z be a Finite coarse excision degree in K-theory. Then there is a natural isomorphism K s Q ∗ (cid:16) \ j ∈ J X j (cid:17) ∼ = K s (cid:16) \ j ∈ J Q ∗ ( X j ⊆ X ) (cid:17) , Proof. For closed Y ⊆ X , consider the following commutative diagram. Its rowsare long exact sequences in K-theory. Its vertical isomorphisms are induced by theinclusion of metric spaces Y ⊆ X as in Remark 5.2.7. The third vertical morphism iswell-defined as follows because the rows are exact: For an operator T ∈ D ∗ ( Y ⊆ X ) whose K-theory class [ T ] maps to [ T ′ ] ∈ D ∗ Y under the second vertical morphism,the third morphism maps [ T ] + K s C ∗ ( Y ⊆ X ) to K s (cid:0) [ T ′ ] + K s C ∗ Y (cid:1) . · · · K s C ∗ ( Y ⊆ X ) K s D ∗ ( Y ⊆ X ) K s D ∗ ( Y ⊆ X ) K s C ∗ ( Y ⊆ X ) K s − C ∗ ( Y ⊆ X ) · · ·· · · K s C ∗ Y K s D ∗ Y K s Q ∗ Y K s − C ∗ Y · · · . ∼ = ∼ = ∼ = All squares commute because C ∗ Y → D ∗ Y is a C*-ideal inclusion and because thevertical isomorphisms arose from taking natural direct limits.By the five lemma, the vertical arrow to K s Q ∗ Y must also be an isomorphism. Itis natural by construction. With the isomorphisms from Propositions 5.3.5, we have K s Q ∗ (cid:16) \ j ∈ J X j (cid:17) ∼ = K s D ∗ (cid:0) T j ∈ J X j (cid:1) K s C ∗ (cid:0) T j ∈ J X j (cid:1) ∼ = K s (cid:0) T j ∈ J D ∗ ( X j ⊆ X ) (cid:1) K s (cid:0) T j ∈ J C ∗ ( X j ⊆ X ) (cid:1) ∼ = K s (cid:16) \ j ∈ J Q ∗ ( X j ⊆ X ) (cid:17) . Notation 5.4.1. Throughout Section 5.4, in addition to Notation 5.2.1, write Z = [ j ∈ J X j . For a subset Y ⊆ X , again, denote the indicator function by χ ( Y ) : X → { , } . Lemma 5.4.2. Let F ∗ be either the functor C ∗ or D ∗ . Then F ∗ ( Z ⊆ X ) ⊆ X j ∈ J F ∗ ( X j ⊆ X ) . Proof. By definition, F ∗ ( Y ⊆ X ) and F ∗ X for closed subsets Y ⊆ X are norm50 .4 Sums of relative algebras completions of finite-propagation operator algebras. It suffices to check the inclusionfor finite-propagation operators in F ∗ ( Z ⊆ X ) ; passing to norm completions willthen prove the claim.In this light, let T ∈ C ∗ ( Z ⊆ X ) be an operator with finite propagation. We willconstruct operators T j for j ∈ J such that T j ∈ F ∗ ( X j ⊆ X ) and P j ∈ J T j = T .Because T has finite propagation, find R prop > such that ̺ ( f ) T ̺ ( g ) = 0 ∈ BH whenever f , g ∈ C X satisfy d ( supp f, supp g ) ≥ R prop . Find R supp > such that T is supported in the R -neighborhood of Z .Cover Z = S j ∈ J X j by the following sets Y j for j ∈ J and their union Y : Y j = N d ( X j , R prop + R supp + 1) = { x ∈ X : d ( x, X j ) ≤ R prop + R supp + 1 } ,Y = [ j ∈ J Y j . Each Y j is closed in X . Certainly, T is supported in Y , again a closed set.Choose a linear order ≺ on the finite set J . Define a partition { P j } j ∈ J of Y via P j = (cid:8) x ∈ Z : x ∈ X j and there are no j ′ ≺ j with x ∈ X j ′ (cid:9) ∪ (cid:8) x ∈ Y − Z : x ∈ Y j and there are no j ′ ≺ j with x ∈ Y j ′ (cid:9) ; this is a partition of Y because, given either x ∈ Z or x ∈ Y − Z , exactly one P j iseligible to contain x . Furthermore, each P j is a Borel set because it may be writtenas a finite union, intersection, and difference of Borel sets X j ′ and Y j ′ .Extend the representation ̺ : C X → BH to the Borel functions of X accordingto Lemma 5.2.8. For all j ∈ J , define operators T j ∈ BH by e T = ̺ (cid:0) χ ( X − Y ) (cid:1) T | J | , T j = ̺ (cid:0) χ ( P j ) (cid:1) T + e T . By Lemma 5.2.9, e T and all T j are in F ∗ ( Z ⊆ X ) . The T j sum to X j ∈ J T j = X j ∈ J ̺ (cid:0) χ ( P j ) (cid:1) T + X j ∈ J ̺ (cid:0) χ ( X − Y ) (cid:1) T | J | = ̺ (cid:0) χ ( Y ) + χ ( X − Y ) | {z } = on X (cid:1) T = T. (5.4.2.1)The summand e T of T j merely clarifies P j ∈ J T j = T ; it has no deeper meaning. Forall functions g ∈ C X , the products ̺ ( g ) e T and e T ̺ ( g ) vanish because χ ( X − Y ) issupported further than R prop + R supp away from Z , whereas e T ∈ F ∗ ( Z ⊆ X ) hasthe same propagation constant R prop and support distance constant R supp as T byLemma 5.2.9. 51 Finite coarse excision Support of T j near X j . For a given j ∈ J , let f ∈ C X have support far enoughaway from X j : d ( supp f, X j ) > R prop + R supp + 1 . We will show ̺ ( f ) T j = T j ̺ ( f ) = 0 . For ̺ ( f ) T j = 0 , we have ̺ ( f ) T j = ̺ (cid:0) f χ ( P j ) (cid:1) T + ̺ ( f ) e T | {z } = = 0 (5.4.2.2)because the pointwise product f χ ( P j ) is zero in C X : The function f is supportedmore than R prop + R supp + 1 away from X j , but P j ⊆ Y j = N d ( X j , R prop + R supp + 1) .For T j ̺ ( f ) = 0 , we similarly show that T j ̺ ( f ) = ̺ (cid:0) χ ( P j ) (cid:1) T ̺ ( f ) + e T ̺ ( f ) | {z } = = 0; (5.4.2.3)to see this, observe that d ( X j , X − P j ) ≤ R supp + 1 by construction of P j and d ( X j , supp f ) > R prop + R supp +1 by the choice of f . The difference between these twovalues is more than R prop , the propagation constant of T , thus ̺ (cid:0) χ ( P j ) (cid:1) T ̺ ( f ) = 0 . Local compactness of T j outside X j . Let f be a function in C ( X − X j ) ⊆ C X .We will show that ̺ ( f ) T j and T j ̺ ( f ) are compact.Recall that ̺ ( f ) T j = ̺ (cid:0) f χ ( P j ) (cid:1) T + 0 , thus it suffices to examine the pointwiseproduct f χ ( P j ) : It may assume nonzero values only in ( Y j − Z ) ∪ X j by definitionof P j . Because X j ⊆ Z , each point from P j falls either into X j or into Y − Z . Wemay decompose f χ ( P j ) as f χ ( P j ) = f χ ( P j ∩ X j ) + f χ ( P j − Z ) . The left summand is the zero function because f vanishes on X j . The right summandmay be nonzero, but vanishes on Z . Outside Z , the original T is locally compact,therefore ̺ (cid:0) f χ ( P j − Z ) (cid:1) T is compact. Since f χ ( P j − Z ) = f χ ( P j ) and furthermore ̺ (cid:0) f χ ( P j ) (cid:1) T = ̺ ( f ) T j , the desired operator ̺ ( f ) T j is compact.The difference ̺ ( f ) T j − T j ̺ ( f ) is compact because T j ∈ D ∗ ( Z ⊆ X ) is pseudo-compact. With ̺ ( f ) T j already proven compact, T j ̺ ( f ) must be compact, too. Summary. The claim follows from 5.4.2.1, 5.4.2.2, 5.4.2.3, and from the local com-pactness of T j outside Z : We have decomposed T into a sum P j ∈ J T j with each T j in F ∗ ( X j ⊆ X ) .52 .4 Sums of relative algebras Lemma 5.4.3. For F ∗ = C ∗ or F ∗ = D ∗ , we have F ∗ ( Z ⊆ X ) ⊇ X j ∈ J F ∗ ( X j ⊆ X ); i.e., the inclusion from Lemma 5.4.2 is an equality of sets.Proof. For each j ∈ J , let T j be an operator in F ∗ ( X j ⊆ X ) such that T j is supportedin an R j -neighborhood of X j . Define T = P j ∈ J T j . This T is supported in the (max j ∈ J R j ) -neighborhood of Z = S j ∈ J X j .For F ∗ = D ∗ , given f ∈ C ( X − Z ) ⊆ C X and j ∈ J , we know that f is also in C ( X − X j ) . The operators ̺ ( f ) T j and T j ̺ ( f ) are compact since T j ∈ F ∗ ( X j ⊆ X ) .The finite sums ̺ ( f ) T and T ̺ ( f ) of compact operators are again compact. Proposition 5.4.4. For all s ∈ Z , there are natural isomorphisms K s C ∗ (cid:16) [ j ∈ J X j (cid:17) ∼ = K s (cid:16) X j ∈ J C ∗ ( X j ⊆ X ) (cid:17) ,K s D ∗ (cid:16) [ j ∈ J X j (cid:17) ∼ = K s (cid:16) X j ∈ J D ∗ ( X j ⊆ X ) (cid:17) . Proof. Combine Lemma 5.4.2 with Lemma 5.4.3 for Z = S j ∈ J X j : C ∗ ( Z ⊆ X ) = X j ∈ J C ∗ ( X j ⊆ X ) , D ∗ ( Z ⊆ X ) = X j ∈ J D ∗ ( X j ⊆ X ) . Theorem 5.2.6 yields the claimed isomorphisms. Proposition 5.4.5. Let s ∈ Z be a degree in K-theory. There is a natural isomor-phism K s Q ∗ (cid:16) [ j ∈ J X j (cid:17) ∼ = K s (cid:16) X j ∈ J Q ∗ ( X j ⊆ X ) (cid:17) . Proof. In the proof of Proposition 5.3.6, we constructed a natural isomorphism forclosed subsets Y ⊆ X , K s D ∗ ( Y ⊆ X ) K s C ∗ ( Y ⊆ X ) ∼ = −→ Q ∗ Y. Finite coarse excision Combine this isomorphism with the natural isomorphisms from Proposition 5.4.4: K s Q ∗ (cid:16) [ j ∈ J X j (cid:17) ∼ = K s D ∗ (cid:0) S j ∈ J X j (cid:1) K s C ∗ (cid:0) S j ∈ J X j (cid:1) ∼ = K s (cid:0) P j ∈ J D ∗ ( X j ⊆ X ) (cid:1) K s (cid:0) P j ∈ J C ∗ ( X j ⊆ X ) (cid:1) ∼ = K s (cid:16) X j ∈ J Q ∗ ( X j ⊆ X ) (cid:17) . We may summarize Propositions 5.3.5, 5.3.6, 5.4.4, and 5.4.5 in a single theorem. Theorem 5.5.1. Let ( X, d ) be a metric space. Let { X β } β ∈ α be a finite or infinitecoarsely excisive cover of X and let J ⊆ α be a finite nonempty subset.Let F ∗ be either the functor C ∗ from the coarse category to C ∗ A or one of thefunctors D ∗ or Q ∗ from the coarse-continuous category to C ∗ A . Let s be a degree inK-theory. Then K s F ∗ (cid:16) \ j ∈ J X j (cid:17) ∼ = K s (cid:16) \ j ∈ J F ∗ ( X j ⊆ X ) (cid:17) ,K s F ∗ (cid:16) [ j ∈ J X j (cid:17) ∼ = K s (cid:16) X j ∈ J F ∗ ( X j ⊆ X ) (cid:17) . These isomorphisms are natural with respect to morphisms (coarse maps for C ∗ , orcoarse and continuous maps for D ∗ and Q ∗ ) to other coarse spaces with compatiblecoarsely excisive covers (Definition 5.1.4). When the cover { X β } β ∈ α has a finite index set α , the algebras become suitablefor our spectral sequence for finite ideal inclusions. Theorem 5.5.2. Let ( X, d ) be a metric space with a finite coarsely excisive cover { X β } β ∈ α . Let F ∗ be either the functor C ∗ from the coarse category to C ∗ A or oneof the functors D ∗ or Q ∗ from the coarse-continuous category to C ∗ A . There is aspectral sequence { E rp,q , d r } r,p,q with E p,q ∼ = M | J | = p +1 K q F ∗ (cid:16) \ j ∈ J X j (cid:17) for ≤ p < | α | , for p < or p ≥ | α | ,where J ranges over all nonempty subsets of α . This spectral sequence convergesstrongly to K ∗ F ∗ X and is functorial with respect to morphisms (coarse maps for C ∗ ,or coarse and continuous maps for D ∗ and Q ∗ ) to other coarse spaces with compatiblecoarsely excisive covers (Definition 5.1.4). .6 Application: K ∗ C ∗ R n Proof. Apply the spectral sequence from Theorem 4.6.1 about finite sums of abstractC*-algebras to the algebras from Theorem 5.5.1 for coarse spaces.Both properties from Theorem 5.5.1 are required here: The intersection propertyguarantees that the E -term looks as stated. The sum property of the relativeC*-algebras guarantees that the spectral sequence converges to the non-relative C*-algebra of the entire space.Functoriality of the spectral sequence follows from functoriality of the spectralsequence from Theorem 4.6.1 for finite sums of abstract C*-ideals and from thenaturality of the isomorphisms in Theorem 5.5.1. K ∗ C ∗ R n Let d and d ∞ denote the usual 1-metric and sup -metric on R n : For x , y ∈ R n , wehave d ( x, y ) = X j For the n -dimensional Euclidean space R n , metrized either withthe 1-metric d or the sup -metric d ∞ , the Roe algebra has the following K-theory: K s C ∗ R n = Z for s − n even, for s − n odd. This is known, but we reprove this with our K-theory spectral sequence for finitecoarsely excisive covers. Remark 5.6.2. Towards the end of Section 5.6, some claims and proofs might looklike straightforward geometry of R n . In particular, since d and d ∞ are equivalentmetrics on R n , they must induce equivalent coarse structures; it would suffice to lookat only one of them.Nonetheless, we will conduct these proofs in detail for both d and d ∞ becausethese results will serve as lemmas for Section 7.3 to compute the K-theory of aC*-ideal of C ∗ Z ∞ under different metrics. Definition 5.6.3 (Flasque) . Let ( X, d ) be a metric space. X is flasque if there is acoarse map f : X → X satisfying the following conditions: • The map f is coarsely equivalent to id ( X ) . • For all K ⊆ X , there exists N ∈ N such that for all n ≥ N , f n ( X ) ∩ K = ∅ . • The powers of f are uniformly coarse: For all R > , there exists S > that,for all n ∈ N at once and x , y ∈ X with d ( x, y ) ≤ R , we have d ( f n x, f n y ) ≤ S .55 Finite coarse excision Lemma 5.6.4. For a metric space X , the product X × R ≥ is flasque.Proof. Consider the self-map f on X × R ≥ with f ( x, t ) = ( x, t + 1) . Shifting pointsby a constant distance, f is coarsely equivalent to the identity, yet the powers f n eventually shift points out of any given bounded set. As an isometry, f is uniformlycoarse: Choose S = R for the third condition in Definition 5.6.3. Proposition 5.6.5. Let X be a flasque space. Then K ∗ C ∗ X = 0 . Proposition 5.6.5 is proven in [Roe88, Proposition 9.4]. To prove Theorem 5.6.1about K ∗ C ∗ R n with a coarsely excisive cover of R n , it is helpful to have many flasqueintersections. Definition 5.6.6 (Block decomposition of R n ) . Cover R n with n + 1 overlappingsubsets, or blocks , X , X , . . . , X n , where X j = ] −∞ , × R n − for j = 0 , [0 , ∞ [ i × ] −∞ , × R n − j − for < j < n, [0 , ∞ [ n for j = n. Example 5.6.7. In the simplest case, R is covered with two overlapping rays, oneextending into either direction. R is covered with three pieces, a left-hand half-space X , a bottom-right-hand quadrant X , and a top-right-hand quadrant X , asin Figure 5.6.8: X X X X X X X Figure 5.6.8: Decomposition of R into 3 blocks and of R into 4 blocks Remark 5.6.9. In the block decomposition R n = X ∪ X ∪· · ·∪ X n , each X j containsat least one flasque factor, therefore K ∗ C ∗ X j = 0 . Likewise, intersecting fewer thanall n + 1 segments produces a flasque intersection with trivial K-theory of the Roealgebra. Only the ( n + 1) -fold intersection is not flasque; it is the compact one-pointset. Its Roe algebra has K-theory Z in even degrees and zero in odd degrees.56 .6 Application: K ∗ C ∗ R n Definition 5.6.10 (Blocky subset of R n ) . Let X be a subset of R n . We call X blocky if both of these conditions hold: • The origin ∈ R n is part of X . • For all x ∈ X , all n coordinates x j of x , and all λ ≥ , varying x j by λ doesn’tleave X ; i.e., this point is a part of X : ( x , x , . . . , x j − , λx j , x j +1 , . . . , x n − ) Example 5.6.11. Blocky subsets of R n are conical, but not all conical subsets areblocky. Consider the upper-right quadrant in R : This is blocky. It remains conical,but not blocky, after rotating around the origin by an eighth-turn. Remark 5.6.12. For a nonempty collection of blocky subsets { X β } β ∈ α of R n , theintersection T β ∈ α X β is again blocky.All sets of the block decomposition { X j } j ≤ n of R n and all their intersections T j ∈ J X j for nonempty J ⊆ { , , . . . , n } satisfy this definition of blocky . This is themotivation behind the following Lemma 5.6.13. Lemma 5.6.13. Let X β , X γ ⊆ R n be blocky sets. Choose R > . Then N ∞ ( X β , R ) ∩ N ∞ ( X γ , R ) = N ∞ ( X β ∩ X γ , R ) , (5.6.13.1) where N ∞ denotes the R -neighborhood under the sup -metric d ∞ .Proof. The direction “ ⊇ ” is immediate: If a point y ∈ R n is at most R away from X β ∩ X γ , then it is at most R away from both X β and X γ independently.To show “ ⊆ ”, fix y ∈ N ∞ ( X β , R ) ∩ N ∞ ( X γ , R ) . We will show that this y isalready in N ∞ ( X β ∩ X γ , R ) . For each coordinate y δ , by definition of d ∞ , we have inf x ∈ X | y δ − x δ | ≤ R for both X = X β and X = X γ . In particular, for both X = X β and X = X γ , inf x ∈ X | y δ − x δ | = if y δ > and { } δ × [0 , + ∞ [ × { } n − δ − ⊆ X, if y δ < and { } δ × ] −∞ , × { } n − δ − ⊆ X, | y δ | ≤ R otherwise . Certainly, X β ∩ X γ is nonempty; at least the origin is part of this intersection. Tofinish the proof, assume that d ∞ ( y, X β ∩ X γ ) > R . Then there exists a δ -th coordinatesuch that inf {| y δ − x δ | : x ∈ X β ∩ X γ } > R. (5.6.13.2)57 Finite coarse excision Then | y δ | > R because X β ∩ X γ is blocky. This forbids the “otherwise”-case for both X = X β and X = X γ . Both of the remaining two cases force the ray from the originin the δ -th dimension that contains (0 , . . . , , y δ , , . . . , to be a subset of both X β and X γ . But now, setting y δ to zero will not affect the distance to the blocky set X β ∩ X γ : d ∞ (cid:0) ( y , . . . , y δ − , y δ , y δ +1 , . . . , y n − ) , X β ∩ X γ (cid:1) = d ∞ (cid:0) ( y , . . . , y δ − , , y δ +1 , . . . , y n − ) , X β ∩ X γ (cid:1) . After setting y δ to , if there are still coordinates remaining that satisfy 5.6.13.2,repeat this argument and set those coordinates to , too, without altering the dis-tance to X β ∩ X γ . Eventually, we find that no coordinates satisfy 5.6.13.2 anymore.Therefore the assumption d ∞ ( y, X β ∩ X γ ) > R is false and y ∈ N ∞ ( X β ∩ X γ , R ) . Corollary 5.6.14. Let { X β } β ∈ α be a collection of blocky subsets of R n such that S β ∈ α X β = R n . Then { X β } β ∈ α is coarsely excisive with respect to the sup -metric d ∞ . In particular, for a nonempty finite index set J ⊆ α and R > , we have \ j ∈ J N ∞ ( X j , R ) = N ∞ (cid:16) \ j ∈ J X j , R (cid:17) . Proof. For | J | = 1 , the claim is trivial. For a | J | -fold intersection with | J | > , useinduction along the cardinality of J : Choose β ∈ J and set J ′ = J − { β } for whichthe claim already holds. Then \ j ∈ J N ∞ ( X j , R ) = N ∞ ( X β , R ) ∩ \ j ∈ J ′ N ∞ ( X j , R )= N ∞ ( X β , R ) ∩ N ∞ (cid:16) \ j ∈ J ′ X j , R (cid:17) = N ∞ (cid:16) \ j ∈ J X j , R (cid:17) , applying Lemma 5.6.13 at the end because T j ∈ J ′ X j is blocky. Proposition 5.6.15. The block decomposition { X j } j ≤ n of R n from Definiton 5.6.6is coarsely excisive under the sup -metric d ∞ : For a nonempty J ⊆ { , , . . . , n } and R > , we have \ j ∈ J N ∞ ( X j , R ) = N ∞ (cid:16) \ j ∈ J X j , R (cid:17) . Proof. Each X j is blocky as a cartesian product of copies of ] −∞ , , [0 , ∞ [ , and R .The result follows from Corollary 5.6.14.58 .6 Application: K ∗ C ∗ R n Lemma 5.6.16 (Relating 1-metric and sup -metric) . Let X ⊆ R n be arbitrary andchoose R > . Denote by N ( X, R ) the neighborhood of X under the 1-metric d andby N ∞ ( X, R ) its neighboorhood under the sup -metric d ∞ . Then N ( X, R ) ⊆ N ∞ ( X, R ) ⊆ N ( X, nR ) . (5.6.16.1) Proof. For x , y ∈ R n arbitrary, we always have d ( x, y ) = X j Proposition 5.6.17. The block decomposition { X j } j ≤ n is coarsely excisive underthe 1-metric d : Given a nonempty J ⊆ { , , . . . , n } and R > , we can set S = nR to ensure \ j ∈ J N ( X j , R ) ⊆ N (cid:16) \ j ∈ J X j , nR (cid:17) . Proof. Combining Proposition 5.6.15 with Lemma 5.6.16, we get \ j ∈ J N ( X j , R ) ⊆ \ j ∈ J N ∞ ( X j , R )= N ∞ (cid:16) \ j ∈ J X j , R (cid:17) ⊆ N (cid:16) \ j ∈ J X j , nR (cid:17) . This finishes the preparations for our proof of Theorem 5.6.1: We would like to59 Finite coarse excision show K s C ∗ R n = Z for s − n even, for s − n odd. Proof of Theorem 5.6.1. The block decomposition { X j } j ≤ n of R n into n + 1 piecesfrom Definition 5.6.6 is coarsely excisive under either the sup -metric by Proposition5.6.15 or the 1-metric by Proposition 5.6.17.Intersecting fewer than n + 1 blocks X j yields a flasque space Y with trivial K ∗ C ∗ Y = 0 . Intersecting all n + 1 points gives the compact one-point set { } with K s C ∗ { } = Z for s even and K s C ∗ { } = 0 for s odd.These results fit into our spectral sequence from Theorem 5.5.2 for coarsely excisivecovers, letting J range over all nonempty subsets of { , , . . . , n } : The first page is E p,q ∼ = M | J | = p +1 K q C ∗ (cid:16) \ j ∈ J X j (cid:17) for ≤ p ≤ n , for p < or p > n ,and the spectral sequence converges to K ∗ C ∗ R n . The E -term has only one nonzerocolumn E n, ∗ from intersecting all n + 1 pieces: · · · Z · · · · · · Z − · · · · · · n − n n + 1 n + 2 p.q This spectral sequence collapses on the first page. There is no extension problem tosolve. We may read K s C ∗ R n directly from the s -th diagonal p + q = s of E ∗ , ∗ : If s − n is even, this K-theory is Z ; otherwise, it vanishes.60 Infinite sums of ideals For A = I + I + · · · + I n , we have a spectral sequence. What happens when A is thenorm closure of a sum over countably many C*-ideals instead of over finitely many?Now A is a direct limit C*-algebra, A = X j ∈ N I j = [ n ∈ N (cid:16) X j We define the infinite simplex ∆ N = ( x j ) j ∈ [0 , N : ∞ X j =0 x j ≤ . This becomes a topological space with the usual product topology. 61 Infinite sums of ideals Let A = P j ∈ N I j be a C*-algebra. Even if we succeed in defining a function space ∆ N J for J ⊆ N and function algebras B J ⊆ B of certain functions ∆ N → A , it willbe hard interpret the function algebras appropriately. In the finite case with n + 1 ideals, the algebra { f : ∆ n → A : f ↾ ∂ ∆ n = 0 } is isomorphic to the n -fold suspension of A . The suspension isomorphism allowedus to relate the K-theory of ideals to the K-theory of A . When we replace ∆ n with ∆ N , we lose the suspension isomorphism and cannot give a convergence theorem fora spectral sequence.For another approach, recall the spectral sequence for ideal inclusions, constructedboth in Section 3 and by C. Schochet in [Sch81]: Theorem 3.1.1 (Spectral sequence for ideal inclusions) . Let A = S p ∈ N I p be a C*-algebra, where the I ⊆ I ⊆ I ⊆ · · · ⊆ I p ⊆ · · · form a chain of closed two-sidedideals. There is a spectral sequence { E rp,q , d r } r,p,q with E p,q = K p + q ( I p /I p − ) . This spectral sequence converges strongly to K ∗ A ; i.e., given s ∈ Z , the groups E ∞ p,q along the diagonal s = p + q pose an extension problem to reconstruct K s A . In our setting, we do not have A = S p ∈ N I p but merely A = P ∞ j =0 I j . A plausibleadaption to our setting might be: • Compute K ∗ (cid:0) P nj =0 I j (cid:1) from I , I , I , . . . , I n using our spectral sequence thattakes finite sums of ideals. • For each n , compute K ∗ (cid:0) P nj =0 I j / P n − j =0 I j (cid:1) with the six-term exact sequence. • Feed these results at once into the spectral sequence from Theorem 3.1.1 tocompute K ∗ A .The downside is the multilayered computation: We build many spectral sequences,solve an extension problem for every single one, and then fit the results into yetanother spectral sequence. This is unlikely to work except in trivial cases where K ∗ (cid:0)P ∞ j =0 I j (cid:1) would have been straightforward to compute by other means, or whenthe K-theory would already equal K ∗ (cid:0) P nj =0 I j (cid:1) for an n ∈ N . Instead, we wouldlike a more robust approach featuring a single spectral sequence { E rp,q , d r } r,p,q withterms E p,q that are easier to describe and compute.62 .2 Linking two chains of ideals Notation 6.2.1. Fix a C*-algebra A = P ∞ j =0 I j where the I j are closed two-sidedideals of A . We denote by E ( n ) rp,q the ( p, q ) -th module in the r -th page of the spectralsequence for the sum of the first n ideals I + I + · · · + I n − : Each of these spectralsequences { E ( n ) rp,q , d ( n ) r } r,p,q is constructed according to our main Theorem 4.6.1about finite sums of C*-ideals.We will construct a morphism of spectral sequences from E ( n ) to E ( n + 1) . Thisrequires several technical propositions. Each spectral sequence arises from a chain ofideals Q ⊆ Q ⊆ Q ⊆ · · · that strongly depends on a fixed number of ideals I , I , . . . , I j , . . . chosen in the beginning of the construction – see Section 4.3. To relate E ( n ) with E ( n + 1) , we first construct morphisms between the two chains of idealsthat lead to these two spectral sequences.Along this way, we diligently track naturality with respect to ∗ -homomorphismsthat preserve ideal decompositions (Definition 4.1.1 and Remark 4.1.2). Lemma 6.2.2. Let I , . . . , I n be C*-ideals. Fix k ∈ { , , . . . , n } . Define I ′ k = 0 ,moreover I ′ j = I j for j = k . Fix J ⊆ { , , . . . , n } − { k } . Then B ( I , . . . , I n ) J = B ( I ′ , . . . , I ′ n ) J = B ( I ′ , . . . , I ′ n ) J ∪{ k } . Proof. The first equality holds because functions in B ( I , . . . , I n ) J and B ( I ′ , . . . , I ′ n ) J are never allowed to take nonzero values in I k or I ′ k , the only ideals that differ in theconstruction.The second equality holds because B J and B J ∪{ k } differ at most by conditionsenforced on the subspace ∆ nk ⊆ ∆ n , but on ∆ nk , functions map to I ′ k = 0 anyway. Proposition 6.2.3. Let I , . . . , I n be C*-ideals that sum to A . Construct the cakealgebra B ( I , . . . , I n ) as in Definition 4.3.1. Now include the zero algebra I n +1 as an extra ideal, leading to a different cake algebra B ( I , . . . , I n , : B ( I , . . . , I n ) ⊆ C (∆ n , A ) ,B ( I , . . . , I n , ⊆ C (∆ n +1 , A ) . Let J ⊆ { , , . . . , n } be an index set. There is a suspension isomorphism S (cid:0) B ( I , . . . , I n ) J (cid:1) ∼ = B ( I , . . . , I n , J ∪{ n +1 } . Figure 6.2.4 gives the geometric idea. Functions on the line vanish on its two endpoints. Functions on the grey four-sided shape vanish on the continuously drawn63 Infinite sums of ideals boundary lines but not on the dashed line.We claim that the suspension algebra of the functions on the line is isomorphic tothe function algebra on the grey shape. (1 , 0) (0 , , ) I I (1 , , 0) (0 , , , , I I I = 0 Figure 6.2.4: Geometric idea of Proposition 6.2.3 Proof of Proposition 6.2.3. By Lemma 6.2.2, we can reduce the case of arbitrary J to J = { , , . . . , n } by replacing I j with for all j / ∈ J , completing the constructionin this proof, then re-inserting the original ideals I j .Define X = n [ j =0 ∆ n +1 j , Y = X ∩ ∂ ∆ n +1 , Z = X ∩ ∆ n +1 n +1 . The set X corresponds to the grey area in the example figure above, Y to the greyarea’s ceiling, Z to its floor. Functions in B ( I , . . . , I n , vanish on Y and Z , butthey may assume nontrivial values in I , . . . , I n on the interior X − Y − Z .The points x in X have n + 2 barycentric coordinates ( x , . . . x n +1 ) . For functionsin B ( I , . . . , I n , , the relative values of the first n + 1 of these coordinates selectideals among I , . . . , I n as the function’s range. The final coordinate x n +2 does notaffect that choice, but x n +2 is not well-suited to see that B ( I , . . . , I n , is isomorphicto a suspension algebra. Instead, we show this with a reparametrization ϕ of X . Set y min = min { y j : y j is a barycentric coordinate of y = ( y , . . . , y n ) } ,ϕ : ∆ n × (cid:20) n + 2 , (cid:21) → ∆ n +1 ,ϕ ( y , . . . , y n , t ) = ( y − ty min , . . . , y n − ty min , ( n + 1) ty min ) . This function ϕ maps continuously to X : ϕ distributes ( n + 1) portions of ty min fromthe first ( n + 1) coordinates to the last coordinate. Thus the sum of all coordinatesremains 1. Furthermore, ϕ cannot map to the interior of ∆ n +1 n +1 because the lastcoordinate cannot be the uniquely smallest: With t ≥ n +2 by definition of ϕ , we64 .2 Linking two chains of ideals have ( n + 1) ty min ≥ y min − ty min . Equality holds exactly for t = n +2 .The map ϕ is surjective onto X : Construct a preimage of ( y , . . . , y n , y n +1 ) bydistributing y n +1 n +1 onto each of the first n +1 coordinates, then choose t . Furthermore, ϕ is injective on the interior of ∆ n × [ n +2 , .On the simplex boundaries, we need not check injectivity. Because B ( I , . . . , I n ) or B ( I , . . . , I n , must vanish on the simplex boundaries, it suffices to verify thatall points of ∂ ∆ n × [ n +2 , map to ∂ ∆ n +1 . This holds because y min = 0 , thus y min − ty min = 0 regardless of the value t ∈ [ n +2 , .On the interior of X , that is, on the interior of ϕ (cid:0) (∆ n ) ◦ × ] n +2 , (cid:1) , functions in B ( I , . . . , I n , are subject to the restrictions from the relations of the first n + 1 barycentric coordinates, but not to any restriction from the final zero ideal or from thecoordinate t ∈ [ n +2 , . On ϕ (cid:0) (∆ n ) ◦ × (cid:8) n +2 , (cid:9)(cid:1) , the functions must be zero. Thusvia ϕ , we see that B ( I , . . . , I n , is isomorphic to the suspension of B ( I , . . . , I n ) .It suffices to parametrize X instead of the entire simplex ∆ n +1 because everyconsidered C*-function living on X has only one possible extension – by zero – to afunction in B ( I , . . . , I n , . Remark 6.2.5. The isomorphism from Proposition 6.2.3 is natural with respectto finite ideal decompositions: A collection of ∗ -homomorphisms i j : I j → I ′ j for j ∈ { , , . . . , n } on the input C*-ideals, together with the constructed isomorphismson either side, leads to a commutative diagram S (cid:0) B ( I , . . . , I n ) J (cid:1) B ( I , . . . , I n , J ∪{ n +1 } S (cid:0) B ( I ′ , . . . , I ′ n ) J (cid:1) B ( I ′ , . . . , I ′ n , J ∪{ n +1 } . ∼ = ∼ = S (cid:0) B ( i , . . . , i n ) (cid:1) B ( i , . . . , i n , Proposition 6.2.6. Let A = I + I + · · · + I n be a sum of C*-ideals. The functionalgebras B J = B ( I , . . . , I n ) J for J ⊆ { , , . . . , n } give rise to cake sums Q p = P | J |≤ p +1 B J by Definition 4.3.6.Let I n +1 = 0 be an extra zero ideal. The function algebras e B J = B ( I , . . . , I n , J for this larger set of ideals and J ⊆ { , , . . . , n, n +1 } give rise to e Q p = P | J |≤ p +1 e B J .Then e Q p ∼ = SQ p for p ∈ Z and e Q n +1 = e Q n . In other words, the chains of ideal Infinite sums of ideals inclusions Q p → Q p +1 and e Q p → e Q p +1 fit into this commutative diagram: SQ SQ · · · SQ n e Q e Q · · · e Q n e Q n +1 . ∼ = ∼ = ∼ = ∼ = Proof. Our index sets will sometimes contain indices in { , , . . . , n } , sometimes in { , , . . . , n, n + 1 } . For clarity, given p ∈ Z , we define these collections of index sets: A ( p ) = { J ⊆ { , , . . . , n } : | J | ≤ p } , B ( p ) = { J ⊆ { , , . . . , n + 1 } : | J | ≤ p } . For p < , A ( p ) and B ( p ) are empty and Q p is the zero algebra.The following argument will work without modification for all p ∈ Z . For a given p ∈ Z , we may write the collection B ( p + 1) as the disjoint union of A ( p + 1) and { J ∪ { n + 1 } : J ∈ A ( p ) } . Applying this to the definition of e Q p , we obtain e Q p = X B ( p +1) e B J = X A ( p +1) e B J + X A ( p ) e B J ∪{ n +1 } . By Lemma 6.2.2, the index n + 1 can be dropped from B J ∪{ n +1 } with no changebecause I n +1 = 0 : e Q p = X A ( p +1) e B J + X A ( p ) e B J . By definition of A ( p ) as a collection of all index sets up to cardinality p , we have A ( p ) ⊆ A ( p + 1) . The sum simplifies to e Q p = X A ( p +1) e B J . None of the sets in A ( p + 1) contain the index n + 1 . This allows us to rewrite e B J as SB J via a natural isomorphism according to Proposition 6.2.3. Taking sumscommutes with suspensions: e Q p = X A ( p +1) e B J ∼ = X A ( p +1) SB J = SQ p . This shows the main result. The extra result e Q n +1 = e Q n follows from e Q n ∼ = SQ n = SQ n +1 ∼ = e Q n +1 .66 .3 Compatibility of suspensions Remark 6.2.7. The constructed isomorphism is natural: It is a composition ofequalities and the natural isomorphism from Proposition 6.2.3. Lemma 6.3.1. Let I be a C*-ideal in A . Then S ( A/I ) ∼ = SA/SI . Explicitly, thereis an isomorphism Φ : SA/SI → S ( A/I ) with Φ( f + SI )( t ) = f ( t ) + I for f : [0 , → A with f (0) = f (1) = 0 . Later, we show naturality of Φ in a separate lemma; first, we construct this iso-morphism. Proof of Lemma 6.3.1. The function Φ is well-defined, additive, and multiplicativebecause I is an ideal; it preserves the involution because I is a C*-ideal. If I containsthe range of Φ( f ) , then f was already in SI , therefore Φ is injective.For surjectivity, we will use A ′ ⊗ C ]0 , ∼ = SA ′ for arbitrary C*-algebras A ′ andthe nuclearity of C ]0 , as proven, e.g., in [WO93]. As a result, the top two rows ofthis commutative diagram become exact: I ⊗ C ]0 , A ⊗ C ]0 , A/I ⊗ C ]0 , 1[ 00 SI SA S ( A/I ) 00 SI SA SA/SI . ∼ = ∼ = ∼ =Φ The square in the bottom right commutes by our explicit construction of Φ . Sincethe bottom row is the standard quotient exact sequence for the inclusion SI → SA ,the ∗ -homomorphism Φ is an isomorphism by the five lemma. Lemma 6.3.2. The isomorphism Φ : SA/SI → S ( A/I ) constructed in Lemma 6.3.1is natural with respect to ∗ -homomorphisms h : A → A ′ that map I into I ′ , where I ′ ⊆ A ′ is a given C*-ideal.Proof. Construct Φ ′ : SA ′ /SI ′ → S ( A ′ /I ′ ) for the C*-ideal I ′ ⊆ A ′ according to67 Infinite sums of ideals Lemma 6.3.1. Then h : A → A ′ gives rise to a diagram: SA/SI S ( A/I ) SA ′ /SI ′ S ( A ′ /I ′ ) . Φ Sh/S ( h ↾ I ) Φ ′ S (cid:0) h/ ( h ↾ I ) (cid:1) This diagram commutes: Consider ( f + SI ) ∈ SA/SI for a given f : [0 , → A with f (0) = f (1) = 0 . The upper right path through the diagram maps this to t f ( t )+ I and then to the class containing t ( h ◦ f )( t ) + h ( I ) , which is t ( h ◦ f )( t ) + I ′ in S ( A ′ /I ′ ) .The lower left path maps f + SI first to h ◦ f + h ( SI )+ SI ′ , which is h ◦ f + SI ′ since h ( I ) ⊆ I ′ , and then onwards also to t ( h ◦ f )( t ) + I ′ according to the constructionof Φ ′ . Proposition 6.3.3. Let → A → B → C → be a short exact sequence of C*-algebras. Then for each s ∈ Z , the following diagram is commutative: K s +1 ( C ) K s ( A ) K s ( SC ) K s − ( SA ) . ∂ s +1 ( C, A ) σ s +1 ( C ) ∼ = ∂ s ( SC, SA ) σ s ( A ) ∼ = Here σ denotes suspension isomorphisms and ∂ denotes the boundary maps in thelong exact K-theory sequences that arise from the original short exact sequence andfrom its suspension → SA → SB → SC → . Even though ∂ is natural with respect to ∗ -homomorphisms, the claim does notfollow immediately from functoriality because the suspension isomorphisms σ ariseonly in K-theory, not on the level of C*-algebras. Proof of Proposition 6.3.3. The Bott isomorphism β ( C ) : K ( C ) K ( SC ) K ( C ) , ∼ = σ ( C ) − which is a composition of two isomorphisms, and the exponential map δ : K ( C ) K ( C ) K ( A ) β ( C ) ∂ ( C, A ) .4 Linking spectral sequences are constructed in [RLL00, Chapters 11–12] explicitly to make the two outermostpaths σ ( A ) ◦ δ and ∂ ( SC, SA ) ◦ σ ( C ) ◦ β ( C ) in the following diagram commute: K ( C ) K ( C ) K ( A ) K ( SC ) K ( SA ) . β ( C ) ∼ = δ ∂ ( C, A ) σ ( C ) ∼ = ∂ ( SC, SA ) σ ( A ) ∼ = Because β ( C ) is an isomorphism, commutativity of the square follows from commu-tativity of the two outermost paths. This commuting square proves the claim for n = 1 , the lowest n of interest.For higher s , the claim follows from this base case by composing the entire diagramwith an ( s − -fold suspension isomorphism. The higher boundary maps in K-theoryare defined precisely to agree with such a suspension. The claim for lower s followsfrom composing with Bott isomorphisms. Having linked the chain of ideal inclusions Q p → Q p +1 with e Q p → e Q p +1 , we cannow link the spectral sequences that arise from the first n ideals along increasingcardinalities n ∈ N . Proposition 6.4.1. Let A = I + I + · · · + I n − be a sum of C*-ideals. Constructthe spectral sequence { E r ∗ , ∗ , d r } r for this n -fold sum as in Theorem 4.6.1.Alternatively, add an extra ideal I n = 0 and construct a second spectral sequence { e E r ∗ , ∗ , e d r } r for the ( n + 1) -fold sum I + I + · · · + I n − + 0 .Then there are isomorphisms E rp,q ∼ = e E rp,q − for all r ≥ and p , q ∈ Z with thefollowing properties: • They are natural with respect to ∗ -homomorphisms that preserve the ideal de-compositions: ∗ -homomorphisms h : A → A ′ for a C*-algebra A ′ = I ′ + I ′ + · · · + I ′ n − + 0 such that, for all j < n , we have h ( I j ) ⊆ I ′ j . • They commute with the differentials d r and e d r .Proof. Recall the first page of the spectral sequence from the main statement of69 Infinite sums of ideals Theorem 4.6.1, adapted to n instead of n + 1 ideals: E p,q ∼ = M | J | = p +1 K q (cid:16) \ j ∈ J I j (cid:17) for ≤ p < n , for p < or p ≥ n .Suspend Q p /Q p − and compensate for this suspension by a degree shift in K-theoryto maintain isomorphy. Apply our isomorphism results from above: E p,q ∼ = K q ( Q p /Q p − ) ∼ = K q − (cid:0) S ( Q p /Q p − ) (cid:1) ∼ = K q − ( SQ p /SQ p − ) ∼ = K q − ( e Q p / e Q p − ) ∼ = e E p,q − . All isomorphisms come from our earlier propositions and are therefore natural. Wedo not have to show anything new here for naturality with respect to h : A → A ′ ofthe isomorphism between the two spectral sequences.Commutation of isomorphisms and differentials follows from the definition of thedifferentials according to Theorem 3.2.5 and Definition 3.3.3: The differentials arecompositions of maps induced by ideal inclusions, maps induced by natural quotientprojections, and boundary maps from long exact sequences in K-theory that arisefrom ideal inclusions. All these maps commute individually with all isomorphismsapplied above; in particular, Proposition 6.3.3 guarantees that the differentials be-have well with suspensions. Definition 6.4.2 (Link between spectral sequences for increasingly-many ideals) . Let A ( n + 1) = I + I + · · · + I n be a sum of n + 1 C*-ideals and denote by A ( n ) ⊆ A ( n + 1) the sum of the first n ideals, I + I + · · · + I n − .For all pages r ≥ and p , q ∈ Z , let { g rp,q } : { E ( n ) rp,q , d ( n ) r } → { e E rp,q − , e d r } be theisomorphism constructed in Proposition 6.4.1 from the spectral sequence for A ( n ) into the spectral sequence for I + I + · · · + I n − + 0 . The ideal inclusions I = I , I = I , . . . , I n − = I n − , → I n induce a morphism { i rp,q } that preserves all degrees, { i rp,q } : { e E rp,q } → { E ( n + 1) rp,q } , of spectral sequences between the intermediate { e E rp,q , e d r } r,p,q and the desired spectral70 .4 Linking spectral sequences sequence { E ( n + 1) rp,q , d ( n + 1) } r,p,q for A ( n + 1) .The link between the spectral sequences for A ( n ) and A ( n + 1) is ℓ ( n ) = { ℓ ( n ) rp,q } = { i rp,q − ◦ g rp,q } : { E ( n ) rp,q } → { E ( n + 1) rp,q − } . Lemma 6.4.3 (Faithfulness of the link ℓ ( n ) ) . Given n ∈ N and p, q ∈ Z , considera set of indices J ⊆ { , , . . . , n − } with | J | = p + 1 and the direct summand V = K q (cid:0) T j ∈ J I j (cid:1) for J in the group E ( n ) p,q .Then ℓ ( n ) p,q maps V isomorphically onto V ′ = K q − (cid:0) T j ∈ J I j (cid:1) in E ( n +1) p,q − Noother summand of E ( n ) p,q besides V maps onto V ′ nontrivially. No other summandin E ( n + 1) p,q − besides V ′ is a nontrivial image of V .Proof. We examine the components of ℓ ( n ) p,q in the notation of Definition 6.4.2.The first component of ℓ ( n ) p,q is g p,q : E ( n ) p,q → e E p,q − . As a suspension isomor-phism, it agrees with quotients, direct sums, and other suspensions. Because it isnatural with respect to ∗ -homomorphisms that preserve ideal decompositions, g p,q cannot permute V with other direct summands for different choices of ideals than J among the first n ideals. Again by naturality, it cannot map to K q − (cid:0) I n ∩ T j ∈ J I j (cid:1) within e E p,q − either: This K-theory group must always vanish regardless of A be-cause, by definition, { e E p,q } r,p,q is the spectral sequence for I n = 0 .On all ideals I j with j = n , the second component i p,q − : e E p,q − → E ( n + 1) p,q − is induced by the identity. It maps g p,q ( V ) necessarily to V ′ because n / ∈ J and hitsno other summands besides V ′ .Thus ℓ ( n ) p,q = i p,q − ◦ g p,q has the claimed isomorphy property and hits no othersummands besides V ′ . It follows that no other summand in E ( n + 1) p,q − besides V ′ may be a nontrivial image of V . Proposition 6.4.4. The link ℓ ( n ) between the spectral sequences { E ( n ) rp,q , d ( n ) r } r,p,q and { E ( n + 1) rp,q , d ( n + 1) r } r,p,q is natural with respect to ∗ -homomorphisms h : A → A ′ for another C*-algebra A ′ = I ′ + I ′ + · · · + I ′ n ′ that is a sum of n ′ + 1 ≥ n + 1 C*-ideals, as long as h ( I j ) ⊆ I ′ j for all j ≤ n .Proof. We have constructed ℓ ( n ) as a composition of two maps that already satisfythis desired naturality with respect to h as shown in the various earlier lemmas. 71 Infinite sums of ideals Theorem 6.5.1 (Spectral sequence for countable sums) . Let A be the direct limit C*-algebra I + I + · · · + I j + · · · of sums of countably many C*-ideals I j ⊆ A . Thereis a spectral sequence { E rp,q , d r } r,p,q with E p,q ∼ = M | J | = p +1 K q (cid:16) \ j ∈ J I j (cid:17) for p ≥ , for p < , (6.5.1.1) where J ranges over all nonempty finite subsets of indices. In general, this is ahalf-page spectral sequence, any term E p,q with p ≥ may be nonzero.This spectral sequence converges strongly to K ∗ A . It is functorial with respect to ∗ -homomorphisms that preserve countable ideal decompositions. We prove Theorem 6.5.1 in two steps: First, in Proposition 6.5.4, we prove exis-tence, that the spectral sequence is well-defined, that it is functorial, and that its E ∗ , ∗ -term matches the description 6.5.1.1. Later, in Proposition 6.6.12, we prove thestrong convergence.Our strategy is to take the direct limit along a directed system of links ℓ ( n ) for n → ∞ , but these links ℓ ( n ) do not connect the spectral sequences E ( n ) and E ( n +1) perfectly. There is an index shift from E ( n ) rp,q to E ( n + 1) rp,q − . As shown before,the shifted index affects the degree of the K-theory; it has no other effect on equation6.5.1.1.Complex K-theory admits Bott isomorphisms β , · · · β ∼ = K s − (cid:16) \ j ∈ J I j (cid:17) β ∼ = K s (cid:16) \ j ∈ J I j (cid:17) β ∼ = K s +2 (cid:16) \ j ∈ J I j (cid:17) β ∼ = · · · ; their naturality allows us to work around the index shift. Remark 6.5.2. The Bott isomorphisms β are natural with respect to ∗ -homomor-phisms and commute with the suspension isomorphisms σ for any C*-algebra A : K s ( A ) K s +2 ( A ) K s − ( SA ) K s +1 ( SA ) . ββσ s σ s +2 Definition 6.5.3 (Degree-amending link λ ( n ) ) . Compose two links with the Bottisomorphism to create a morphism λ ( n ) of bidegree (0 , , called the degree-amending .5 Main theorem for countably many ideals link , between two spectral sequences: λ ( n ) = β ◦ ℓ ( n + 1) ◦ ℓ ( n ) : { E ( n ) rp,q } r,p,q → { E ( n + 2) rp,q } r,p,q . Proposition 6.5.4. Let A = I + I + · · · + I j + · · · be a sum of C*-ideals. Thespectral sequence postulated in Theorem 6.5.1 with E p,q ∼ = M | J | = p +1 K q (cid:16) \ j ∈ J I j (cid:17) for p ≥ , for p < ,exists and is functorial with respect to ∗ -homomorphisms that preserve countable idealdecompositions.Proof. Take the direct limit along λ (2 n ) for n → ∞ . This yields again a spectralsequence. Functoriality of this spectral sequence with respect to ideal inclusions fol-lows from all earlier constructions that, as we have remarked repeatedly, are naturalwith respect to ∗ -homomorphisms that preserve countable ideal decompositions.It remains to show that the choice of offset, i.e., n or n + 1 , and the position of β among the links ℓ have no effect on the direct limit of degree-amending links; i.e.,the following system λ ′ (2 n ) of morphisms produces the same direct limit: λ ′ ( n ) = ℓ ( n + 1) ◦ β ◦ ℓ ( n ) : { E ( n ) rp,q } → { E ( n + 2) rp,q } . The position of β is irrelevant because the Bott isomorphism is natural with respectto both ∗ -homomorphisms and suspension isomorphisms. In Definition 6.4.2, we havedefined ℓ ( n ) as a composition of several suspension isomorphisms and several directmorphisms between C*-algebras. Thus for n → ∞ , the systems λ (2 n ) and λ ′ (2 n ) produce naturally isomorphic direct limits, without even requiring index shifts.To compare the systems λ (2 n ) and λ (2 n + 1) for n → ∞ , unroll the compositions λ into their definitions, and take the direct limit – necessarily yielding the same limitas before – across the unrolled system: colim n →∞ λ (2 n + 1) = colim (cid:0) · · · → • ℓ ( n +1) −→ • ℓ ( n +2) −→ • β −→ • ℓ ( n +3) −→ • → · · · (cid:1) = colim (cid:0) · · · → • ℓ ( n +1) −→ • β −→ • ℓ ( n +2) −→ • ℓ ( n +3) −→ • → · · · (cid:1) = colim (cid:0) · · · → • ℓ ( n +2) −→ • ℓ ( n +3) −→ • β −→ • → · · · (cid:1) = colim n →∞ λ (2 n ) , because the morphisms ℓ commute with β and because removing the first element of73 Infinite sums of ideals a directed system will not change the limit.Thus our limit spectral sequence is well-defined as colim n λ (2 n ) . Finally, because E ( n ) rp,q ∼ = M | J | = p +1max J Φ : G/H ∼ = colim n →∞ ( G n /H n ) that is natural with respect to morphisms between systems ( G n ) n ∈ N and ( e G n ) n ∈ N thatpreserve their inclusions and their respective systems of subgroups ( H n ⊆ G n ) n ∈ N and ( e H n ⊆ e G n ) n ∈ N .Proof. For each n ∈ N , consider the following diagram D n . All of the horizontalarrows in D n are natural projections and the square commutes due to naturality of74 .6 Convergence the projections: G n G n /H n G G/H n G/H. α n γ n /H n γ n δ n π n To construct a direct limit of the sequence of entire diagrams ( D n ) n ∈ N , link twodiagrams D n and D n +1 by the following system of morphisms: • i n : G n → G n +1 ; • the composition ( τ n +1 ◦ i n /H n ) : G n /H n → G n +1 /H n → G n +1 /H n +1 wherethe natural projection τ n +1 : G n +1 /H n → G n +1 /H n +1 is well-defined because H n ⊆ H n +1 ; • the natural projection G/H n → G/H n +1 from dividing by H n +1 /H n ; and • the identities on G and G/H , respectively.All squares that arise from linking two diagrams D n and D n +1 via these morphismscommute due to naturality of projections; e.g., α n +1 ◦ i n = τ n +1 ◦ i n /H n ◦ α n . Takethe direct limit along ( D n ) n ∈ N ; the limit object is again a commutative diagram: G colim n ( G n /H n ) G colim n ( G/H n ) G/H. colim n α n colim n ( γ n /H n )colim n δ n colim n π n (6.6.1.1)Furthermore, for each n ∈ N , there is a short exact sequence → H n → G n α n −→ G n /H n → . Taking direct limits of abelian groups is an exact functor, resulting in a short exactsequence of direct limits which fits into the following diagram as the top row: H G colim n ( G n /H n ) 00 H G G/H . colim n α n colim n ( π n ◦ δ n ) colim n ( π n ◦ γ n /H n ) Infinite sums of ideals The right square commutes because its morphisms already commuted in the earlierdiagram 6.6.1.1 of direct limits.Finally, because both rows are short exact sequences, the five lemma guaranteesthat colim n ( π n ◦ γ n /H n ) is the desired isomorphism Φ . Naturality of Φ follows fromthe constructions in this proof: Both taking direct limits and taking quotients isnatural. Notation 6.6.2. For a sum A = I + I + · · · + I j + · · · of C*-ideals, write A ( n ) = I + I + · · · + I n − for the sum of the first n ideals and let, for s ∈ Z , a ( n ) s : K s A ( n ) → K s A ( n + 1) be the map induced in K-theory by the inclusion of C*-algebras A ( n ) ⊆ A ( n + 1) .Given A ( n ) , define the cake algebras B ( n ) J = B ( I , I , . . . , I n − ) J for index sets J ⊆ { , , . . . , n − } as in Definition 4.3.1 and the sums of cake algebras as inDefinition 4.3.6, Q ( n ) p = X | J |≤ p +1 B ( n ) J for p ∈ Z and J ⊆ { , , . . . , n − } . Remark 6.6.3. By Lemma 4.3.9, we have a chain of ideals Q ( n ) p ⊆ Q ( n ) p +1 acrossall p ∈ Z . The chain stabilizes with Q ( n ) p = 0 for p < and Q ( n ) p = B ( n ) { , ,...,n − } for p ≥ n − .Furthermore, P n − p =0 Q ( n ) p = B ( n ) { , ,...,n − } ∼ = S n − A ( n ) by Theorem 4.4.11. Notation 6.6.4 ( i ( n, p ) s ) . We recall the filtrations on the spectral sequences forfinitely many ideals: The convergence target of { E ( n ) rp,q , d ( n ) r } is K ∗ A ( n ) , filteredby F p K s A ( n ) ∼ = im i : K s − n +1 Q ( n ) p → K s − n +1 S n − A ( n ) (6.6.4.1) ∼ = im i ( n, p ) s : K s − n +1 Q ( n ) p → K s A ( n ) (6.6.4.2)as in Definition 3.4.1 for p , s ∈ Z . In the construction of the spectral sequence inSection 3, the symbol i may also stand for various other maps in K-theory.Here in Section 6.6, we will write i ( n, p ) s for the maps in 6.6.4.2 that define afiltration of K s A ( n ) , not of K s − n +1 S n − A ( n ) , for a given p . Normally, we wouldindex maps in K-theory with the degree of the domain, but for i ( n, p ) s , it will beeasiest to track the K-theory degree s of the desired convergence target K s A ( n ) .76 .6 Convergence We reserve i (without annotation in parentheses) to discuss internals of Section 3. Remark 6.6.5. Because Q ( n ) p is an ideal in S n − A ( n ) , not in A ( n ) , we have shiftedthe K-theoretic degree from K s Q ( n ) p to K s − n +1 Q ( n ) p to compensate. This shift issimilar to the shift in the proof of Theorem 4.6.1 about the spectral sequence forsums of finitely many C*-ideals. Definition 6.6.6 (Link between filtrations) . Fix n ∈ N and p ∈ Z . Besides Q ( n ) p ,recall the C*-algebra e Q ( n ) p for the n + 1 ideals I , I , . . . , I n − , as in Proposition6.2.6. For all s ∈ Z , define the link between the filtrations of K ∗ A ( n ) and K ∗ A ( n + 1) , ψ ( n, p ) s : K s − n +1 Q ( n ) p → K s − n Q ( n + 1) p , as the composition K s − n +1 Q ( n ) p K s − n SQ ( n ) p K s − n e Q ( n ) p , K s − n Q ( n + 1) p ; ψ ( n, p ) s ∼ = ∼ = a ( n ) + 0 here the left arrow is the suspension isomorphism, the middle arrow is the naturalisomorphism constructed in Proposition 6.2.6, and the right arrow is induced by theinclusion of the ( n + 1) -fold ideal decomposition A ( n ) + 0 into A ( n + 1) . Lemma 6.6.7. For all n ∈ N and p , s ∈ Z , the following diagram commutes: K s − n +1 Q ( n ) p K s − n Q ( n + 1) p K s A ( n ) K s A ( n + 1) . ψ ( n, p ) s i ( n, p ) s i ( n + 1 , p ) s a ( n ) s (6.6.7.1) Proof. Replace ψ ( n, p ) s by its definition as a composition of three maps to get thetop row of the following diagram where i ′ denotes the map induced by the idealinclusion Q ( n ) p ⊆ S n − A ( n ) with appropriate K-theoretic degree shifts: K s − n +1 Q ( n ) p K s − n SQ ( n ) p K s − n e Q ( n ) p , K s − n Q ( n + 1) p K s A ( n ) K s A ( n ) K s A ( n + 1) . ∼ = ∼ = a ( n ) s + 0 i ( n, p ) s i ′ i ( n + 1 , p ) s a ( n ) s Infinite sums of ideals The left square commutes by definition of i ′ because the top arrow is the suspensionisomorphism.The right side commutes because, by Remark 6.2.7, the isomorphism at the topis natural with respect to ∗ -homomorphisms that preserve ( n + 1) -fold ideal decom-positions: The two ∗ -homomorphisms here are the ideal inclusion that induces a ( n ) and the ideal inclusion Q ( n + 1) p → S n A ( n + 1) that induces i ′ and i ( n + 1 , p ) s andrestricts to e Q ( n ) p ; these two ideal inclusions commute with each other already onthe level of C*-algebras. Definition 6.6.8 (Filtration of K ∗ A ) . Fix p , s ∈ Z and compose diagram 6.6.7.1with itself across all n ≥ : K s Q (1) p K s − Q (2) p K s − Q (3) p · · · K s A (1) K s A (2) K s A (3) · · · . i (1 , p ) s i (2 , p ) s i (3 , p ) s a (1) s a (2) s a (3) s ψ (1 , p ) s ψ (2 , p ) s ψ (3 , p ) s The direct limit of K s A ( n ) for n → ∞ along a ( n ) is K s A by continuity of K-theory.Take the direct limit of K s − n +1 Q ( n ) p for n → ∞ along ψ ( n, p ) s and consider, byfunctoriality of the direct limit, the direct limit of the vertical arrows i ( n, p ) s , colim n →∞ i ( n, p ) s : (cid:16) colim n →∞ K s − n +1 Q ( n ) p (cid:17) → K s A. With this map, define the filtration of K ∗ A , { F p K ∗ A } p ∈ Z , by F p K s A = im (cid:16) colim n →∞ i ( n, p ) s (cid:17) ⊆ K s A. Lemma 6.6.9. The filtration { F p K s A } p ∈ Z of K s A from Definition 6.6.8 is an in-creasing filtration.Proof. For all p ≤ p ′ , we have Q ( n ) p ⊆ Q ( n ) p ′ . Using i as in the notation of Section3, i (cid:0) K s − n +1 Q ( n ) p (cid:1) ⊆ K s − n +1 Q ( n ) p ′ . Because the i ( n, p ) s arise from the directedsystem of morphisms i from Remark 3.4.3, K s Q ( n ) p i −→ K s Q ( n ) p +1 i −→ · · · i −→ K s Q ( n ) n = K s Q ( n ) n +1 = · · · , merely via isomorphisms that implement a K-theoretic degree shift, we may pullback our direct limit construction for colim n i ( n, p ) s via these isomorphisms to the78 .6 Convergence system of i . Passing to the direct limit morphism along n → ∞ gives (cid:16) colim n →∞ i (cid:17)(cid:16) colim n →∞ K s − n +1 Q ( n ) p (cid:17) ⊆ colim n →∞ K s − n +1 Q ( n ) p ′ . Thus the filtration is increasing: F p K s A = (cid:16) colim n →∞ i ( n, p ) s (cid:17)(cid:16) colim n →∞ K s − n +1 Q ( n ) p (cid:17) ⊆ (cid:16) colim n →∞ i ( n, p ′ ) s (cid:17)(cid:16) colim n →∞ K s − n +1 Q ( n ) p ′ (cid:17) = F p ′ K s A. Lemma 6.6.10. The p -indexed filtration of K ∗ A from Definition 6.6.8 is Hausdorff,exhaustive, and complete according to Definition 2.7.2.Proof. For p < , all terms K s − n +1 Q ( n ) p vanish regardless of s and n , thus theirdirect limit also vanishes. This renders the filtration Hausdorff and complete.For any class [ x ] ∈ K s A , there is n ∈ N such that K s A ( n ) contains the preimageof [ x ] . We can choose p = n to make i ( n, n ) s : K s − n +1 Q ( n ) n → K s A ( n ) surjective,thereby including that preimage of [ x ] in the range of i ( n, n ) s . Commutativity of thediagram in Definition 6.6.8 shows that [ x ] is in the range of colim n i ( n, p ) s . Thus thefiltration is exhaustive. Remark 6.6.11. The biggest problem in passing to direct limits for n → ∞ werethe iterated suspensions S n − , but these have already been handled by the degree-amending links λ (2 n ) from Definition 6.5.3 via degree shifts and Bott isomorphisms.Thus throughout Section 6.6, we may rest assured that any direct limits of modulesor differentials on E rp,q for r ≥ or r = ∞ remain well-defined for the spectralsequence { E rp,q , d r } r,p,q for countably many C*-ideals. Proposition 6.6.12 (Convergence of the limit spectral sequence) . Let A be a C*-algebra with A = I + I + · · · + I j + · · · , a sum of C*-ideals. The limit spectralsequence constructed in Theorem 6.5.1, defined as the direct limit along the system λ (2 n ) with E rp,q = colim n →∞ (cid:16) β ◦ ℓ (2 n + 1) ◦ ℓ (2 n ) : { E (2 n ) rp,q } → { E (2 n + 2) rp,q } (cid:17) , converges strongly to K ∗ A .Proof. For all n ∈ N , the spectral sequence { E ( n ) rp,q , d ( n ) r } converges strongly: E ( n ) ∞ p,q ∼ = F p K p + q A ( n ) /F p − K p + q A ( n ) . Infinite sums of ideals The terms E ∞ p,q are the direct limits of these E ( n ) ∞ p,q along the morphisms induced on E (2 n ) ∞ p,q by the λ (2 n ) because these λ (2 n ) had all desirable properties – naturalitywith respect to ∗ -homomorphisms that preserve countable ideal decompositions andcommutativity with the differentials. Furthermore, K-theory is continuous. Alongthe system of λ (2 n ) , colim n →∞ (cid:0) F p K p + q A (2 n ) /F p − K p + q A (2 n ) (cid:1) ∼ = colim n →∞ (cid:0) E (2 n ) ∞ p,q , λ (2 n ) (cid:1) ∼ = E ∞ p,q . The filtration { F p K ∗ A } p ∈ Z of K ∗ A is Hausdorff, exhaustive, and complete by Lemma6.6.9 and E ∞ p,q is isomorphic to F p K p + q A/F p − K p + q A by Lemma 6.6.1. Thereforethe limit spectral sequence { E rp,q , d r } r,p,q converges strongly to K ∗ A .This concludes the proof of Theorem 6.5.1 about the existence, well-definedness,functoriality, and strong convergence of the limit spectral sequence for countablymany C*-ideals. In most geometrical applications, if a C*-algebra may be written as a sum of easilycomputable ideals, this sum will be a countable sum. We have described a spectralsequence for this case. Still, it seems reasonable to generalize the cardinality of thealgebra decomposition. Theorem 6.7.1 (Spectral sequence for arbitrary sums) . Let α be an arbitrary indexset: finite, countable, or uncountable. Let A = P β ∈ α I β be the norm closure of asum of | α | -many C*-ideals I β ⊆ A . There is a spectral sequence { E rp,q , d r } r,p,q with E p,q ∼ = M | J | = p +1 K q (cid:16) \ j ∈ J I j (cid:17) for p ≥ , for p < ,where J ranges over all nonempty finite index subsets J ⊆ α . In general, this is ahalf-page spectral sequence, any term E p,q with p ≥ may be nonzero.This spectral sequence converges strongly to K ∗ A . It is functorial with respect to ∗ -homomorphisms that preserve α -indexed ideal decompositions. This generalizes Theorem 6.5.1 about countable α . To prove Theorem 6.7.1 foruncountable α , we will adapt our construction for countable index sets to suitabledirect limits that capture all of α .80 .7 Uncountable sums of ideals Definition 6.7.2 (Directed system of finite sets) . For a set α , we define its directedsystem of finite sets , Fin( α ) = { J ⊆ α : | J | is finite } ; this system is partially ordered by the subset relation ⊆ . Remark 6.7.3. This is indeed a directed system: Given arbitrary J , J ′ ∈ Fin( α ) ,both are equal to or smaller than their union J ∪ J ′ , still a finite set.The direct limit of Fin( α ) in the category of sets is α . All chains, i.e., linearsubsystems, in Fin( α ) are either finite or have the order type of ( N , ≤ ) .We may consider the partially ordered set Fin( α ) itself a thin category: Its ele-ments become objects. Comparable sets J ⊆ J ′ are linked with a unique morphism J → J ′ . Notation 6.7.4 (Algebras for subsets J ⊆ α ) . For the C*-algebra A = P β ∈ α I β asin the statement of Theorem 6.7.1 and J ∈ Fin( α ) , define a subalgebra A ( J ) of A by A ( J ) = X j ∈ J I j . Let J ′ ∈ Fin( α ) be another subset with J ⊆ J ′ . For s ∈ Z , let a ( J, J ′ ) s : K s A ( J ) → K s A ( J ′ ) be the map induced in K-theory by the inclusion of C*-algebras A ( J ) ⊆ A ( J ′ ) . Remark 6.7.5 (Directed system in K-theory) . For each s ∈ Z , consider the functorfrom Fin( α ) to abelian groups that maps J to K s A ( J ) and a comparable pair ofsets J ⊆ J ′ to a ( J, J ′ ) s : K s A ( J ) → K s A ( J ′ ) . This turns { K s A ( J ) : J ∈ Fin( α ) } with the system of morphisms { a ( J, J ′ ) s : J ⊆ J ′ ∈ Fin( α ) } into a directed system.Because K-theory is continuous, colim J ∈ Fin( α ) K s A ( J ) = K s A. Notation 6.7.6 (Spectral sequence for J ) . For J ∈ Fin( α ) , the finite sum of ideals A ( J ) has a spectral sequence { E ( J ) rp,q , d ( J ) r } r,p,q according to Theorem 4.6.1 thatconverges strongly to A ( J ) and has first-page terms of the form E ( J ) p,q ∼ = M | L | = p +1 L ⊆ J K q (cid:16) \ j ∈ L I j (cid:17) for ≤ p < | J | , for p < or p ≥ | J | , 81 Infinite sums of ideals where L ranges over all nonempty subsets of J . Remark 6.7.7 (Directed system of spectral sequences) . Let J ⊆ J ′ ∈ Fin( α ) be twosets such that | J | and | J ′ | differ by an even number. Then the spectral sequences { E ( J ) rp,q , d ( J ) r } r,p,q and { E ( J ′ ) rp,q , d ( J ′ ) r } r,p,q fit into a directed system of spectralsequences connected by degree-amending links shaped like λ (2 n ) from Definition6.5.3. These morphisms have bidegree (0 , .Let F ⊆ Fin( α ) be the subsystem of Fin( α ) of all sets J ∈ Fin( α ) with evencardinality. Then F and Fin( α ) have the same direct limit α in the category of sets.Consider the category of spectral sequences of the form { E ( J ) rp,q , d ( J ) r } r,p,q for J ∈ F with morphisms shaped like λ (2 n ) : Passing from J ∈ Fin( α ) to the spectralsequence { E ( J ) rp,q , d ( J ) r } r,p,q becomes a functor between directed systems. Remark 6.7.8. It does not matter whether the sets J ∈ F have even or odd cardi-nalities. As we have seen in the proof of Proposition 6.5.4, the limit spectral sequencealong ( N , ≤ ) does not depend on whether we consider the subsystem linked by λ (2 n ) or that linked by λ (2 n + 1) . Lemma 6.7.9. Let F ⊆ Fin( α ) be the subsystem of Fin( α ) of all sets J ∈ Fin( α ) with even cardinality. Fix p ∈ Z with p ≥ and fix q ∈ Z .Consider all groups E ( J ) p,q for J ∈ F : This is a directed system of abelian groups;the morphisms are restrictions of degree-amending links of the shape of λ (2 n ) fromDefinition 6.5.3. Then colim J ∈ F E ( J ) p,q = colim J ∈ F M | L | = p +1 L ⊆ J K q (cid:16) \ j ∈ L I j (cid:17) = M | L | = p +1 L ⊆ α K q (cid:16) \ j ∈ L I j (cid:17) . Proof. For J ⊆ J ′ ∈ F , the morphisms E ( J ) p,q → E ( J ′ ) p,q are well-defined becausedegree-amending links have bidegree (0 , .These morphisms were defined as compositions of Bott isomorphisms and links(Definition 6.4.2) between spectral sequences; they preserve all information: Given L ⊆ J of cardinality p + 1 , the direct summand K q (cid:0) T j ∈ L I j (cid:1) from E ( J ) p,q mapsisomorphically onto its copy in the direct sum E ( J ′ ) p,q . Given L ⊆ J ′ such that L is not a subset of J , the direct summand K q (cid:0) T j ∈ L I j (cid:1) is not in the range.In the category of abelian groups, the direct limit may be constructed from a largedirect sum of all objects, dividing by relations according to the morphisms. Here thelinks behave like inclusions, enforcing trivial relations.Finally, F is cofinal in Fin( α ) : For any given set J ⊆ α , the system F contains aset J ′ with J ⊆ J ′ . Therefore the desired direct limit is the direct sum taken overall subsets of α that have cardinality p + 1 .82 .7 Uncountable sums of ideals With these preparations, we may now prove our main theorem. Proof of Theorem 6.7.1. Let F ⊆ Fin( α ) be the subsystem of Fin( α ) of all sets J ∈ Fin( α ) with even cardinality.Consider the directed system of spectral sequences { E ( J ) rp,q , d ( J ) r } r,p,q along all J ∈ F from Remark 6.7.7. Lemma 6.7.9 guarantees that the page E ∗ , ∗ of the limitspectral sequence has the desired structure.All constructions in earlier sections behaved well with the differentials d ( J ) r . Herein Section 6.7, we have applied various functorial direct limit constructions. There-fore the limit spectral sequence has the desired differentials.Likewise, functoriality of the spectral sequence with respect to ideal decomposi-tions follows from all earlier sections and the functoriality of direct limits.Finally, we must prove the strong convergence. All direct limit results from Section6.6 about strong convergence along the directed system ( N , ≤ ) continue to hold forour directed system F because all infinite chains in F have the order type ( N , ≤ ) :Whenever the symbol n ∈ N determines a K-theoretic degree shift in Section 6.6,this may be replaced with n = | J | for J ∈ F . Even though the diagram in Definition6.6.8 of the morphism colim n i ( n, p ) s relies on the linearity of ( N , ≤ ) , the constructionitself is worded purely with direct limits: Objects in this category are morphisms ofthe form i ( n, p ) s and morphisms in this category are commutative diagrams. Thisconstruction does not require linearity of the underlying system, yet provides thedesired filtration on E ∞∗ , ∗ . 83 Infinite coarse excision The spectral sequence for infinite sums of C*-ideals allows us to strengthen Theorem5.5.2, the spectral sequence for finite coarsely excisive covers, to infinite coarselyexcisive covers. Proposition 7.1.1. Let ( X, d ) be a coarse space and let { X β } β ∈ α be a coarsely exci-sive cover of ( X, d ) . The index set α may be finite, countably infinite, or uncountable.Let F ∗ be either the functor C ∗ from the coarse category to C ∗ A or one of the functors D ∗ or Q ∗ from the coarse-continuous category to C ∗ A . For each β ∈ α , consider theC*-ideal F ∗ X β ∼ = F ∗ ( X β ⊆ X ) of F ∗ X .Then the direct limit of finite sums of ideals, [ J ⊆ α | J |∈ N (cid:16) X j ∈ J F ∗ ( X j ⊆ X ) (cid:17) , (7.1.1.1) is a C*-ideal of F ∗ X .Proof. Let A denote the C*-algebra in expression 7.1.1.1.For each β ∈ α , the algebra F ∗ ( X β ⊆ X ) is a C*-ideal in F ∗ X . The inclusion F ∗ ( X β ⊆ X ) → F ∗ X factors through any finite sum P j ∈ J F ∗ ( X j ⊆ X ) when β ∈ J ,and the inclusion of that finite sum in F ∗ X factors again through A ; thus the closed A is a sub-C*-algebra of F ∗ X .To check that A is a C*-ideal in F ∗ X , it remains to show that A is an algebraictwo-sided ideal. Given a ∈ A and b ∈ F ∗ X , find a sequence of finite sets ( J n ) n ∈ N with J n ⊆ α and a sequence ( a n ) n ∈ N with a n ∈ P j ∈ J n F ∗ ( X j ⊆ X ) that convergesto a . Finite sums of C*-ideals F ∗ ( X j ⊆ X ) are again C*-ideals, thus both ( a n b ) n ∈ N and ( ba n ) n ∈ N stay within the closed A . These sequences converge in A to ab and ba respectively because multiplication is continuous. Remark 7.1.2. For finite decompositions, the direct limit of finite sums from ex-pression 7.1.1.1 equals F ∗ X by Theorem 5.5.1. Theorem 7.2.1 (Spectral sequence for coarsely excisive covers) . Let ( X, d ) be acoarse space and let { X β } β ∈ α be a coarsely excisive cover of ( X, d ) . Let F ∗ be eitherthe functor C ∗ from the coarse category to C ∗ A or one of the functors D ∗ or Q ∗ from Infinite coarse excision the coarse-continuous category to C ∗ A . There is a spectral sequence { E rp,q , d r } r,p,q with E p,q ∼ = M | J | = p +1 K q F ∗ (cid:16) \ j ∈ J X j (cid:17) for p ≥ , for p < ,where J ranges over all nonempty finite subcollections of indices in α . For finite α , this spectral sequence converges strongly to K ∗ F ∗ X . In general, the spectral se-quence converges strongly to the K-theory of S J P j ∈ J F ∗ ( X j ⊆ X ) , a C*-ideal of F ∗ X , where J ranges over all finite subcollections of indices in α . The spectral se-quence is functorial with respect to morphisms (coarse maps for C ∗ , or coarse andcontinuous maps for D ∗ and Q ∗ ) to other coarse spaces with compatible coarselyexcisive covers (Definition 5.1.4).Proof. Apply the spectral sequence from Theorem 6.7.1 about arbitrary sums ofabstract C*-algebras to the algebras from Theorem 5.5.1 for coarse spaces.By Theorem 6.7.1, the spectral sequence converges strongly to the K-theory of thenorm closure of finite sums of the input ideals. These ideals are F ∗ X β ∼ = F ∗ ( X β ⊆ X ) and we have described the norm closure of their finite sums in Proposition 7.1.1.The special case for finite coarsely excisive covers follows from Remark 7.1.2. Remark 7.2.2 (Warning about uncountable decompositions) . Let ( X, d ) be a coarsespace. The important C*-algebras C ∗ X , D ∗ X , and Q ∗ X are defined via very amplerepresentations ̺ : C X → BH for a separable Hilbert space H . Separability ofthe Hilbert space is crucial for several isomorphism theorems. Ampleness of therepresentation guarantees that no two functions f = f ′ ∈ C X may be representedon that separable Hilbert space by operators ̺ ( f ) and ̺ ( f ′ ) that differ only by acompact operator.If the coarsely excisive cover { X β } β ∈ α of ( X, d ) has an uncountable index set α ,the separability requirement may force F ∗ X β = F ∗ X β ′ for many β = β ′ , or may forceoutright triviality of ideals. It appears hard to construct interesting examples foruncountable coarse excision.With Theorem 7.2.1, we can strengthen the following Mayer-Vietoris result. Let ( X, d ) be a coarse space and X , X ⊆ X such that { X , X } is a coarsely excisivecover. All rows and columns in the following diagram are long exact sequences; the86 .2 Corollaries for coarsely excisive covers index j in L j ranges over { , } : · · · K s +1 C ∗ X K s C ∗ ( X ∩ X ) L j K s C ∗ X j K s C ∗ X · · ·· · · K s +1 D ∗ X K s D ∗ ( X ∩ X ) L j K s D ∗ X j K s D ∗ X · · ·· · · K s +1 Q ∗ X K s Q ∗ ( X ∩ X ) L j K s Q ∗ X j K s Q ∗ X · · ·· · · K s C ∗ X K s − C ∗ ( X ∩ X ) L j K s − C ∗ X j K s − C ∗ X · · · . The columns are exact by definition of C ∗ , D ∗ , and Q ∗ . Commutativity follows fromthe naturality of the Mayer-Vietoris sequence. Proposition 7.2.3. Let ( X, d ) be a coarse space and let { X β } β ∈ α be a coarsely exci-sive cover of ( X, d ) . Consider the following diagram with exact columns; horizontalarrows are induced by (direct sums of ) inclusions, and unions range over all finiteindex sets J ⊆ α : L β ∈ α K s C ∗ X β K s (cid:0)S J P j ∈ J C ∗ ( X j ⊆ X ) (cid:1) K s C ∗ X L β ∈ α K s D ∗ X β K s (cid:0)S J P j ∈ J D ∗ ( X j ⊆ X ) (cid:1) K s D ∗ X L β ∈ α K s Q ∗ X β K s (cid:0)S J P j ∈ J Q ∗ ( X j ⊆ X ) (cid:1) K s Q ∗ X L β ∈ α K s − C ∗ X β K s − (cid:0)S J P j ∈ J C ∗ ( X j ⊆ X ) (cid:1) K s − C ∗ X. Then this diagram commutes.Proof. The columns are exact again by definition of C ∗ , D ∗ , and Q ∗ . Commutativityfollows from continuity of K-theory – the algebras in the center column are directlimits – and from functoriality of the spectral sequence in Theorem 6.5.1 with respectto morphisms between C*-algebras; here, direct sums of inclusion morphisms. 87 Infinite coarse excision Z ∞ Consider the free Z -module Z ∞ = L N Z of N -indexed tuples ( x , x , . . . , x n , . . . ) withonly finitely many entries different from 0. This space can be metrized in differentways. Definition 7.3.1 (Weight functions, weighted 1-metric) . Let w : N → R > be anarbitrary function, for example w : n n + 1 , w : n , or w : n n + 1 . We call w a weight function . Given a weight function w , define a metric d w on Z ∞ : d w (cid:0) ( x , x , . . . x n , , , . . . ) , ( y , y , . . . , y n ′ , , , . . . ) (cid:1) = ∞ X j =0 w ( j ) | x j − y j | . Example 7.3.2. For the constant weight function w : n , the metric d w coincideswith the usual 1-metric d . Remark 7.3.3 (Topological properties of Z ∞ ) . For w : n n + 1 , the metric d w isproper: With minimum distance k + 1 between points in the k -th dimension, any ballof finite diameter is finite, thus compact. For w : n or w : n n +1 , closed d w -balls with finite radius larger than 1 are not compact anymore. Under w : n n +1 ,the space ( Z ∞ , d w ) is not even locally compact.More than with the topological properties of these spaces, we are concerned withtheir coarse properties.The identity on any coarse space is a coarse map: Choose S = R in Definition2.5.2. But the identity Z ∞ → Z ∞ fails to be a coarse map when the two spaces aremetrized according to two different weight functions among n n + 1 , n , and n n +1 . The identity ceases to be uniformly expansive: Points with distance in dimension n may have distance n + 1 or even ( n + 1) in the target space. Noconstant S > can serve as an upper bound across all dimensions n .If the identity fails as a coarse equivalence, can other maps substitute? The answeris no, for a similar reason: Proposition 7.3.4 (Coarse properties of Z ∞ ) . The weight functions n n + 1 and n generate different coarse structures on Z ∞ ; i.e., there is no coarse equivalenceaccording to Definition 2.5.6. .3 The coarse space Z ∞ Proof. Assume that there is a pair of coarse equivalences f : ( Z ∞ , d n n +1 ) → ( Z ∞ , d n ) ,g : ( Z ∞ , d n ) → ( Z ∞ , d n n +1 ) . Then let S > satisfy all of these conditions: • For all x ∈ Z ∞ , we have d n n +1 ( x, gf x ) ≤ S , this is possible because f and g are a pair of coarse equivalences. • Whenever x , x ′ ∈ Z ∞ with d n ( x, x ′ ) ≤ , then d n n +1 ( gx, gx ′ ) ≤ S , this ispossible because g is a coarse map. • For simplicity, S ∈ N .Let x and x ′ have d n ( x, x ′ ) ≤ . Then d n n +1 ( gx, g ≤ S and in all dimensions S , ( S + 1) , ( S + 2) , . . . , the coordinates of g ( x ) and g ( x ′ ) must be identical.Any two points y , y ′ can be linked by a finite sequence of hops between two pointseach with d n -distance ≤ . By induction, g ( y ) and g ( y ′ ) agree in all coordinatesfrom dimension S onwards.Let z ∈ ( Z ∞ , d n n +1 ) have different coordinates than g ( y ) in dimension S . Then ( g ◦ f )( z ) and z have distance at least S + 1 . This is not allowed when f and g arecoarse equivalences. Remark 7.3.5. When we replace the weight function n with n n +1 , thesame argument shows that the coarse structure induced by that weight function isnot equivalent to ( Z ∞ , d n n +1 ) either. Notation 7.3.6. Let X ⊆ Z ∞ be a set. For a metric d w as above and R > , wewrite N w ( X, R ) instead of N d w ( X, R ) for the R -neighborhood of X under the metric d w according to Definition 2.6.1. Lemma 7.3.7. For all three weight functions w : n n + 1 , w : n , and w : n n +1 , the decomposition { X j } j ∈ N with X j = Z j ≥ × Z ≤ × Z ∞ defines a coarsely excisive cover of ( Z ∞ , d w ) . The block decomposition of R n from Definition 5.6.6 was similar, but covered onlya finite-dimensional space. 89 Infinite coarse excision Proof of Lemma 7.3.7. Consider a finite nonempty index set J ⊆ N and the finitesubcollection { X j } j ∈ J of the decomposition { X j } j ∈ N . We have to show that, for R > , there exists S > with \ j ∈ J N w ( X j , R ) ⊆ N w (cid:16) \ j ∈ J X j , S (cid:17) . Let n denote the highest index in J . We may write \ j ∈ J X j = Y × Y × . . . × Y n − × Z ≤ × Z ∞ , where Y j = { } for j ∈ J , otherwise Y j = Z ≥ . For easier notation, we will write Y n for Z ≤ .We are interested in the R -neighborhood of each X j for R > and in the S -neighborhood of the intersection for a suitable S . Let x = ( x , x , x , . . . ) be apoint in Z ∞ . The coordinates after the n -th coordinate do not matter anymore: Forthe intersection T j ∈ J X j , the distance of x to T j ∈ J X j depends only on the earlycoordinates up to the n -th.Since the metric d w on Z ∞ is a weighted 1-metric – distance is the weighted sum ofthe dimension-wise distances – it makes sense to decompose ( Z ∞ , d w ) into a productof metric spaces, Z n +1 × Z ∞ , of the first n + 1 dimensions and the remainder Z ∞ that is irrelevant for the chosen J . Let p and q be the projections for this productdecomposition, p : Z ∞ → Z n +1 , p ( x , x , x , . . . ) = ( x , x , . . . , x n ) ,q : Z ∞ → Z ∞ , q ( x , x , x , . . . ) = ( x n +1 , x n +2 , . . . ) . Both projections admit one-sided inverse by padding all other coordinates with zeros.We pull back the metric d w to either factor along these inverses. For x ∈ Z ∞ and aspace Y that is either Y = X j for a j ∈ J or Y = T j ∈ J X j , we then have d w ( x, Y ) = d w (cid:0) px, p ( Y ) (cid:1) + d w (cid:0) qx, q ( Y ) (cid:1)| {z } = 0 ; (7.3.7.1)the rightmost summand vanishes because both q ( X j ) and q (cid:0) T j ∈ J X j (cid:1) are the entirerange q ( Z ∞ ) by construction. On the finite-dimensional remainder Z n +1 , the weightfunction w admits an upper and a lower bound: There is a constant M ≥ n + 1 suchthat M ≤ w ( k ) ≤ M for all k < n + 1 . For x , y ∈ Z n +1 , Lemma 5.6.16 shows d w ( x, y ) ≤ M d ( x, y ) ≤ M d ∞ ( x, y ) ≤ M d ( x, y ) ≤ M d w ( x, y ) . (7.3.7.2)90 .3 The coarse space Z ∞ The finite-dimensional space Z n +1 embeds isometrically into R n +1 . This embeddingis not necessarily the inclusion Z n +1 ⊆ R n +1 ; rather, depending on w , its imagelattice is scaled differently per dimension. Still, we can now apply Proposition 5.6.15for the sup -metric d ∞ on R n +1 : \ j ∈ J N ∞ (cid:0) p ( X j ) , R (cid:1) = N ∞ (cid:16) \ j ∈ J p ( X j ) , R (cid:17) . Together with 7.3.7.1 and 7.3.7.2, we conclude that S = M R certifies the desiredcoarse excisiveness in the infinite-dimensional space Z ∞ : \ j ∈ J N w ( X j , R ) ⊆ N w (cid:16) \ j ∈ J X j , M R (cid:17) . Proposition 7.3.8. Let { X j } j ∈ N be the coarsely excisive cover of Z ∞ as in Lemma7.3.7 and let A denote the direct limit C*-algebra A = [ J ⊆ N | J |∈ N X j ∈ J C ∗ ( X j ⊆ Z ∞ ) = X j ∈ N C ∗ ( X j ⊆ Z ∞ ) . Under any of the three considered weight functions, the algebra A then has trivialK-theory: K ∗ A = 0 .Proof. We may use our spectral sequence from Theorem 6.5.1 because the cover { X j } j ∈ N is coarsely excisive.Each X j is flasque (Definition 5.6.3) because it contains the flasque factor Z ≤ .For finite J ⊆ N , the intersection T j ∈ J X j is again flasque: Let n be the largestindex in J . We may describe this intersection by listing its one-dimensional factors:First, there are n factors that we may ignore. Then there is the flasque factor Z ≤ .All further factors form a copy of Z ∞ . Because of the flasque factor, we concludethat K ∗ C ∗ (cid:0) T j ∈ J X j (cid:1) = 0 .Since this holds for all finite J ⊆ N , Theorem 6.5.1 gives a spectral sequence withfirst page E ∗ , ∗ = 0 , converging to K ∗ A = 0 .The countable intersection T j ∈ N X j is a single point, but infinite intersections donot appear in the spectral sequence. 91 Infinite coarse excision Let α be either N or a finite cardinality with α > . Let X β = ( R ≥ , be a ray forall β ∈ α , pointed at the origin. Define X = _ β ∈ α X β , a finite or countable wedge sum of the half-open rays glued together at their origins.Define a coarse structure on X by the metric d ( x, y ) = | x − y | if x and y lie in the same ray, | x | + | y | otherwise.If local compactness were desired, we could remove the common point . This wouldnot change any large-scale properties of X . But in this example, local compactnessis irrelevant. Lemma 7.4.1. Let Y β = X ∪ X β for all β ∈ α . Cover X by { Y β : β ∈ α } .This is a coarsely excisive cover of the wedge sum X .Proof. We have to check: For all finite { Y β (0) , Y β (1) , . . . , Y β ( n − } and all R > ,there exists S > such that the n -fold intersection of the R -neighborhoods lies inthe S -neighborhood of the intersection: \ j For the wedge sum X = W β ∈ α X β with each X β = ( R ≥ , aray pointed at the origin and the coarsely excisive cover { Y β } β ∈ α from Lemma 7.4.1,let A be the direct limit C*-algebra of sums P j ∈ J C ∗ ( Y j ⊆ X ) over finite sets J ⊆ α . .4 Wedge sum of rays Then K s A = for s even, L β ∈ αβ =0 Z for s odd.Proof. The set Y remains a flasque ray. Each other Y β is coarsely equivalent to R and therefore has K-theory K C ∗ Y n = 0 and K C ∗ Y n = Z . This determines thecolumn E , ∗ of the first page.Each finite intersection of at least two different Y n is the flasque space Y = X .The K-theory of its Roe algebra vanishes. Therefore the E ∗ , ∗ -term looks as follows: ... L β ∈ αβ =0 Z − L β ∈ αβ =0 Z − p.q This spectral sequence collapses on the first page. We may read the K-theory of A ⊆ C ∗ X from the only nonzero column: If α is countably infinite, the dimension ofthe free Z -module in odd degrees is countably infinite; if α is finite, the dimension is α − .For finite α , an alternative proof to compute A = C ∗ X by induction repeats theMayer-Vietoris principle α − times for two-fold coverings: Glue single rays, oneafter another, to the wedge sum that starts with a single ray. 93 Generalizations Instead of K-theory of C*-algebras over C , we may examine KO-theory for a C*-algebra A over R , denoted KO ∗ A . All basic definitions carry over without change,turning KO ∗ into a Z -graded covariant continuous functor into abelian groups.The major difference is the degree of the Bott isomorphism: Instead of K s A ∼ = K s +2 A , real Bott periodicity admits a natural isomorphism β : KO s A ∼ = KO s +8 A for all s ∈ Z . As a result, for C*-ideals I ⊆ A , the six-term exact sequence · · · → K I → K A → K ( A/I ) ∂ ◦ β −→ K I → K A → K ( A/I ) ∂ −→ K I → · · · becomes a 24-term exact sequence in the real case: · · · → KO A → KO ( A/I ) ∂ ◦ β −→ KO I → KO A → KO ( A/I ) ∂ −→ KO I → · · · . Looking back to the constructions of the various spectral sequences, we relied onthe Bott isomorphism merely for the construction of the degree-amending links λ ( n ) : E ( n ) rp,q → E ( n + 2) rp,q from Definition 6.5.3. These morphisms and Proposi-tion 6.5.4 about the existence of the limit spectral sequence can be adapted to workwith KO-theory: Define λ R ( n ) : E ( n ) rp,q → E ( n + 8) rp,q by chaining 8 links { ℓ ( n ) rp,q } r,p,q from Definition 6.4.2 – instead of only 2 such links inthe case of K-theory – with the real Bott isomorphism. The direct limit along these λ R (8 n ) for n → ∞ does not depend on the position of the Bott isomorphism withinthe chain nor on whether we consider the directed system along λ R (8 n ) , λ R (8 n + 1) , . . . , or λ R (8 n + 7) .Convergence is proven as in Section 6.6 and generalized to uncountable ideal de-compositions as in Section 6.7. This yields a well-defined spectral sequence thatcomputes the KO-theory of C*-ideal sums: The statement from Theorem 6.7.1 holdswhen we replace K-theory with KO-theory. Let ( X, d ) be a metric space. Let G be a countable discrete group that acts on X freely and properly by d -isometries. This action extends to C X via ( gf )( x ) = f ( g − x ) for g ∈ G , f ∈ C X , and x ∈ X . In addition to a very ample representa-tion ̺ : C X → BH for a separable Hilbert space H , let U : G → H be a unitary95 Generalizations representation with U ( g ) ̺ ( f ) = ̺ ( gf ) U ( g ) .This gives rise to C*-algebras C ∗ G X and D ∗ G X by changing the usual definitionsof C ∗ X and D ∗ X : The norm closure is taken of only the G -invariant operators in BH that satisfy all other requirements of C ∗ X and D ∗ X , respectively. Furthermore,define Q ∗ G X = D ∗ G X/C ∗ G X .In [Sie12, Definition 3.6], for a G -invariant subspace Y ⊆ X that we may assume tobe closed, P. Siegel constructs the relative C*-algebras C ∗ G ( Y ⊆ X ) and D ∗ G ( Y ⊆ X ) by imposing on operators in C ∗ G X and D ∗ G X the conditions from Section 5.2 forsupport near Y and local compactness outside Y . Finally, define Q ∗ G ( Y ⊆ X ) as thequotient D ∗ G ( Y ⊆ X ) /C ∗ G ( Y ⊆ X ) . There is a long exact sequence for s ∈ Z , · · · → K s C ∗ G X → K s D ∗ G X → K s Q ∗ G X → K s − C ∗ G X → · · · . Furthermore, Q ∗ G X ∼ = Q ∗ X G . The sequence may thus be rewritten with the K-homology of X G instead of Q ∗ G X according to Remark 2.4.12.Let the functor F ∗ G stand for either C ∗ G , D ∗ G , or Q ∗ G . According to [Sie12, Propo-sitions 3.8, 3.9], for closed G -invariant subspaces Y ⊆ X and s ∈ Z , we have K s F ∗ G ( Y ⊆ X ) ∼ = K s F ∗ G Y. For G -invariant coarsely excisive covers { X , X } of X , we have F ∗ G ( X ⊆ X ) + F ∗ G ( X ⊆ X ) = F ∗ G X,F ∗ G ( X ⊆ X ) ∩ F ∗ G ( X ⊆ X ) = F ∗ G ( X ∩ X ⊆ X ) . This leads to a Mayer-Vietoris exact sequence. We may expect a generalization ofour spectral sequence to arbitrary G -invariant coarsely excisive covers { X β } β ∈ α of X . The definitions of C ∗ G ( Y ⊆ X ) and D ∗ G ( Y ⊆ X ) treat the G -invariance in theleast intrusive way possible. There should be no difficulty in adapting the equationsto finite selections J ⊆ α of the coarsely excisive cover: K s (cid:16) X j ∈ J F ∗ G ( X j ⊆ X ) (cid:17) ∼ = K s F ∗ G (cid:16) [ j ∈ J X j (cid:17) ,K s (cid:16) \ j ∈ J F ∗ G ( X j ⊆ X ) (cid:17) ∼ = K s F ∗ G (cid:16) \ j ∈ J X j (cid:17) , with F ∗ G standing for C ∗ G , D ∗ G , or Q ∗ G . The constructions will be similar to thoseleading to our Theorem 5.5.1 for these equations where G is trivial.This provides a G -invariant version of our spectral sequence from Theorem 7.2.1for coarsely excisive covers.96 eferences References [Boa99] J. Michael Boardman. Conditionally Convergent Spectral Sequences. In Homotopy Invariant Algebraic Structures (Baltimore, MD, 1998) , volume 239of Contemp. Math. , pages 49–84. Amer. Math. Soc., Providence, RI, 1999.[CE73] Henri P. Cartan and Samuel Eilenberg. Homological Algebra . PrincetonMathematical Series. Princeton University Press, 1973.[Dav96] Kenneth R. Davidson. 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