Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds
TTopological Aspects of the Equivariant Index Theory ofInfinite-Dimensional LT -Manifolds Doman TakataThe University of TokyoJuly 20, 2020
Abstract
Let T be the circle group and let LT be its loop group. We formulate and investigate sev-eral topological aspects of the LT -equivariant index theory for proper LT -spaces, where proper LT -spaces are infinite-dimensional manifolds equipped with “proper cocompact” LT -actions.Concretely, we introduce “ R KK -theory for infinite-dimensional manifolds”, and by using it,we formulate an infinite-dimensional version of the KK -theoretical Poincar´e duality homomor-phism, and an infinite-dimensional version of the R KK -theory counterpart of the assembly map,for proper LT -spaces.The left hand side of the Poincar´e duality homomorphism is formulated by the “ C ∗ -algebra ofa Hilbert manifold” introduced by Guoliang Yu. Thus, the result of this paper suggests that thisconstruction carries some topological information of Hilbert manifolds. In order to formulate theassembly map in a classical way, we need crossed products, which require an invariant measureof a group. However, there is an alternative formula to define them using generalized fixed-pointalgebras. We will adopt it as the definition of “crossed products by LT ”. Mathematics Subject Classification (2010) . Key words : infinite-dimensional manifolds, loop groups, Dirac operators, KK -theory, R KK -theory, representable K -theory, C ∗ -algebras of Hilbert manifold, index theory. Contents C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Equivariant KK -theory and equivariant R KK -theory . . . . . . . . . . . . . . . . . 82.3 Bott periodicity and the C ∗ -algebra S . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Spin c -manifolds . . . . . 213.3 A convenient description of the topological assembly map . . . . . . . . . . . . . . . 253.4 Twisted equivariant version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 a r X i v : . [ m a t h . K T ] J u l Index problem on proper LT -spaces 39 LT . . . . . . . . . . . . . . . . . . . . . . . 434.3 R KK -theory for non-locally compact action groupoids . . . . . . . . . . . . . . . . . 464.4 Loop group equivariant KK -theory and the main result . . . . . . . . . . . . . . . . 53 C ∗ -algebra of a Hilbert manifold and Poincar´e duality homomorphism . . . . . . . 565.2 The index element of [T4] and a study on C ∗ -algebras of Hilbert manifolds . . . . . 605.3 The Poincar´e duality homomorphism for proper LT -spaces . . . . . . . . . . . . . . 83 LT -spaces 88 LT -spaces . . . . . . . . . 886.2 The “descent of the Dirac element” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3 The topological assembly map for proper LT -spaces . . . . . . . . . . . . . . . . . . 1036.4 Next problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 The Atiyah-Singer index theorem states that the analytic index of the Dirac operator on a closedmanifold is determined by topological data [ASi1, ASi2, ASe]. The overall goal of our research isto establish an infinite-dimensional version of this theorem. As a first step of this project, we havebeen studying infinite-dimensional manifolds equipped with LT -actions, where T := S and LT isits loop group. In the present paper, we study topological aspects of this problem.Let us begin with the KK -theoretical description of a non-compact version of the index theoremgiven by Kasparov [Kas1, Kas3]. The details will be explained in Section 3. Fact 1.1 ([Kas1, Kas3]) . Let X be a complete Riemannian manifold and let G be a locally compactsecond countable Hausdorff group acting on X in an isometric, proper and cocompact way. Let W be a G -equivariant Clifford bundle, and let D be a G -equivariant Dirac operator on W .(1) D has an analytic index ind( D ) as an element of KK ( C , C (cid:111) G ).(2) The KK -element [ D ] := [( L ( X, W ) , D )] ∈ KK G ( C ( X ) , C ) determines ind( D ) with a KK -theoretical procedure µ G : KK G ( C ( X ) , C ) → KK ( C , C (cid:111) G ) , that is to say, ind( D ) = µ G ([ D ]). This procedure is called the analytic assembly map .(3) [ D ] is determined by the topological data [ σ ClD ] = [( C ( X, W ) , ∈ R KK G ( X ; C ( X ) , Cl τ ( X )),where Cl τ ( X ) is the section algebra of the Clifford algebra of the tangent bundle. More generally,there is a homomorphism called the Poincar´e duality homomorphism
PD : KK G ( C ( X ) , C ) → R KK G ( X ; C ( X ) , Cl τ ( X ))and it is isomorphic.(4) Consequently, ind( D ) is completely determined by the topological data [ σ ClD ]. The concreteformula giving the index is called the topological assembly map , and it is denoted by ν G : R KK G ( X ; C ( X ) , Cl τ ( X )) → KK ( C , C (cid:111) G ) . ν G . For this aim, we need to overcome many problems including the following: For an infinite-dimensional manifold, the C -algebra is trivial; For a non-locally compact space, wecan not define R KK -theory in a classical way. For the former problem, we will use a “ C ∗ -algebra of a Hilbert manifold X ” denoted by A ( X ),introduced in [Yu], which is a wide generalization of a C ∗ -algebra of a Hilbert-Hadamard spacedefined in [GWY]. Strictly speaking, it is an infinite-dimensional analogue of, not C ( X ) butthe “graded suspension of Cl τ ( X )”. However, Cl τ ( X ) is KK -equivalent to C ( X ) if X is even-dimensional and Spin c . Thus, we can remove C ( X ) from the index theorem for such manifolds,and we can reformulate it with the graded suspension of Cl τ ( X ) which can be generalized to Hilbertmanifolds.For the latter one, by the same KK -equivalence, we have an isomorphism R KK ( X ; C ( X ) , Cl τ ( X )) ∼ = R KK ( X ; C ( X ) , C ( X )). Moreover, it is isomorphic to RK ( X ), which is the representable K -theory of Segal. While R KK ( X ; C ( X ) , C ( X )) is defined by C ( X ), RK ( X ) is defined in a purelytopological way. Thus, it might be possible to generalize R KK -theory to an infinite-dimensionalmanifold in a topological way. With this observation, we will formulate R KK -theory for infinite-dimensional spaces, via fields of C ∗ -algebras and Hilbert modules.We can now state the main results of the present paper. A proper LT -space is, roughly speaking,an infinite-dimensional manifold equipped with a proper cocompact LT -action (see Section 4.1 forthe details). We would like to study the index problem of a Spinor bundle twisted by a τ -twisted LT -equivariant line bundle L . Theorem 1.2.
We can formulate a “ KK -theoretical Poincar´e duality homomorphism” for a Hilbertmanifold satisfying a bounded geometry type condition equipped with a proper isometric group action.When we apply this construction to a proper LT -space, we obtain an appropriate answer: ThePoincar´e duality homomorphism assigns to the “index element twisted by L constructed in [T4]”the R KK -element corresponding to L . Theorem 1.3.
We can formulate a “topological assembly map” for proper LT -spaces. This mapassigns to the R KK -element corresponding to L the “analytic index of the Dirac operator twistedby L ” constructed in [T3]. Corollary 1.4.
By the composition of the above two homomorphisms, the “analytic index of theDirac operator twisted by L ” is assigned to the “index element twisted by L ”. These results are obviously a part of the LT -equivariant index theorem. Moreover, they sug-gest that the C ∗ -algebra of a Hilbert manifold carries some topological information of the Hilbertmanifold. For a locally compact Hausdorff space, such a result is in some sense automatic, becausethe category of locally compact Hausdorff spaces is equivalent to that of commutative C ∗ -algebras.On the other hand, in a non-locally compact setting, such results are highly non-trivial and quiteinteresting.Let us explain possible applications of the LT -equivariant index theorem and the ideas used inthe present paper (we have not completed the index theorem for proper LT -spaces, and we willsummarize what we should do after the present paper in order to compete it in Section 6.4).We begin with the quantization problem of Hamiltonian loop group spaces [AMM, Mei1, Son,LMS, LS]. A Hamiltonian loop group space is an infinite-dimensional symplectic manifold equipped3ith a loop group action and a proper moment map taking values in the dual Lie algebra of theloop group, where the loop group action on it is the gauge action. Since the moment map isequivariant and proper, and since the gauge action is proper and cocompact, the loop group actionon a Hamiltonian loop group space is automatically proper and cocompact. Proper LT -spaces areobvious generalizations of Hamiltonian loop group spaces.In general, a quantization is a procedure making a Hilbert space from a classical system. Thequantization of a Hamiltonian G -space given by Bott is the equivariant index of the Spin c -Diracoperator twisted by the pre-quantum line bundle. In [Son], this procedure was generalized toHamiltonian LG -spaces by a formal and infinite-dimensional argument, using the one-to-one corre-spondence between Hamiltonian LG -spaces and quasi-Hamiltonian G -spaces given in [AMM]. Theloop group equivariant index theory will give a KK -theoretical description of this result, and theresult of the present paper will enable us to compute the quantization in the topological language.Another application is a noncommutative geometrical reformulation of the Freed-Hopkins-Telemenisomorphism (FHT isomorphism for short) [FHT1, FHT2, FHT3]. This result states that theGrothendieck completion of the semigroup consisting of isomorphism classes of positive energyrepresentations of the loop group of G (so called the Verlinde ring), is isomorphic to the twistedequivariant K -group of G with respect to the conjugation action. This isomorphism is constructedas follows: For a given positive energy representation, the authors defined an LG -equivariant con-tinuous field of Fredholm operators parameterized by L g ∗ ; Since the groupoid L g ∗ //LG is locallyequivalent to G//G , they can pushforward the field to
G//G ; Since a positive energy representationis projective, the obtained field defines an element of the twisted K -group. Since the twisted K -theory of L g ∗ //LG should be isomorphic to the “twisted LG -equivariant K -homology of L g ∗ ” bythe Poincar´e duality, the FHT isomorphism might be related to the LG -equivariant index theory.See [Loi] for a research on this line.Finally, we add a comment on the Witten genus. The Witten genus of a compact manifold M is defined “using the fixed-point formula for the S -equivariant topological index of its free loopspace LM ” [Wit]. This theory is very interesting, but the definition is not quite satisfying, in thata Dirac operator whose index is the Witten genus has not been given.Although there are no direct connections between the Witten genus and the LT -equivariantindex theory, the observations of the present paper can be quite useful for a noncommutativegeometrical formulation of the Witten genus. In fact, if an S -equivariant index can be defined inan infinite-dimensional way, it should be realized as a homomorphism χ from the S -equivariant K -homology of LM to R ( S ) ∗ = Hom( R ( S ) , Z ) or something, where R ( S ) ∗ can be identified with theset of formal power series (cid:80) n ∈ Z a n z n . Although K -homology theory does not work for non-locallycompact spaces, we can define a substitute for it in the language of noncommutative geometrywith the C ∗ -algebra of a Hilbert manifold. Moreover, by the Poincar´e duality homomorphismconstructed in the present paper, it might be possible to give a topological formula of χ . If thisformula coincides with the definition of the Witten genus, the index homomorphism χ will realizethe Witten genus. This realization has an advantage compared with the original one, in that χ isdefined as a homomorphism from an invariant of not M but LM . It might be possible to use someglobal structure of LM .From now on, we explain a few backgrounds of the present paper. Since we have explainedpossible applications, we focus on technical sides.One of the main objects of the present paper is a C ∗ -algebra of a Hilbert manifold constructedby Guoliang Yu. It is a wide generalization of the construction of [HKT]. In this paper, the4uthors defined a C ∗ -algebra for a Hilbert space using a finite-dimensional approximation and aninductive limit. In order to construct it, “the suspension of Clifford algebra-valued function algebra”was appropriate, as we have mentioned. This construction was used to prove the Baum-Connesconjecture for a-T-menable groups in [HK], and it was generalized to Hilbert bundles and Fredholmmanifolds in [Tro, DT]. Several years ago, the construction was reformulated and generalized toHilbert-Hadamard spaces in [GWY]. Guoliang Yu generalized this construction to Hilbert manifoldswith positive injectivity radius, and we will use it in the present paper.These two kinds of constructions have different advantages. The constructions of [HKT, HK,Tro, DT] are rather algebraic, and hence they are convenient to compute several invariants. Theconstruction of [GWY, Yu] are rather geometric, and hence they look more natural and they areconvenient to define group actions. In the present paper, we use both constructions.There are many results on substitutes for “ L -spaces” and “Dirac operators” on infinite-dimensionalspaces related to loop groups, for example [FHT2, Lan, Son, Was]. These results are based onrepresentation theoretical ideas: The L -space on a compact group can be written using its repre-sentation theory by the Peter-Weyl theorem, and direction derivatives are written as infinitesimalrepresentations. See also [Kos] for algebraic Dirac operators.In [HKT, HK], the authors constructed the “ L -space on a Hilbert space” using the cre-ation/annihilation operators. In [T1], we found out that it is strongly related to the above repre-sentation theoretical construction: The Peter-Weyl type construction for the infinite-dimensionalHeisenberg group at fixed level gives a similar construction to [HKT, HK]. We will use it in thebody of the present paper.There are probabilistic approaches using the Wiener measure. However, we would like to considera Hilbert space like the “ L -space defined by the Lebesgue measure”. Thus, we do not discuss sucha theory in the following.An analytic index over a geometrical object X should be a group homomorphism from the“ K -homology of X ”. In this sense, the construction of an analytic index theory for X is formally equivalent to the construction of a C ∗ -algebra substituting for a “function algebra of X ” and agroup homomorphism from the K -homology of this C ∗ -algebra. However, the content of this theoryis the value of the analytic index homomorphism at a certain K -homology element. Thus, theexplicit construction of an element of the K -homology group is quite important. In order to definesuch a K -homology element for an infinite-dimensional space, one needs to define substitutes foran “ L -space” and a “Dirac operator”, which is compatible with the “function algebra” . The Diracelement of [HK], and the index element and the Dirac element of [T4] are such constructions. Theconstructions of [T4] are less general, but much more direct, than that of [HK]. We will reformulatethe index element of [T4] and we will use it in the present paper.Let us explain the organization of the present paper.In Section 2, we will review several standard facts on KK -theory. In particular, we will comparea classical formulation of the Bott periodicity and the reformulation using the Bott homomorphismexplained in [HKT, Section 2]. This comparison will be necessary to generalize the local Bottelement to infinite-dimensional manifolds.In Section 3, we will describe the statement of the Kasparov index theorem in detail, and wewill reformulate it for even-dimensional Spin c -manifolds. Since C ( X ) is KK -equivalent to Cl τ ( X )for an even-dimensional Spin c -manifold X , we can remove C ( X ) from the description of the indextheory for such X . For the same reason, we can replace the right hand side of the Poincar´e dualityhomomorphism with R KK G ( X ; C ( X ) , C ( X )) ∼ = RK G ( X ), which is easy to generalize to proper5 T -spaces. In addition, we will give an explicit definition of the topological assembly map, and thedescription of it under the above reformulation.In Section 4, we will start the study of infinite-dimensional spaces. First, we will precisely set upthe problem. The remainder of this section will be devoted to explain this setting in detail. Second,we will review the representation theory of LT , and we will recall the substitute for the “ τ -twistedgroup C ∗ -algebra of LT ” from [T1]. Third, we will introduce R KK -theory for non-locally compactaction groupoids. The detailed study will be done in [NT], and we will just give definitions and afew necessary results. Finally, we will re-introduce loop group equivariant KK -theory, which wasintroduced in [T4] in a naive way, and the detailed studies were put off. Thanks to the betterdefinition given there, we will be able to prove desired properties on Kasparov products on thistheory. With this collect LT -equivariant KK -theory, we will describe the results of the presentpaper.In Section 5, we will give the definition of a C ∗ -algebra of a Hilbert manifold, following [Yu], andwe will define an infinite-dimensional analogue of the Poincar´e duality homomorphism. In order toindicate that this construction is appropriate, we apply it to proper LT -spaces. For this aim, wewill reformulate the index element of [T4] using the C ∗ -algebra of a Hilbert manifold given in [Yu].This reconstruction will contain a detailed study of this C ∗ -algebra. Then, we will compute thePoincar´e duality homomorphism for this index element.In Section 6, we will construct an infinite-dimensional analogue of the topological assembly map.The key of the construction will be the fixed-point algebra description of the crossed products andthe descent homomorphisms obtained in Section 3. We will compute the topological assembly mapfor proper LT -spaces. Finally, we will explain what we should do after the present paper, in orderto complete the LT -equivariant index theory. Notations • For a complex Hilbert space V , a complex bilinear form is denoted by angle brackets (cid:104)• , •(cid:105) ,and an inner product is denoted by parentheses ( • | • ). More generally, for a C ∗ -algebra A ,an A -valued inner product on a Hilbert A -module E , is denoted by ( • | • ) A or ( • | • ) E . • For a C ∗ -algebra B and a Hilbert B -module E , the set of compact operators on E is denotedby K B ( E ) or simply by K ( E ), and that of bounded operators is denoted by L B ( E ) or simplyby L ( E ). Associated to it, for a Hilbert space V , K ( V ) = K C ( V ) denotes the set of all compactoperators and L ( V ) = L C ( V ) denotes the set of all bounded operators. • A group action on a set X (e.g. a manifold, a vector space, a C ∗ -algebra) is often denoted bythe common symbols α . The automorphism corresponding to g ∈ G is denoted by α g . Whenwe need to emphasize X , we denote it by α Xg . • A G -action induced by left translation on a function algebra on G , is often denoted by “lt”,namely lt g ( f )( x ) := f ( g − x ). Similarly, rt g ( f )( x ) := f ( xg ). • The left and right regular representations are denoted by L and R , respectively. • ∗ -homomorphisms to define left module structures on Hilbert modules, are often denotedby the common symbol π when it is not confusing. Associated to it, the ∗ -homomorphism A (cid:111) G → L ( E (cid:111) G ) induced by the descent homomorphism by G from π : A → L ( E ), isdenoted by π (cid:111) lt. • For a Lie group G , its Lie algebra is denoted by Lie( G ) or g . • The symbol (cid:98) ⊗ means the graded tensor product of Hilbert spaces, C ∗ -algebras or Hilbertmodules. We often use this symbol even for trivially graded objects.6 The algebraic tensor product is denoted by (cid:98) ⊗ alg , where for Hilbert spaces, C ∗ -algebras orHilbert modules V and V , V (cid:98) ⊗ V stands for (cid:110)(cid:80) finite v i (cid:98) ⊗ v i | v ij ∈ V j (cid:111) . V (cid:98) ⊗ V is the comple-tion of V (cid:98) ⊗ alg V with respect to an appropriate norm. • For a Z -graded vector space V = V (cid:98) ⊕ V , we denote the grading by ∂ and the graded homo-morphism by (cid:15) , that is to say, for v ∈ V i ( i = 0 or 1), ∂v := i and (cid:15)v = ( − i v . • We always use graded commutators: For a Z -graded vector space V and two even or oddoperators F , F ∈ End( V ), we define [ F , F ] := F F − ( − ∂F ∂F F F . • For a group G equipped with a U (1)-central extension 1 → U (1) i −→ G τ p −→ G →
1, a τ -twisted G -representation on a vector space V is a group homomorphism ρ : G τ → U ( V ) so that ρ ◦ i ( z ) = z id V for z ∈ U (1). Similarly, we define the concept of τ -twisted G -actions onHilbert modules or vector bundles. • We frequently use some objects which are substitutes for something that we can not define ina classical way. Such substitutes are denoted by the standard symbol with an underline. Forexample, the Hilbert space substituting for the “ L -space of an infinite-dimensional manifold M ” is denoted by L ( M ). • Infinite-dimensional objects are written in \ mathcal (e.g. an infinite-dimensional manifoldsare written as X , M , · · · ), and fields are written in \ mathscr (e.g. continuous fields of Banachspaces are written as A , B , · · · ). The converse is not true: We use S to denote the C ∗ -algebra C ( R ) equipped with a Z -grading homomorphism (cid:15)f ( t ) := f ( − t ). • ε and δ are used to describe Assumption 5.6. Thus, we use (cid:15) and ∆ as small positive realnumbers. (cid:15) is often used as a grading homomorphism, but we believe it is not confusing. • All the locally compact group appearing in the present paper are assumed to be amenable forsimplicity. Thus, the crossed products are uniquely determined. C ∗ -algebras In this subsection, we define several basic C ∗ -algebras appearing in the present paper. Definition 2.1. (1) For a Euclidean space V , we define the Clifford algebra of V by Cliff ± ( V ) := T C ( V ) / (cid:10) v ∓ | v | · (cid:11) , where T C ( V ) is the tensor algebra over C . They inherit the Z -grading from T C ( V ), because (cid:10) v ∓ | v | · (cid:11) is contained in the even-part.(2) The metric of V induces metrics on Cliff ± ( V ). By left multiplications Cliff ± ( V ) → End(
Cliff ± ( V )), and the Z -graded C ∗ -algebra structures on End( Cliff ± ( V )), we define a Z -graded C ∗ -algebra structures on both of Cliff ± ( V ). Note that v ∗ = ± v in Cliff ± ( V ) for each v ∈ V .(3) A Z -graded Hermite vector space S is called a Clifford module of V , if it is equippedwith an even ∗ -homomorphism c : Cliff − ( V ) → End( S ). A Clifford module of V is called a Spinor if it is an irreducible representation of
Cliff − ( V ).It is known that an even-dimensional vector space V has two different Spinors up to equiva-lence [Fur]. We fix one of them, and we denote it by ( S, c ). The Clifford multiplication c givesan isomorphism Cliff − ( V ) ∼ = End( S ) as Z -graded C ∗ -algebras. Associated to it, we obtain thefollowing. Lemma 2.2. (1)
The dual space S ∗ = Hom( S, C ) is a ∗ -representation space of Cliff + ( V ) by c ∗ ( v ) · f := ( − ∂f f ◦ c ( v ) . It gives an isomorphism c ∗ : Cliff + ( V ) ∼ = End( S ∗ ) . We can define a right Hilbert Cliff + ( V ) -module structure on S , by identifying S with S ∗∗ . Infact, the operations are given by s · v := s ◦ c ∗ ( v ) and ( s | s ) Cliff + := s ∗ (cid:98) ⊗ s ∈ End( S ∗ ) ∼ = Cliff + ( V ) for s, s , s ∈ S and v ∈ V .Example . Let V be a 2-dimensional Euclidean space equipped with a complex structure J . Itdefines a complex base { z, z } of V ⊗ C . Let S be C (cid:98) ⊕ C and we define c : V ⊗ C → End( S ) by c ( z ) := (cid:18) −√
20 0 (cid:19) , c ( z ) := (cid:18) √ (cid:19) . One can easily check that the restriction of c to V extends to Cliff − ( V ). The tensor product ofcopies of this example plays an important role in the present paper.There is a family version of these constructions. Let X be a Riemannian manifold. A pair ofa Hermitian vector bundle S and a bundle homomorphism c : Cliff + ( T X ) → End( S ) preservingthe Z -grading is called a Spinor bundle S over X . If X admits such a bundle, X is called a Spin c -manifold. For example, an almost complex manifold is Spin c , whose Spinor bundle is givenby the same construction of the above example.We define a C ∗ -algebra Cl τ ( X ) := C ( X, Cliff + ( T X )). The following is a family version ofLemma 2.2.
Lemma 2.4.
Let ( S, c ) be a Spinor bundle over X . By the family version of Lemma 2.2, C ( X, S ) admits a Hilbert Cl τ ( X ) -module structure. Moreover, a Hilbert C ( X ) -module C ( X, S ∗ ) admits aleft Cl τ ( X ) -module structure c ∗ : Cl τ ( X ) → L C ( X ) ( C ( X, S ∗ )) . The following easy Z -graded C ∗ -algebras are related to almost all objects appearing in thepresent paper. Definition 2.5. (1) We define a Z -graded C ∗ -algebra S by C ( R ) with the grading homomorphism (cid:15)f ( t ) := f ( − t ). In other words, an even and odd function are an even and odd element, respectively.We define an unbounded multiplier X on S by Xf ( t ) := tf ( t ).(2) Similarly, we define a Z -graded C ∗ -algebra S ε for ε > C ( − ε, ε ) with the gradinghomomorphism (cid:15)f ( t ) := f ( − t ). We define a multiplier X on S ε by Xf ( t ) := tf ( t ). KK -theory and equivariant R KK -theory We use the unbounded picture of equivariant KK -theory. We refer to [Blac, Kas2] for the boundedpicture. For the following definition, see [T4]. Definition 2.6.
Let G be a locally compact second countable Hausdorff group, and let A , B beseparable G - C ∗ -algebras. An unbounded G -equivariant Kasparov ( A, B ) -module is a triple( E, π, D ) satisfying the following condition: • E is a countably generated Z -graded G -equivariant Hilbert B -module; • π : A → L B ( E ) is an even G -equivariant ∗ -homomorphism; • D : E ⊇ dom( D ) → E is a densely defined odd regular self adjoint operator satisfying thefollowing conditions: 8 There exists a ∗ -subalgebra A (cid:48) ⊆ A such that every a ∈ A (cid:48) satisfies the following condi-tions: π ( a ) preserves dom( D ) and [ π ( a ) , D ] is bounded; – π ( a )(1 + D ) − ∈ K B ( E ); – G preserves dom( D ). The map G (cid:51) g (cid:55)→ g ( D ) − D is L B ( E )-valued, and it is norm-continuous.When D satisfies g ( D ) = D , the Kasparov module ( E, π, D ) is said to be actually equivariant .The central tool of KK -theory is the Kasparov product. There is a criterion to be a Kasparovproduct in the unbounded picture [Kuc]. For bimodules E , E and e ∈ E , we define T e : E → E (cid:98) ⊗ E by e (cid:55)→ e (cid:98) ⊗ e . Proposition 2.7 ([Kuc]) . Let G be a locally compact second countable Hausdorff group, and let A , A , B be separable G - C ∗ -algebras. Let ( E , π , D ) be an unbounded G -equivariant Kasparov ( A, A ) -module, and ( E , π , D ) be an unbounded G -equivariant Kasparov ( A , B ) -module. x and y are the corresponding KK -elements of ( E , π , D ) and ( E , π , D ) , respectively. A G -equivariantKasparov ( A, B ) -module ( E (cid:98) ⊗ E , D ) is a representative of the Kasparov product of x and y , if thefollowing conditions are fulfilled: • For all v in some dense subset of AE , the graded commutator (cid:20)(cid:18) D D (cid:19) , (cid:18) T v T ∗ v (cid:19)(cid:21) is bounded on dom( D ⊕ D ) . We call it the connection condition . • dom( D ) is contained in dom( D (cid:98) ⊗ id) . • There exists κ ∈ R such that (cid:0) id (cid:98) ⊗ D ( e ) (cid:12)(cid:12) D ( e ) (cid:1) + (cid:0) D ( e ) (cid:12)(cid:12) id (cid:98) ⊗ D ( e ) (cid:1) ≥ κ ( e | e ) for any e ∈ dom( D ) . We call it the positivity condition . We recall R KK -theory which is a generalization of KK -theory. A generalization of it to infinite-dimensional manifolds is the central object of the present paper. Definition 2.8.
Let G be a locally compact second countable Hausdorff group and let X be a σ -compact locally compact Hausdorff space equipped with a continuous G -action.(1) A C ∗ -algebra A is a C ( X ) - C ∗ -algebra if it admits a ∗ -homomorphism φ : C ( X ) → Z ( M ( A )) satisfying φ ( C ( X )) A = A . It is also called an X (cid:111) { e } - C ∗ -algebra . In the following, weregard this homomorphism as a left module structure and φ ( f ) a is denoted by f · a .(2) In addition, if A is a G - C ∗ -algebra and φ : C ( X ) → Z ( M ( A )) is G -equivariant, A iscalled a G - C ( X ) - C ∗ -algebra . It is also called an X (cid:111) G - C ∗ -algebra or an X (cid:111) G -equivariant C ∗ -algebra . Remarks . (1) The above concept is generalized to groupoid equivariant C ∗ -algebras in [LG].Our terminology “ X (cid:111) G -equivariant C ∗ -algebra” comes from it. In fact, a continuous G -action on X defines a topological groupoid X (cid:111) G .(2) When we need to emphasize a C ∗ -algebra A does not have a C ( X )-structure (or we ignoreit), we call A a single C ∗ -algebra . This is because we regard a C ( X )-algebra as a field of C ∗ -algebras parameterized by X in the present paper.9 efinition 2.10 ([Kas2, Definition 2.19]) . Let G be a locally compact second countable Hausdorffgroup and let X be a σ -compact locally compact Hausdorff space equipped with a continuous G -action. Let A, B be X (cid:111) G - C ∗ -algebras. A G -equivariant Kasparov ( A, B )-module (
E, π, F ) is saidto be X (cid:111) G -equivariant if it satisfies the condition π ( f · a )( e · b ) = π ( a )( e · ( f · b )) . A homotopy between two X (cid:111) G -equivariant Kasparov ( A, B )-modules is an X (cid:111) G -equivariantKasparov ( A, BI )-module by the obvious X (cid:111) G -algebra structure on BI . The set of homotopyclasses of X (cid:111) G -equivariant Kasparov ( A, BI )-modules is denoted by R KK G ( X ; A, B ).We can define the concept of Kasparov products on R KK -theory [Kas2, Proposition 2.21]. Wecan deal with R KK -theory in the unbounded picture in an obvious way. We omit the details.We recall several fundamental and necessary constructions, emphasizing the relationship between KK -theory and R KK -theory. Definition 2.11.
Let G be a locally compact second countable Hausdorff group and let X be a σ -compact locally compact Hausdorff space equipped with a continuous G -action. Let A, B be X (cid:111) G -equivariant C ∗ -algebras.(1) We define the forgetful homomorphism fgt : R KK G ( X ; A, B ) → KK G ( A, B )by forgetting the C ( X )-module structure.(2) Let D, D , D be G -equivariant C ∗ -algebras. We define σ D : R KK G ( X ; A, B ) → R KK G ( X ; D (cid:98) ⊗ A, D (cid:98) ⊗ B )by ( E, π, F ) (cid:55)→ ( D (cid:98) ⊗ E, id (cid:98) ⊗ π, id (cid:98) ⊗ F ), and we define σ X,A : KK G ( D , D ) → R KK G ( X ; D (cid:98) ⊗ A, D (cid:98) ⊗ A )by ( E, π, F ) (cid:55)→ ( E (cid:98) ⊗ A, π (cid:98) ⊗ id , F (cid:98) ⊗ id). Remarks . (1) When we define a C ( X )-algebra structure on the tensor product of two C ( X )-algebras A and B by that of A , we denote it by A (cid:58) (cid:98) ⊗ B with a wave underline. For example, we willencounter both of C ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) and C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X ) to define local Bott elements.(2) We can also define σ D : R KK G ( X ; A, B ) → R KK G ( X ; A (cid:98) ⊗ D, B (cid:98) ⊗ D ) in an obvious way.When we distinguish these two homomorphisms, we use the symbol σ iD , where D is tensored at the i -th factor, although there are no differences between them.(3) All the single C ∗ -algebras which we will encounter in the present paper are nuclear, and thetensor products are uniquely determined.At first sight, R KK -theory is almost the same with KK -theory. However, R KK -theory hasa strong topological flavor. Roughly speaking, an element of R KK G ( X ; A, B ) is a “ G -equivariantfamily of Kasparov modules parameterized by X ”. We briefly explain this idea. With this observa-tion, we will formulate a non-locally compact version of R KK -theory in Section 4.3.Let B be a C ( X )- C ∗ -algebra. Then, we can define a C ∗ -algebra B x by B/C x B , where C x isthe ideal of C ( X ) consisting of f ∈ C ( X ) such that f ( x ) = 0. Thus, B is, roughly speaking,10he “continuous section algebra” of this field. The field given in this say is called an upper semi-continuous field of C ∗ -algebras over X (u.s.c. field for short). Conversely, a u.s.c. field of C ∗ -algebras over X defines a C ( X )- C ∗ -algebra, and these two operations are mutually inverse. See[Nil, Blan] for details on this result.If E is a Hilbert B -module, we can define the fiber of E at x by E x := E/EB x . Then, a B -module homomorphism F on E defines a B x -module homomorphism on E x , because F preserves EB x . It is denoted by F x .Let A be a C ( X )- C ∗ -algebra. Suppose that a homomorphism π : A → L B ( E ) satisfying π ( f a )( eb ) = π ( a )( e ( f b )) is given. Then, the induced operator π ( a ) x is determined by a x . Infact, if a x = 0, we can approximate it by finite sums of f a (cid:48) for f ∈ C x and a (cid:48) ∈ A . Since π ( f a (cid:48) )( eb ) = π ( a (cid:48) )( e ) · f b ∈ EB x , π ( a )( E ) ⊆ EB x , and hence π ( a ) x = 0. Consequently, π determinesa family of ∗ -homomorphisms { π x : A x → L B x ( E x ) } x ∈ X .Combining the preceding three paragraphs, we find that an X (cid:111) { e } -equivariant Kasparov module( E, π, F ) defines a “continuous family” { ( E x , π x , F x ) } x ∈ X , where E x is a Hilbert B x -module, π x : A x → L B x ( E x ) is a ∗ -homomorphism, and F x ∈ L B x ( E x ) satisfies the condition to be a Kasparovmodule. It is obvious that we can do the same constructions for equivariant situations.In order to describe R KK -element purely in the language of fields of Kasparov modules, thatis to say, in order to obtain an R KK -element from a field of Kasparov modules, we need to definethe concept of (u.s.c.) fields of adjointable operators and compact operators in the language offields of modules (strictly speaking, if we define the field of adjointable operators in a natural way,it is not upper semi-continuous). We postpone this problem until Section 4.3. Although we willuse the terminology “continuous field of Kasparov modules” in Section 3 to define several basic R KK -elements, this terminology stands for “the field defined from a Kasparov module” in theabove way. C ∗ -algebra S We prepare the Bott periodicity theorem and its reformulation using a ∗ -homomorphism givenin [HKT]. We will use it in order to generalize the “local Bott element” to infinite-dimensionalmanifolds. Definition 2.13.
Let V be a finite-dimensional Euclidean space. O ( V ) denotes the compact Liegroup consisting of orthogonal transformation on V .(1) We define the Bott element [ b V ] ∈ KK O ( V ) ( C , Cl τ ( V )) by the bounded transformation of( Cl τ ( V ) , , C ) , where 1 : C → C ( E ) ⊆ L C ( E ) ( C ( E )) is the ∗ -homomorphism given by z (cid:55)→ z id, and C is anunbounded multiplier on Cl τ ( V ) defined by the pointwise Clifford multiplication: For h ∈ Cl τ ( V )and v ∈ V , we define Ch ( v ) := v · h ( v ). Note that C is O ( V )-invariant and the above KK -elementis O ( V )-equivariant.(2) We define the Dirac element [ d V ] ∈ KK O ( V ) ( Cl τ ( V ) , C ) by the bounded transformationof (cid:0) L ( V, Cliff + ( T V )) , π, D (cid:1) , where π : Cl τ ( V ) → L C ( L ( V, Cliff + ( T V ))) is the ∗ -homomorphism given by left multiplication,and D is the Dirac operator given by (cid:88) i (cid:98) e i ∂∂x i { e i } and the corresponding coordinate { x i } , where (cid:98) e i v := ( − ∂v v · e i . Notethat D is O ( V )-invariant and the above KK -element is O ( V )-equivariant. Lemma 2.14. [ b V ] (cid:98) ⊗ Cl τ ( V ) [ d V ] = C and [ d V ] (cid:98) ⊗ C [ b V ] = Cl τ ( V ) . Consequently, Cl τ ( V ) is KK O ( V ) -equivalent to C . This result is called the Bott periodicity theorem .Proof.
One can check that the unbounded Kasparov module( L ( V, Cliff + ( T V )) , , D + C )satisfies the conditions to be a Kasparov product of [ b V ] and [ d V ]. By the spectral theory ofthe harmonic oscillator, we notice that ker( D + C ) is spanned by e − (cid:107) v (cid:107) ·
1, where “1” is themultiplicative identity of the Clifford algebra. Since it is rotation invariant, the Kasparov productis C in the sense of equivariant KK -theory.In order to prove the opposite direction, we use the well-known rotation trick. We refer to [Ati,HKT]. Let “flip” be the automorphism on Cl τ ( V ) (cid:98) ⊗ Cl τ ( V ) defined by f (cid:98) ⊗ f (cid:55)→ ( − ∂f ∂f f (cid:98) ⊗ f .It is homotopic to j ∗ (cid:98) ⊗ id by the rotation homotopy, where j : V → V is defined by x (cid:55)→ − x . Thus,[ d V ] (cid:98) ⊗ C [ b V ] = (cid:0) Cl τ ( V ) (cid:98) ⊗ C [ b V ] (cid:1) (cid:98) ⊗ Cl τ ( V ) (cid:98) ⊗ Cl τ ( V ) (cid:110) [flip] (cid:98) ⊗ Cl τ ( V ) (cid:98) ⊗ Cl τ ( V ) (cid:0) Cl τ ( V ) (cid:98) ⊗ C [ d V ] (cid:1)(cid:111) = (cid:0) Cl τ ( V ) (cid:98) ⊗ C [ b V ] (cid:1) (cid:98) ⊗ Cl τ ( V ) (cid:98) ⊗ Cl τ ( V ) (cid:0) [ j ∗ ] (cid:98) ⊗ C [ d V ] (cid:1) = [ j ∗ ] (cid:98) ⊗ C (cid:0) [ b V ] (cid:98) ⊗ Cl τ ( V ) [ d V ] (cid:1) = [ j ∗ ] . Since [ j ∗ ] (cid:98) ⊗ Cl τ ( V ) [ j ∗ ] = Cl τ ( V ) , we have the formula [ d V ] (cid:98) ⊗ C [ b V ] (cid:98) ⊗ Cl τ ( V ) [ j ∗ ] = Cl τ ( V ) . Since the leftinverse coincides with the right inverse, we obtain the result.We now explain another proof of the Bott periodicity following [HKT], and we prove the rela-tionship between these two proofs. We define A ( V ) by the graded tensor product S (cid:98) ⊗ Cl τ ( V ). Definition 2.15.
The Bott homomorphism β : S → A ( V ) is defined by the functional calculus f (cid:55)→ f ( X (cid:98) ⊗ id + id (cid:98) ⊗ C ) . It defines a KK -element [ β ] ∈ KK O ( V ) ( S , A ( V )).We will prove that [ β ] = σ S ([ b V ]) for even-dimensional V . Before that, we prove that S is KK -equivalent to the trivially graded C ∗ -algebra C . More concretely, we will construct four KK -elements [ d ± ] ∈ KK ( S , C ) and [ b ± ] ∈ KK ( C , S ), and we will prove that [ d S ] := [ d + ] ⊕ [ d − ] ∈ KK ( S , C ⊕ C ) and [ b S ] := [ b + ] ⊕ [ b − ] ∈ KK ( C ⊕ C , S ) are mutually inverse, where these ⊕ ’s are defined by the additivity of KK -theory with respect to both variables. For this aim, itsuffices to prove that [ b ± ] (cid:98) ⊗ [ d ± ] = C , and [ b ± ] (cid:98) ⊗ [ d ∓ ] = 0 (double signs are in the same order) and[ d S ] (cid:98) ⊗ C [ b S ] = S . We will prove the former two statements directly, and prove the last one by aneasy algebraic argument.Let us introduce several notations. On function spaces ( C or L ) on R , we define Z -gradinghomomorphisms (cid:15) by (cid:15) ( f )( t ) := f ( − t ). L ( R ) gr denotes the Z -graded Hilbert space L ( R ) by (cid:15) . Definition 2.16.
We define [ b ± ] ∈ KK ( C , S ) by the bounded transformations of( S , , ± X )12nd [ d ± ] ∈ KK ( S , C ) by the bounded transformations of (cid:18) L ( R ) gr , µ, ± ddx ◦ (cid:15) (cid:19) , where µ : S → L C ( L ( R ) gr ) is given by left multiplication. Lemma 2.17. [ b ± ] (cid:98) ⊗ [ d ± ] = C , and [ b ± ] (cid:98) ⊗ [ d ∓ ] = 0 .Proof. It is easy to prove that the representatives of [ b ± ] (cid:98) ⊗ [ d ± ] and [ b ± ] (cid:98) ⊗ [ d ∓ ] are given by (cid:18) L ( R ) gr , , ± (cid:18) ddx ◦ (cid:15) + X (cid:19)(cid:19) and (cid:18) L ( R ) gr , , ∓ (cid:18) ddx ◦ (cid:15) − X (cid:19)(cid:19) respectively. It suffices to prove that ind( ddx ◦ (cid:15) + X ) = 1 and ind( ddx ◦ (cid:15) − X ) = 0. Under thedecomposition with respect to the Z -grading L ( R ) gr = L ( R ) ev (cid:98) ⊕ L ( R ) od , the operators can bewritten as ddx ◦ (cid:15) + X = (cid:18) ddx − X ddx + X (cid:19) and ddx ◦ (cid:15) − X = (cid:18) ddx + X ddx − X (cid:19) . Thanks to the spectral theory of the harmonic oscillator, the index of the former operator is 1, andthat of the latter one is 0, because the creation operator is injective on the whole of L ( R ), and theannihilation operator has one-dimensional kernel which is contained in L ( R ) ev .It is clear that KK ( C , C ) ∼ = M ( Z ) as a ring. Therefore,[ b S ] (cid:98) ⊗ S [ d S ] = (cid:18) [ b + ] (cid:98) ⊗ S [ d + ] [ b + ] (cid:98) ⊗ S [ d − ][ b − ] (cid:98) ⊗ S [ d + ] [ b − ] (cid:98) ⊗ S [ d − ] (cid:19) = (cid:18) (cid:19) = C ∈ KK ( C , C ) . It is clear that [ d S ] (cid:98) ⊗ C [ b S ] = [ d + ] (cid:98) ⊗ C [ b + ] + [ d − ] (cid:98) ⊗ C [ b − ]. We will prove that it is S by computing KK ( S , S ) in detail. Lemma 2.18. KK ( S , S ) is isomorphic to Z and the base is given by { [ d + ] (cid:98) ⊗ C [ b + ] , [ d + ] (cid:98) ⊗ C [ b − ] , [ d − ] (cid:98) ⊗ C [ b + ] , [ d − ] (cid:98) ⊗ C [ b − ] } . S is given by [ d + ] (cid:98) ⊗ C [ b + ] + [ d − ] (cid:98) ⊗ C [ b − ] = [ d S ] (cid:98) ⊗ C [ b S ] .Proof. We prove that S can be written as an extension of easy C ∗ -algebras. We define ev : S → C by f (cid:55)→ f (0). It preserves the grading, because any odd function vanishes at 0. Next, we define ι : Cl τ (0 , ∞ ) → S as follows. First, we notice that Cl τ (0 , ∞ ) is isomorphic to C (0 , ∞ ) (cid:98) ⊕ C (0 , ∞ )with the product ( f (cid:98) ⊕ f ) · ( g (cid:98) ⊕ g ) := ( f g + f g ) (cid:98) ⊕ ( f g + f g ) . Under this identification, ι is defined by f (cid:98) ⊕ f (cid:55)→ [ t (cid:55)→ f ( | t | ) + sign( t ) f ( | t | )]. Then,0 → Cl τ (0 , ∞ ) ι −→ S ev −−→ C → KK ( S , S ), we note that KK ( C , S ) and KK ( S , C ) are isomorphic to Z ,and bases of these groups are given by { [ b + ] , [ b − ] } , and { [ d + ] , [ d − ] } , respectively. In fact, theformer statement is obvious from the above short exact sequence and the six-term exact sequencefor K -theory and K -homology. The latter one is obvious from the fact that [ b ± ] (cid:98) ⊗ S [ d ± ] = 1 and[ b ± ] (cid:98) ⊗ S [ d ∓ ] = 0 and the Kasparov product gives a bilinear form KK ( C , S ) × KK ( S , C ) → Z . By using the six-term exact sequence again, we notice that KK ( S , S ) ∼ = Z . we have four grouphomomorphisms from KK ( S , S ) to Z given by f ± , ± : x (cid:55)→ [ b ± ] (cid:98) ⊗ S x (cid:98) ⊗ S [ d ± ] (double signs are inarbitrary order). It is easy to see that { [ d ± ] (cid:98) ⊗ C [ b ± ] } and { f ± , ± } are dual bases (double signs are inarbitrary order).We must prove that S = [ d + ] (cid:98) ⊗ C [ b + ] + [ d − ] (cid:98) ⊗ C [ b − ]. It is equivalent to the four equations f ± , ± ( S ) = f ± , ± ([ d + ] (cid:98) ⊗ C [ b + ] + [ d − ] (cid:98) ⊗ C [ b − ])(double signs are in arbitrary order). They are satisfied by Lemma 2.17. Proposition 2.19.
When V is even-dimensional, σ S ([ b V ]) = [ β ] as elements of KK O ( V ) ( S , A ( V )) .Proof. Thanks to the results so far, it suffices to prove the following four equalities: (cid:8) [ b ± ] (cid:98) ⊗ S σ S ([ b V ]) (cid:9) (cid:98) ⊗ A ( V ) (cid:8) [ d ± ] (cid:98) ⊗ C [ d V ] (cid:9) = (cid:8) [ b ± ] (cid:98) ⊗ S [ β ] (cid:9) (cid:98) ⊗ A ( V ) (cid:8) [ d ± ] (cid:98) ⊗ C [ d V ] (cid:9) and (cid:8) [ b ± ] (cid:98) ⊗ S σ S ([ b V ]) (cid:9) (cid:98) ⊗ A ( V ) (cid:8) [ d ∓ ] (cid:98) ⊗ C [ d V ] (cid:9) = (cid:8) [ b ± ] (cid:98) ⊗ S [ β ] (cid:9) (cid:98) ⊗ A ( V ) (cid:8) [ d ∓ ] (cid:98) ⊗ C [ d V ] (cid:9) , (double signs are in the same order). Obviously, (cid:8) [ b ± ] (cid:98) ⊗ S σ S ([ b V ]) (cid:9) (cid:98) ⊗ A ( V ) (cid:8) [ d ± ] (cid:98) ⊗ C [ d V ] (cid:9) = C and (cid:8) [ b ± ] (cid:98) ⊗ S σ S ([ b V ]) (cid:9) (cid:98) ⊗ A ( V ) (cid:8) [ d ∓ ] (cid:98) ⊗ C [ d V ] (cid:9) = 0 . We need to compute the right hand sides. Let us compute (cid:8) [ b ± ] (cid:98) ⊗ S [ β ] (cid:9) . In the unbounded picture,they are given by (cid:0) A ( V ) , , ± ( X (cid:98) ⊗ id + id (cid:98) ⊗ C ) (cid:1) , respectively. Thus, we have the formulas [ b + ] (cid:98) ⊗ S [ β ] = [ b + ] (cid:98) ⊗ C [ b V ] and [ b − ] (cid:98) ⊗ S [ β ] = [ b − ] (cid:98) ⊗ C ( Cl τ ( V ) , , − C ).This means that (cid:8) [ b ± ] (cid:98) ⊗ S [ β ] (cid:9) (cid:98) ⊗ A ( V ) (cid:8) [ d ∓ ] (cid:98) ⊗ C [ d V ] (cid:9) = 0 . Moreover, (cid:8) [ b ± ] (cid:98) ⊗ S [ β ] (cid:9) (cid:98) ⊗ A ( V ) (cid:8) [ d ± ] (cid:98) ⊗ C [ d V ] (cid:9) = [( Cl τ ( V ) , , ± C )] (cid:98) ⊗ Cl τ ( V ) [ d V ] = ind( D ± C ) . Thanks to the spectral theory of the harmonic oscillator, we can compute the kernel of theseoperators: ker( D + C ) = C e − (cid:107) v (cid:107) · D − C ) = C e − (cid:107) v (cid:107) · vol. The index of the formerone is 1 as we have mentioned, and that of the latter one is ( − dim( V ) . Thus, the statement holdswhen V is even-dimensional. Remark . The most natural homomorphism between KK G ( A, B ) and KK G ( S (cid:98) ⊗ A, S (cid:98) ⊗ B ) is σ S ,which will appear in the following section. Unfortunately, it can be not an isomorphism, because KK G ( A, B ) and KK G ( S (cid:98) ⊗ A, S (cid:98) ⊗ B ) can be not isomorphic. Roughly speaking, this operation isgiven by KK G ( A, B ) (cid:51) x (cid:55)→ (cid:18) x x (cid:19) ∈ M ( KK G ( A, B )) ∼ = KK G ( S (cid:98) ⊗ A, S (cid:98) ⊗ B )14or G - C ∗ -algebras A and B .There is a trick to solve this non-triviality using the crossed product by Z (although we do notconsider this kind of tricks after this remark). By the grading homomorphism (cid:15) S , we can define a Z -action on S . We can also define a Z -action on C by ( a, b ) (cid:55)→ ( b, a ). Then, [ b S ] and [ d S ] are Z -equivariant KK -elements. In fact, the Z -action on S ⊕ S is given by ( f , f ) (cid:55)→ ( (cid:15) S ( f ) , (cid:15) S ( f )),and similarly for L ( R ) gr ⊕ L ( R ) gr . Thus, by the operation j Z and a KK -equivalence C (cid:111) Z ∼ = C ,we have a KK -equivalence S (cid:111) Z ∼ = C . Thus, the operation σ S ε (cid:111)Z : KK G ( A, B ) → KK G ( S ε (cid:111) Z (cid:98) ⊗ A, S ε (cid:111) Z (cid:98) ⊗ B )is isomorphic.Suppose that a C ∗ -algebra B can be regarded as a “graded suspension of A ”. The C ∗ -algebra ofa Hilbert space [HKT] is an example of such a C ∗ -algebra. If we can define a Z -action on B whichcan be regarded as “ (cid:15) S (cid:98) ⊗ id on S (cid:98) ⊗A ” in some sense, it will be possible define the “ K -homology of A ” by KK ( B (cid:111) Z , S (cid:111) Z ), by “using” the preceding paragraph. In this section, we recall the index theorem for complete Riemannian manifolds with proper cocom-pact actions [Kas1, Kas3]. We will also prove several necessary results. Then, we will reformulateit for even-dimensional
Spin c -manifolds. In particular, we will rewrite the KK -theoretical topo-logical index following the idea explained at the end of Section 2.2. Finally, we will give a twistedequivariant versions of them. We recall the precise statement of the index theorem [Kas1, Kas3] and we prepare necessary KK -elements. Let X be a complete Riemannian manifold and let G be a locally compact secondcountable Hausdorff group acting on X in an isometric, proper, cocompact way. Let W be a G -equivariant Clifford bundle: A G -equivariant Hermitian bundle over X equipped with a G -equivariant bundle homomorphism c : T X → End( W ) such that c ( v ) is skew-adjoint and c ( v ) = −(cid:107) v (cid:107) id W . Take a G -equivariant Dirac operator D on W . In this setting, we have the followingthree objects. µ denotes the modular function of G . Definition 3.1. (1) A dense subspace C c ( X, W ) of L ( X, W ) admits a pre-Hilbert C c ( G )-modulestructure ( s | s ) C(cid:111) G ( g ) := µ ( g ) − / (cid:90) X (cid:0) s ( x ) (cid:12)(cid:12) g · [ s ( g − · x )] (cid:1) W x dx,s · b := (cid:90) G µ ( g ) − / ( g · s ) b ( g − ) dg for s , s , s ∈ C c ( X, W ) and b ∈ C c ( G ). The pair of the completion of this pre-Hilbert module andthe extension of the operator D : C ∞ c ( X, W ) (cid:8) defines an element of KK ( C , C (cid:111) G ) and it is calledthe analytic index of D , and it is denoted by ind C(cid:111) G ( D ). See the beginning of [Kas3, Section 5].(2) The triple ( L ( X, W ) , π, D ) is an unbounded G -equivariant Kasparov ( C ( X ) , C )-module,where π is given by left multiplication. The corresponding KK -element is denoted by [ D ] ∈ KK G ( C ( X ) , C ), and it is called the index element of D . See [Kas3, Lemma 3.7].153) The triple ( C ( X, W ) , π,
0) is an unbounded X (cid:111) G -equivariant Kasparov ( C ( X ) , Cl τ ( X ))-module, where π is given by left multiplication. Note that C ( X, W ) admits a left
Cliff − ( T X )-module structure, and thus it admits a right Hilbert Cl τ ( X )-module structure, thanks to Lemma2.4. The corresponding R KK -element is denoted by [ σ ClD ] ∈ R KK G ( X ; C ( X ) , Cl τ ( X )), and it iscalled the Clifford symbol element of D . See [Kas3, Definition 3.8].The following is the precise statement of the index theorem. The task of the remainder of thissubsection is to explain the details on this result, and to prove several related equalities. Theorem 3.2 ([Kas1, Kas3]) . (1) ind C(cid:111) G ( D ) is determined by [ D ] by a KK -theoretical procedurecalled the analytic assembly map . (2) KK G ( C ( X ) , C ) and R KK G ( X ; C ( X ) , Cl τ ( X )) are isomorphic ( Poincar´e duality ). Un-der this isomorphism, [ D ] corresponds to [ σ ClD ] . (3) Consequently, ind
C(cid:111) G ( D ) is completely determined by [ σ ClD ] . The explicit formula to compute ind C(cid:111) G ( D ) from [ σ ClD ] is called the topological assembly map in the present paper. Let us begin with Poincar´e duality. The geometrical setting defines the following KK -elements.See also Definition 2.13 (2). Definition 3.3. (1) The Dirac element of X is defined by[ d X ] := [( L ( X, Cliff + ( T X )) , π, D )] ∈ KK G ( Cl τ ( X ) , C ) , where D = (cid:80) i (cid:98) e i ∇ LC e i for an orthonomal base { e i } of each tangent space and the Levi-Civitaconnection ∇ LC , and π is given by the Clifford multiplication by the left. See [Kas3, Definition 2.2].(2) There is a positive real ε such that the injectivity radius at any x ∈ X is greater than 2 ε ,because the group action is cocompact. Let U x be the ε -ball centered at x in X and let Θ x be thevector field on U x defined byΘ x ( y ) := − log y ( x ) = ( d exp x ) log x ( y ) Eul x (log x ( y )) = “ −→ xy ” , where log x : U x → T x X is the local inverse of exp x : T x X → X , and Eul x is the Euler vectorfield on T x X centered at the origin. The local Bott element [Θ X, ] is defined by the element of R KK G ( X ; C ( X ) , C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X )) represented by the family of Kasparov modules (cid:0) { Cl τ ( U x ) } x ∈ X , { x } x ∈ X , { ε − Θ x } x ∈ X (cid:1) , where 1 x denotes the homomorphism C ( X ) x ∼ = C → L Cl τ ( X ) ( Cl τ ( U x )) given by z (cid:55)→ z id, andΘ x denotes left multiplication by Θ x , namely [Θ x f ]( y ) := Θ x ( y ) f ( y ) . This family is organizedinto a single Kasparov module ( C ( U, Cliff + ( T fiber U ) , π, Θ), where U := { ( x, y ) | x ∈ X, y ∈ U x } is a fiber bundle over X equipped with the diagonal G -action g · ( x, y ) = ( g · x, g · y ), T fiber U = (cid:96) x ∈ X (cid:96) u ∈ U x T u U x , and π is left multiplication. When we define a C ( X )-algebra structure on C ( X ) (cid:98) ⊗ Cl τ ( X ) by the product over the second tensor multiple, we obtain another local Bottelement [Θ X, ] ∈ R KK G ( X ; C ( X ) , C ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ). See [Kas3, Definition 2.3]. Remark . Local Bott elements come from other R KK -elements of the “narrower” C ∗ -algebra:[Θ X, ] ∈ R KK G (cid:18) X ; C ( X ) , C ( U ) · (cid:18) C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X ) (cid:19)(cid:19) , X, ] ∈ R KK G (cid:18) X ; C ( X ) , C ( U ) · (cid:18) C ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:19)(cid:19) . Strictly speaking, the local Bott elements given in the Definition should be denoted by [Θ X, ] (cid:98) ⊗ [ ι U,X × X ],and similarly for [Θ X, ], where the canonical inclusion C ( U ) · (cid:18) C ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:19) (cid:44) → C ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) is denoted by ι U,X × X . In the present paper, by an abuse of notation, we use the same symbols torepresent them unless we need to emphasize the difference, namely “[Θ X, ] (cid:98) ⊗ [ ι U,X × X ] = [Θ X, ]” and“[Θ X, ] (cid:98) ⊗ [ ι U,X × X ] = [Θ X, ]”. Fact 3.5 (Theorem 4.1 and Theorem 4.6 of [Kas3]) . We define a homomorphism PD : KK G ( C ( X ) , C ) →R KK G ( X ; C ( X ) , Cl τ ( X )) byPD( x ) := [Θ X, ] (cid:98) ⊗ X,C ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:8) σ X,Cl τ ( X ) ( x ) (cid:9) , and we call it the Poincar´e duality homomorphism .(1) PD is isomorphic and its inverse is given by the following composition: R KK G ( X ; C ( X ) , Cl τ ( X )) fgt −→ KK G ( C ( X ) , Cl τ ( X )) − (cid:98) ⊗ [ d X ] −−−−−→ KK G ( C ( X ) , C ) . (2) PD([ D ]) coincides with [ σ ClD ].Let us explain the assembly maps.
Definition 3.6 (Theorem 3.14 of [Kas2]) . (1) Let A and B be X (cid:111) G - C ∗ -algebras, and let ( E, π, F )be an X (cid:111) G -equivariant Kasparov ( A, B )-module. C c ( G, E ) admits a pre-Hilbert C c ( G, B )-modulestructure and a ∗ -homomorphism π (cid:111) lt : C c ( G, A ) → L C c ( G,B ) ( C c ( G, E )) given by the followingoperations: For e, e , e ∈ C c ( G, E ), a ∈ A (cid:111) G , b ∈ B (cid:111) G and g ∈ G ,( e | e ) C c ( G,B ) ( g ) := (cid:90) G g (cid:48)− . (cid:0) e ( g (cid:48) ) (cid:12)(cid:12) e ( g (cid:48) g ) (cid:1) E dg (cid:48) , [ e · b ]( g ) := (cid:90) G e ( g (cid:48) ) ga (cid:48) . [ b ( g (cid:48)− g )] dg (cid:48) ,π (cid:111) lt( a )( e )( g ) := (cid:90) G π ( a ( g (cid:48) )) g (cid:48) . [ e ( g (cid:48)− g )] dg (cid:48) . By the completion with respect to the above inner product, we obtain a Hilbert B (cid:111) G -module E (cid:111) G and a ∗ -homomorphism π (cid:111) lt : A (cid:111) G → L B (cid:111) G ( E (cid:111) G ). These operations are compatiblewith the C ( X/G )-algebra structures on A (cid:111) G and B (cid:111) G . We define an operator (cid:101) F on E (cid:111) G by [ (cid:101) F e ]( g ) := F [ e ( g )]. Then, the triple ( E (cid:111) G, π (cid:111) lt , (cid:101) T ) is an ( X/G ) (cid:111) { e } -equivariant Kasparov( A (cid:111) G, B (cid:111) G )-module. The correspondence ( E, π, F ) (cid:55)→ ( E (cid:111) G, π (cid:111) lt , (cid:101) F ) induces a homomorphism j G : R KK G ( X ; A, B ) → R KK ( X/G, A (cid:111)
G, B (cid:111) G )and it is called the descent homomorphism .(2) By the same construction, we can define a homomorphism j G : KK G ( A, B ) → KK ( A (cid:111) G, B (cid:111) G )for arbitrary G - C ∗ -algebras A, B . It is called the descent homomorphism .17he following is obvious by definition.
Lemma 3.7.
The following diagram commutes: R KK G ( X ; A, B ) j G −−−−→ R KK ( X/G, A (cid:111)
G, B (cid:111) G ) fgt (cid:121) (cid:121) fgt KK G ( A, B ) j G −−−−→ KK ( A (cid:111) G, B (cid:111) G ) . Since the G -action on X is proper and cocompact, there is a compactly supported continuousfunction c : X → R ≥ satisfying (cid:82) G c ( g − · x ) dg = 1 for every x ∈ X . Such a function is called a cut-off function . It is easy to see that all such functions are homotopic by the linear homotopy. Definition 3.8. (1) A cut-off function c defines a projection c of C ( X ) (cid:111) G by { c ( g ) } ( x ) := µ ( g ) − / (cid:112) c ( x ) c ( g − · x ). The corresponding KK -element is denoted by [ c X ] ∈ KK ( C , C ( X ) (cid:111) G ).(2) The analytic assembly map µ G : KK G ( C ( X ) , C ) → KK ( C , C (cid:111) G ) is defined by µ G ( x ) := [ c X ] (cid:98) ⊗ C ( X ) (cid:111) G { j G ( x ) } . (3) The topological assembly map ν G : R KK G ( X ; C ( X ) , Cl τ ( X )) → KK ( C , C (cid:111) G ) isdefined by ν G ( y ) := [ c X ] (cid:98) ⊗ C ( X ) (cid:111) G { fgt ◦ j G ( y ) } (cid:98) ⊗ Cl τ ( X ) (cid:111) G { j G ([ d X ]) } . In Theorem 3.2 (3), we stated that there is an explicit formula to compute the analytic indexof D from its symbol. It is the topological assembly map. The proof is quite formal. Proposition 3.9 ([Kas3, Theorem 5.6]) . µ G ( x ) = ν G ◦ PD( x ) .Proof. we have the following commutative diagram: KK G ( C ( X ) , C ) − (cid:98) ⊗ [ d X ] ←−−−−− KK G ( C ( X ) , Cl τ ( X )) fgt ←−−−− R KK G ( X ; C ( X ) , Cl τ ( X )) j G (cid:121) (cid:121) j G (cid:121) j G KK ( C ( X ) (cid:111) G, C (cid:111) G ) − (cid:98) ⊗ j G [ d X ] ←−−−−−−− KK ( C ( X ) (cid:111) G, Cl τ ( X ) (cid:111) G ) fgt ←−−−− R KK ( X/G ; C ( X ) (cid:111) G, Cl τ ( X ) (cid:111) G ) [ c X ] (cid:98) ⊗− (cid:121) [ c X ] (cid:98) ⊗− (cid:121) KK ( C , C (cid:111) G ) − (cid:98) ⊗ j G [ d X ] ←−−−−−−− KK ( C , Cl τ ( X ) (cid:111) G ) . The top right square commutes by Lemma 3.7. The top left square commutes thanks to [Kas2,Theorem 3.11]. The bottom left square commutes thanks to the associativity of the Kasparovproduct. Thus, ν G ( y ) = µ G (fgt( y ) (cid:98) ⊗ [ d X ]) for every y ∈ R KK G ( X ; C ( X ) , Cl τ ( X )). Thanks to Fact3.5 (2), we obtain the result.Related to Fact 3.5 (1), we have the following result. We will use a corollary of it later. For thedetails on differential geometrical facts used in the following, see [KN1, KN2, Sak] for example. Lemma 3.10 (See [Kas3, Theorem 2.4]) . [Θ X, ] ∈ R KK G (cid:18) X ; C ( X ) , C ( U ) · (cid:18) C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X ) (cid:19)(cid:19) and [ d X ] are mutually inverse in the following sense: [Θ X, ] (cid:98) ⊗ X,C ( U )[ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X )] (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9) = X,C ( X ) (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9) (cid:98) ⊗ X,C ( X ) [Θ X, ] = X,C ( U ) · ( Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) ) . roof. The former is proved in [Kas2, Theorem 4.8]. We deduce the latter one from the former oneby the rotation trick.Let U × X U := (cid:8) ( u, v, x ) ∈ X | u, v ∈ U x (cid:9) . On the C ( X )-algebra C ( U × X U ) · (cid:26) Cl τ ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:27) , we can define two automorphisms: flip ∗ is defined by the pullback of the diffeomorphism flip :( u, v, x ) (cid:55)→ ( v, u, x ), and j ∗ is defined by that of j : ( u, v, x ) (cid:55)→ ( − u, v, x ), where − u := exp x ( − log x ( u )).For the moment, we suppose that [flip ∗ ] = [ j ∗ ] in the R KK -group, which will be proved in the fol-lowing paragraphs. We use a similar argument of the proof of Lemma 2.14. (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9) (cid:98) ⊗ X,C ( X ) [Θ X, ]= (cid:26) X,C ( U ) · ( Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) ) (cid:98) ⊗ X,C ( X ) [Θ X, ] (cid:27) (cid:98) ⊗ X,C ( U × X U ) · [ Cl τ ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) ] (cid:26) [flip ∗ ] (cid:98) ⊗ X,C ( U × X U ) · [ Cl τ ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) ] X,C ( U ) · ( Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) ) (cid:98) ⊗ X,C ( X ) [ ι U,X × X ] (cid:98) ⊗ (cid:8) σ X,C ( X ) [ d X ] (cid:9)(cid:27) = (cid:26) X,C ( U ) · ( Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) ) (cid:98) ⊗ X,C ( X ) [Θ X, ] (cid:27) (cid:98) ⊗ X,C ( U × X U ) · [ Cl τ ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) ] (cid:26) [ j ∗ ] (cid:98) ⊗ X,C ( U × X U ) · [ Cl τ ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) ] [ ι U,X × X ] (cid:98) ⊗ (cid:8) σ X,C ( X ) [ d X ] (cid:9)(cid:27) = [ j ∗ ] (cid:98) ⊗ X,C ( U × X U ) · [ Cl τ ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) ] (cid:26) [Θ X, ] (cid:98) ⊗ X,C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X ) (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9)(cid:27) = [ j ∗ ] . Thus, (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9) (cid:98) ⊗ X,C ( X ) [Θ X, ] is an isomorphism in R KK -theory. Since the leftand right inverse are the same, we obtain the result.Let us prove the equality [flip ∗ ] = [ j ∗ ]. We would like to use the “rotation homotopy”. Takeother neighborhoods of the diagonal U √ ε := (cid:8) ( u, v, x ) ∈ X | ρ ( u, x ) + ρ ( v, x ) < ε (cid:9) and U ε := (cid:8) ( u, v, x ) ∈ X | ρ ( u, x ) + ρ ( v, x ) < ε (cid:9) . Then, we can define continuous maps flip U √ ε , j U √ ε , flip U ε and j U ε in an obvious way. Each spaceadmits a fiber bundle structure over X by the projection onto the third factor, and it is obviousthat U ε ⊆ U × X U ⊆ U √ ε . On U √ ε , we can consider the rotation homotopy U √ ε × [0 , π/ (cid:51) (( u, v, x ) , t ) (cid:55)→ ( u cos t − v sin t, u sin t + v cos t, x ) ∈ U √ ε , between flip U √ ε and j U √ ε , where u cos t − v sin t is defined by the exponential map on the fiber U √ ε at x : exp U √ ε | x ( x,x ) (log x ( u ) cos t − log x ( v ) sin t ), and similarly for u sin t + v cos t . Thus, [flip ∗ U √ ε ] = [ j ∗ U √ ε ]in the R KK -group.The new neighborhoods of the diagonal are related to U × X U by U ε i −→ U × X U i −→ U √ ε .Thus, we have G -equivariant commutative diagrams19 × X U i −−−−→ U √ ε flip (cid:121) (cid:121) flip U √ ε U × X U i −−−−→ U √ ε , and U × X U i −−−−→ U √ εj (cid:121) (cid:121) j U √ ε U × X U i −−−−→ U √ ε . These induce a G -equivariant homotopy equivalence i ◦ flip = flip U √ ε ◦ i ∼ j U √ ε ◦ i = i ◦ j, and hence [flip ∗ ] (cid:98) ⊗ [ i ∗ ] = [ j ∗ ] (cid:98) ⊗ [ i ∗ ] . Thus, it suffices to prove that [ i ∗ ] is invertible. It suffices to finda homotopy inverse. Let r : U √ ε → U ε be the map ( u, v, x ) (cid:55)→ (2 − / u, − / v, x ). It is equivariantand continuous because the exponential maps are equivariant with respect to the isometric groupaction. We define k := i ◦ r : U √ ε → U × X U . It is a homotopy inverse of i . In fact, both of i ◦ k and k ◦ i are given by the same formula ( u, v, x ) (cid:55)→ (2 − / u, − / v, x ). These are homotopicto the identity by H t ( u, v, x ) := ((1 − t + 2 − / t ) u, (1 − t + 2 − / t ) v, x ). Corollary 3.11.
Let [ − Θ X, ] be the R KK -element defined by (cid:0) { Cl τ ( U x ) } x ∈ X , { x } x ∈ X , {− ε − Θ x } x ∈ X (cid:1) . If X is even-dimensional and orientable, [ − Θ X, ] = [Θ X, ] .Proof. By the above proposition, it suffices to prove that[ − Θ X, ] (cid:98) ⊗ X,C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X ) (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9) = [Θ X, ] (cid:98) ⊗ X,C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X ) (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9) . The right hand side is given by the bounded transformation of (cid:32)(cid:8) L ( U x , Cliff + ( T U x )) (cid:9) x ∈ X , { x } x ∈ X , (cid:40) D − ε − Θ x (cid:112) − ε − Θ x (cid:41) x ∈ X (cid:33) , where D is the Dirac operator given by the same formula of Definition 3.3.Let us compare it with [Θ X, ] (cid:98) ⊗ X,C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X ) (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9) . Fix an orientation of X . On the Clifford algebra of an oriented vector space, we can define the volume element by e e · · · e dim( V ) for an oriented orthonormal base { e , e , · · · , e dim( V ) } . The same construction oneach tangent space defines a global section on Cliff + ( T U x ), and it is denoted by { vol x } x ∈ X . Asimple computation on the Clifford algebra gives the following formulas:vol ◦ e i = ( − dim( X ) − e i vol and vol ◦ (cid:98) e i = ( − dim( X ) (cid:98) e i ◦ vol . Note that vol x ∈ Cl τ ( U x ) commutes with ∇ LC v for any tangent vector v . Therefore, since dim( X )is even, (cid:32) D − ε − Θ x (cid:112) − ε − Θ x (cid:33) ◦ vol x = vol x ◦ (cid:32) D + ε − Θ x (cid:112) − ε − Θ x (cid:33) . Thus, the family { vol x } x ∈ X gives an isomorphism between [ − Θ X, ] (cid:98) ⊗ X,C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X ) (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9) and [Θ X, ] (cid:98) ⊗ X,C ( X ) (cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗ Cl τ ( X ) (cid:8) [ ι U,X × X ] (cid:98) ⊗ σ X,C ( X ) [ d X ] (cid:9) . .2 A reformulation of the index theorem for even-dimensional Spin c -manifolds There are no C ∗ -algebras which play roles of C -algebras for infinite-dimensional manifolds. How-ever, there are constructions of C ∗ -algebras which play roles of “the suspension of the Cliffordalgebra-valued function algebra of infinite-dimensional manifolds” [HKT, HK, Tro, DT, GWY, Yu].Although C and Cl τ are different, they are R KK -equivalent for even-dimensional Spin c -manifolds. Thus, there is a possibility to generalize KK -theoretical results on an even-dimensional Spin c -manifolds formulated in the language of C -algebra, to some infinite-dimensional manifoldsby using the C ∗ -algebras of Hilbert manifolds. This is the fundamental idea of [T4]. In this pa-per, the author rewrote the latter half of the inverse of the Poincar´e duality homomorphism, andformulate a substitute for it for proper LT -spaces. Moreover, he computed it for a special case.In this subsection, we follow the same idea. Basically, we need to replace a single C ∗ -algebra C ( X ) with Cl τ ( X ). Moreover, since it is easy to introduce the concept of an infinite-dimensionalversion of the “ C ( X )-algebra C ( X )”, we will remove the “ C ( X )-algebra Cl τ ( X )” from the theory.Let us begin with the R KK -elements to identify C and Cl τ at the KK -theory level. Asmentioned in Fact 2.2, an irreducible left Cliff − ( V )-module S admits a right Hilbert Cliff + ( V )-module structure, and S ∗ admits a left Cliff + ( V )-module structure. Thus, the following two R KK -elements make sense. Definition 3.12.
Let X be an even-dimensional Riemannian Spin c -manifold equipped with anisometric proper cocompact G -action. We suppose that the G -action lifts to a Spinor bundle ( S, c ).Then its dual ( S ∗ , c ∗ ) is also G -equivariant.(1) C ( X, S ) has a Hilbert Cl τ ( X )-module structure by the above observation, and it admitsa left C ( X )-module structure given by [ π ( f ) s ]( x ) := f ( x ) s ( x ) for f ∈ C ( X ), s ∈ C ( X, S ) and x ∈ X . We define an R KK G -element [ S ] by[ S ] := ( C ( X, S ) , π, ∈ R KK G ( X ; C ( X ) , Cl τ ( X )) . (2) C ( X, S ∗ ) has a Hilbert C ( X )-module structure, and it admits a left Cl τ ( X )-module struc-ture given by [ c ∗ ( f ) s ]( x ) := c ∗ ( f ( x )) s ( x ) for f ∈ Cl τ ( X ), s ∈ C ( X, S ∗ ) and x ∈ X . We define an R KK G -element [ S ∗ ] by[ S ∗ ] := ( C ( X, S ∗ ) , c ∗ , ∈ R KK G ( X ; Cl τ ( X ) , C ( X )) . Fact 2.2 shows that Cl τ ( X ) ∼ = End( S ∗ ). Using it, one can prove the following result. Lemma 3.13. (1)
These two R KK G -elements give an R KK G -equivalence: [ S ] (cid:98) ⊗ X,Cl τ ( X ) [ S ∗ ] = 1 X,C ( X ) and [ S ∗ ] (cid:98) ⊗ X,C ( X ) [ S ] = 1 X,Cl τ ( X ) . (2) Consequently, fgt([ S ]) and fgt([ S ∗ ]) gives a KK G -equivalence between C ( X ) and Cl τ ( X ) . Using these R KK G - and KK G - equivalences, we reformulate the Kasparov index theorem. Wesuppose that the injectivity radius is greater than 2 ε everywhere (it is always possible if X admits anisometric cocompact group action), and we put A ( X ) := S ε (cid:98) ⊗ Cl τ ( X ). Although S ε is not essentialfor the following definition (and it is possible to replace ε with another number), it plays an essentialrole to reformulate the theory further using the Bott homomorphisms of Definition 2.15.21 efinition 3.14 (See also Section 2.5 of [T4]) . Let X be an even-dimensional complete Riemannian Spin c -manifold and let G be a locally compact second countable Hausdorff group acting on X inan isometric, proper and cocompact way. We assume that the G -action lifts to the Spinor bundle.Suppose that the injectivity radius is greater than 2 ε everywhere. Let W be a G -equivariant Cliffordbundle and let D be a G -equivariant Dirac operator on W .(1) We reformulate KK -elements appearing in the index theorem as follows: • [ (cid:101) D ] := σ S ε (cid:0) fgt[ S ∗ ] (cid:98) ⊗ [ D ] (cid:1) ∈ KK G ( A ( X ) , S ε ); • [ (cid:103) σ ClD ] := σ S ε (cid:0) [ σ ClD ] (cid:98) ⊗ [ S ∗ ] (cid:1) ∈ R KK G ( X ; S ε (cid:98) ⊗ C ( X ) , S ε (cid:98) ⊗ C ( X )); • [ (cid:102) d X ] := σ S ε ([ d X ]) ∈ KK G ( A ( X ) , S ε ); • [ (cid:102) c X ] := σ S ε ([ c X ]) ∈ KK ( S ε , S ε (cid:98) ⊗ [ C ( X ) (cid:111) G ]); • [ (cid:93) Θ X, ] := σ S ε ([Θ X, ]) ∈ R KK G ( X ; S ε (cid:98) ⊗ C ( X ) , C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗A ( X )); and • [ (cid:93) Θ X, ] := σ S ε (cid:16) [Θ X, ] (cid:98) ⊗ σ C ( X ) ([ S ∗ ]) (cid:98) ⊗ σ X,C ( X ) (fgt[ S ]) (cid:17) ∈ R KK G ( X ; S ε (cid:98) ⊗ C ( X ) , A ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ).(2) The corresponding homomorphisms to PD, µ G and ν G are denoted with tilde: • (cid:102) PD( (cid:101) x ) := [ (cid:93) Θ X, ] (cid:98) ⊗ (cid:16) σ X,C ( X ) ( (cid:101) x ) (cid:17) : KK ( A ( X ) , S ε ) → R KK G ( X ; S ε (cid:98) ⊗ C ( X ) , S ε (cid:98) ⊗ C ( X )); • (cid:102) µ G ( (cid:101) x ) := [ (cid:102) c X ] (cid:98) ⊗ σ S ε (fgt ( j G ([ S ]))) (cid:98) ⊗ j G ( (cid:101) x ) : KK ( A ( X ) , S ε ) → KK ( S ε , S ε (cid:111) G ); and • (cid:102) ν G ( (cid:101) y ) := [ (cid:102) c X ] (cid:98) ⊗ (fgt ◦ j G ([ (cid:101) y ])) (cid:98) ⊗ σ S ε (cid:8) j G (fgt[ S ] (cid:98) ⊗ [ d X ]) (cid:9) : R KK G ( X ; S ε (cid:98) ⊗ C ( X ) , S ε (cid:98) ⊗ C ( X )) → KK ( S ε , S ε (cid:111) G ). Proposition 3.15.
These reformulated objects satisfy the following index theorem type equalities: [ (cid:103) σ ClD ] = (cid:102)
PD[ (cid:101) D ] , [ (cid:101) D ] = fgt (cid:16) [ S ∗ ] (cid:98) ⊗ [ (cid:103) σ ClD ] (cid:98) ⊗ [ S ] (cid:17) (cid:98) ⊗ [ (cid:102) d X ] and (cid:102) µ G ([ (cid:101) D ]) = (cid:102) ν G ([ (cid:103) σ ClD ]) . Proof.
The following facts show the statement: The Kasparov product is associative; j G commuteswith the Kasparov product; fgt commutes with j G ; Fact 3.5; σ S ε commutes with the Kasparovproduct; σ S ε commutes with σ X,C ( X ) ’s; [ S ] (cid:98) ⊗ [ S ∗ ] = 1 C ( X ) ; [ S ∗ ] (cid:98) ⊗ [ S ] = 1 Cl τ ( X ) ; σ S ε ( µ G ([ D ])) = (cid:102) µ G ([ (cid:101) D ]) and σ S ε (cid:0) ν G ([ σ ClD ]) (cid:1) = (cid:102) ν G ([ (cid:103) σ ClD ]). We leave the details to the reader.
Remark . The formulas on the inverse of the Poincar´e duality and the analytic assembly mapare not satisfying, because the factor [ S ] or [ S ∗ ] appears. Since there is no C in the theory ofinfinite-dimensional manifolds, what we can formulate is not “fgt” itself but the composition of itand [ S ∗ ] (cid:98) ⊗ − (cid:98) ⊗ [ S ], and similarly for j . We do not study infinite-dimensional versions of them inthe present paper. We will mention this problem in Section 6.4 in order to explain what the nextproblem is. Notations 3.17.
From now on, the reformulated Poincar´e duality homomorphims and the assemblymaps are denoted without tildes: PD([ (cid:101) D ]) := (cid:102) PD([ (cid:101) D ]), µ G ([ (cid:101) D ]) := (cid:102) µ G ([ (cid:101) D ]) and ν G ([ (cid:103) σ ClD ]) := (cid:102) ν G ([ (cid:103) σ ClD ]) .
22n order to generalize the reformulated KK -elements to infinite-dimensional manifolds, we willrewrite the local Bott elements using Proposition 2.19 and Corollary 3.11. We begin with theexplicit description of [ (cid:93) Θ X, ] in the language of fields of Kasparov modules. By an abuse of notation, A ( U x ) denotes S ε (cid:98) ⊗ Cl τ ( U x ) (note that we define A ( X ) only for manifolds whose injectivity radiusis bounded below). Lemma 3.18. [ (cid:93) Θ X, ] is represented by σ S ε (cid:0) { Cl τ ( U x ) } x ∈ X , { } x ∈ X , { ε − Θ x } x ∈ X (cid:1) , or equivalently σ S ε [Θ X, ] after the isomorphism A ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ∼ = C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) (cid:98) ⊗A ( X ) .Proof. We first explicitly describe the collection terms σ C ( X ) ([ S ∗ ]) and σ X,C ( X ) ◦ fgt([ S ]) in thelanguage of fields. Noticing that the C ( X )-algebra C ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) is given by the family { C ( X ) (cid:98) ⊗ Cliff + ( T x X ) } x ∈ X , we find that σ C ( X ) ([ S ∗ ]) ∈ R KK ( X ; C ( X ) (cid:98) ⊗ Cl τ ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) , C ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) )is given by the family (cid:0) { C ( X ) (cid:98) ⊗ S ∗ x } x ∈ X , { id (cid:98) ⊗ c ∗ x } x ∈ X , { } x ∈ X (cid:1) as a family of Kasparov modules. Similarly, σ X,C ( X ) ◦ fgt([ S ]) ∈ R KK ( X ; C ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) , Cl τ ( X ) (cid:98) ⊗ C ( X ) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) )is given by (cid:0) { C ( X, S ) (cid:98) ⊗ C x } x ∈ X , { µ (cid:98) ⊗ } x ∈ X , { } x ∈ X (cid:1) as a family of Kasparov modules, where µ : C ( X ) → L ( C ( X, S )) is given by the pointwisemultiplication.Therefore, at the level of fields of modules, thanks to
Cliff + ( T x X ) (cid:98) ⊗ Cliff + ( T x X ) S ∗ x ∼ = S ∗ x , (cid:2) C ( U x ) (cid:98) ⊗ Cliff + ( T x X ) (cid:3) (cid:98) ⊗ C ( X ) (cid:98) ⊗ Cliff + ( T x X ) (cid:2) C ( X ) (cid:98) ⊗ S ∗ x (cid:3) (cid:98) ⊗ C ( X ) (cid:98) ⊗ C x (cid:2) C ( X, S ) (cid:98) ⊗ C x (cid:3) ∼ = (cid:2) C ( U x ) (cid:98) ⊗ S ∗ x (cid:3) (cid:98) ⊗ C ( X ) (cid:98) ⊗ C x (cid:2) C ( X, S ) (cid:98) ⊗ C x (cid:3) ∼ = C ( U x , S ) (cid:98) ⊗ S ∗ x . The isomorphism is given by the formula[ f (cid:98) ⊗ u ] (cid:98) ⊗ [ g (cid:98) ⊗ v ] (cid:98) ⊗ [ h (cid:98) ⊗ w ] (cid:55)→ ( − ∂h ( ∂u + ∂v ) f gh (cid:98) ⊗ c ∗ ( u ) vw for f ∈ C ( U x ), u ∈ Cliff + ( T x X ), g ∈ C ( X ), v ∈ S ∗ x , h ∈ C ( X, S ) and w ∈ C x .One can easily check that the family of operators { ε − Θ x ( • ) (cid:98) ⊗ id (cid:98) ⊗ id } x ∈ X satisfies the conditionsto be the triple Kasparov product. Under the above identification, this operator is transformed into1 (cid:98) ⊗ ε − Θ x ( • ). More concretely1 (cid:98) ⊗ ε − Θ x ( • )[ f (cid:98) ⊗ s ]( y ) := ( − ∂f f ( y ) (cid:98) ⊗ c ∗ ( ε − Θ x ( y ))( s ) . Next, we identify the trivial bundle U x × S ∗ x with S ∗ | U x by the parallel transformations along thegeodesics starting from x . We choose a connection ∇ S ∗ on S ∗ such that it is compatible with theLevi-Civita connection ∇ LC and the Clifford multiplication c ∗ in the following sense : For a section s ∈ C ∞ ( X, S ∗ ) and vector fields v, w ∈ X ( X ), we have ∇ S ∗ v [ c ∗ ( w ) s ] = c ∗ ( ∇ LC v w ) s + c ∗ ( w )( ∇ S ∗ v s ).There always exists such a connection because there exists a Spin structure on each small open23et and we can use the patchwork argument in this context [BGV, Corollary 3.41]. Let P xy bethe parallel transformations of T X , along the unique minimal geodesic traveling from x to y , withrespect to ∇ LC . We use the same symbol for that of S ∗ with respect to ∇ S ∗ . Thanks to thecondition imposed on ∇ S ∗ , we have P xy [ c ∗ ( v ) s ] = c ∗ ( P xy ( v )) P xy ( s ) for v ∈ T x X and s ∈ S ∗ x .We must compute Θ y ( x ) under this identification. Let γ : [0 , → X be the minimal geodesictraveling from x to y . By definition, Θ y ( x ) = − ˙ γ (0), Θ x ( y ) = ˙ γ (1) and P xy ( ˙ γ (0)) = ˙ γ (1). Thus,we have P xy (Θ y ( x )) = − Θ x ( y ). Consequently, we have P xy ( c ∗ (Θ y ( x )) s ) = − c ∗ (Θ x ( y )) P xy ( s ). Since c ∗ corresponds to the left Clifford multiplication under the identification S ∗ (cid:98) ⊗ S ∼ = Cliff + ( T X ), thefollowing diagram commutes: C ( U x , S ) (cid:98) ⊗ S ∗ x P x −−−−→ C ( U x , S ∗ (cid:98) ⊗ S ) (cid:98) ⊗ c ∗ (Θ • ( x )) (cid:121) (cid:121) − Θ x ( • ) C ( U x , S ) (cid:98) ⊗ S ∗ x P x −−−−→ C ( U x , S ∗ (cid:98) ⊗ S ) , where P x is defined by P x [ f (cid:98) ⊗ s ]( y ) := ( − ∂f ( y ) ∂s P xy ( s ) (cid:98) ⊗ f ( y )and the right vertical arrow − Θ x ( • ) means left multiplication by − Θ x ( • ) under the identification C ( U x , S ∗ (cid:98) ⊗ S ) ∼ = Cl τ ( U x ). Thus, [ (cid:93) Θ X, ] is represented by σ S ε (cid:0) { Cl τ ( U x ) } x ∈ X , { } x ∈ X , {− ε − Θ x } x ∈ X (cid:1) . Since X is orientable (this is because X is Spin c ), we can use Corollary 3.11, and we finish theproof.We can further rewrite the above by a family of the Bott homomorphisms of Definition 2.15.We define a ∗ -homomorphism β x : S ε → A ( X ) by β x ( f ) φ ( y ) := (cid:40) f ( X (cid:98) ⊗ (cid:98) ⊗ Θ x ( y )) φ ( y ) y ∈ U x y / ∈ U x . In order to prove that this is a continuous section, let us study f ( X (cid:98) ⊗ (cid:98) ⊗ Θ x ( y )). We can divide f into two parts: f ( t ) = f ( t ) + tf ( t ) for f , f ∈ C [0 , ε ). Then, the value of f ( X (cid:98) ⊗ (cid:98) ⊗ Θ x ( • ))at ( s, y ) ∈ ( − ε, ε ) × X is given by f ( s + ρ ( x, y ) ) + ( s · x ( y )) f ( s + ρ ( x, y ) ) , where ρ is the distance function on X and “1” in the second term is the multiplicative identity ofthe Clifford algebra. Since f and f vanishes on t ≥ ε , β x ( f ) φ ( y ) vanishes on the boundary of U x , and consequently β x ( f ) φ is continuous on X . Proposition 3.19. [ (cid:93) Θ X, ] is represented by the field of Kasparov modules ( {A ( X ) } x ∈ X , { β x } x ∈ X , { } x ∈ X ) . Proof.
By the same argument of Proposition 2.19 and the above result, we notice that [ (cid:93) Θ X, ] isrepresented by ( {A ( U x ) } x ∈ X , { β x } x ∈ X , { } x ∈ X ). It is homotopic to ( {A ( X ) } x ∈ X , { β x } x ∈ X , { } x ∈ X )by β x ( S ε ) A ( X ) ⊆ A ( U x ). 24he following example explain the geometrical meaning of the reformulated index theorem. Example . We have reformulated the Poincar´e duality homomorphism, the index element andthe Clifford symbol element. Let us compute them for the following case.
Let E be a G -equivariantHermitian vector bundle with a G -invariant metric connection ∇ E over X . Let D E be the Diracoperator defined by D E := (cid:88) id E (cid:98) ⊗ c ( v n ) ◦ ∇ E (cid:98) ⊗ Sv n : C ∞ c ( X, E (cid:98) ⊗ S ) (cid:8) , where { v n } is an orthonomal base of the tangent space. In this situation, the reformulated KK -elements are given by the following: • [ (cid:103) D E ] = σ S ε (cid:0) L ( X, E (cid:98) ⊗ Cliff + ( T X )) , π, D (cid:48) E (cid:1) , where D (cid:48) E = (cid:80) id (cid:98) ⊗ c ( v n ) ◦ ∇ E (cid:98) ⊗ Cliff + ( T X ) v n and π denotes the Clifford multiplication on the left c ∗ ; and • [ (cid:103) σ ClD E ] = σ S ε ( C ( X, E ) , π, KK -element corresponding to the Dirac operator twisted by E the R KK -element corresponding to itscoefficient E . The goal of this subsection is to describe all the ingredients of the topological assembly map inmore convenient form. This is essential to define a substitute for the topological assembly map forproper LT -spaces.Let us begin with the descent homomorphism for R KK -theory. The key ingredient is the formulato describe crossed products of X (cid:111) G - C ∗ -algebras using the generalized fixed-point algebras [EE,Theorem 2.14]. It is easy to give the field description of this result. Using it and its Hilbert moduleversion, we can describe the descent homomorphism in the language of fields.We first give a review of the proof of [EE, Section 2]. In order to explain it, we need the conceptof generalized fixed-point algebras . Let G be a locally compact second countable Hausdorffgroup and let X be a σ -compact locally compact Hausdorff space equipped with a proper G -action.Let A be an X (cid:111) G - C ∗ -algebra, whose action is denoted by α A : G → Aut( A ). With the G -actionon X × X given by g : ( x, y ) (cid:55)→ ( gx, gy ), we can define a C ( X × G X )- C ∗ -algebra C ( X × G A ) , bythe set of all continuous functions f : X → A satisfying the following conditions: it is G -invariant f ( x ) = α g [ f ( g − · x )]; and the norm function X × G X (cid:51) [( x, y )] (cid:55)→ (cid:107) f ( x )( y ) (cid:107) ∈ R ≥ vanishes atinfinity, where we regard A as the section algebra over X . The generalized fixed-point algebra of A is defined by the restriction of the C ( X × G X )- C ∗ -algebra C ( X × G A ) to the orbit space of thediagonal set ∆( X ) /G ∼ = X/G : A G,α := C ( X × G A ) | ∆( X ) /G . Example . When X = G and A = C ( G ), whose action is defined by left translation “lt”, C ( G × G C ( G )) is C ( G ). The isomorphism is induced by the homeomorphism G × G (cid:51) [( g, h )] (cid:55)→ h − g ∈ G . The restriction to ∆( G ) /G is the fiber at [( e, e )], which is C . Thus, the generalizedfixed-point algebra C ( G ) G, lt is, as everyone expects, given by C = C ( G/G ).Using the concept of generalized fixed-point algebras, we can describe crossed products. Weneed the following ingredients: the unitary representation R of G on L ( G ) given by R g φ ( x ) := φ ( xg ) (cid:112) µ ( g ); the action Ad R : G → Aut( K ( L ( G ))) given by Ad R g ( k ) := R g ◦ k ◦ R − g for25 ∈ K ( L ( G )). By using it, we can define a G -action on K ( L ( G )) (cid:98) ⊗ A by (Ad R (cid:98) ⊗ α ) g ( k (cid:98) ⊗ a ) :=Ad R g ( k ) (cid:98) ⊗ α Ag ( a ) for a ∈ A , k ∈ K ( L ( G )) and g ∈ G . Proposition 3.22 ([EE]) . Let G be a locally compact second countable Hausdorff group and let X be a σ -compact locally compact Hausdorff space equipped with a proper G -action. We assumethat G is amenable, for simplicity. Let A be an X (cid:111) G - C ∗ -algebra, whose action is denoted by α A : G → Aut( A ) . Then, the crossed product A (cid:111) α A G is isomorphic to the generalized fixed-pointalgebra ( K ( L ( G )) (cid:98) ⊗ A ) G, Ad R (cid:98) ⊗ α as C ( X/G ) -algebras. In the language of u.s.c. fields of C ∗ -algebras and integral kernels, this result is described asfollows. Proposition 3.23.
Let A := ( { A x } x ∈ X , Γ A ) be the u.s.c. field associated to A . Then, A (cid:111) G ∼ = C (cid:16) X × G, Ad R (cid:98) ⊗ α A (cid:8) K ( L ( G )) (cid:98) ⊗ A x (cid:9) x ∈ X (cid:17) , where a section of the field (cid:8) K ( L ( G )) (cid:98) ⊗ A x (cid:9) x ∈ X is continuous if and only if it can be approximatedby finite sums of k (cid:98) ⊗ a for k ∈ K ( L ( G )) and a ∈ Γ A (See also Definition 4.32 for details).Let f ∈ C c ( G, A ) . The integral kernel of the corresponding equivariant section X → (cid:8) K ( L ( G )) (cid:98) ⊗ A x (cid:9) x ∈ X is given by x (cid:55)→ (cid:104) ( g, h ) (cid:55)→ µ ( h ) − α Ag − [ f ( gh − )( gx )] (cid:105) , where µ is the modular function of G . When we regard it as a function on X × G × G , we denoteit by k f ( g, h ; x ) .Remark . Note that f ( gh − )( gx ) is the evaluation of f ( gh − ) ∈ A at g · x . Thanks to “ α Ag − ”, k f ( g, h ; x ) ∈ A x .Under the above isomorphism, let us compute [ c X ] ∈ KK ( C , C ( X ) (cid:111) G ). Recall that [ c X ] isdefined by the projection element c ( g )( x ) = µ ( g ) − / (cid:112) c ( x ) c ( g − · x ) of C ( X ) (cid:111) G for a cut-offfunction c : X → R ≥ . Thus, the integral kernel corresponding to c is given by k c ( g, h ; x ) = µ ( h ) − c ( gh − )( g · x ) = µ ( g ) − / (cid:112) c ( g · x ) · µ ( h ) − / (cid:112) c ( h · x ) . Put c x ( g ) := µ ( g ) − / (cid:112) c ( g · x ). It is an L -unit vector; in fact, (cid:107) c x (cid:107) L = (cid:82) c x ( g ) µ ( g ) − dg = (cid:82) c ( g · x ) µ ( g ) − dg = (cid:82) c ( g − · x ) dg = 1. Thus, the operator given by the above integral kernel isthe rank one projection to C √ c x . This projection is denoted by P √ c x . Since the Hilbert K ( L ( G ))-module corresponding to a rank one projection is always isomorphic to L ( G ) ∗ by k (cid:55)→ √ c x ∗ ◦ k ,we obtain the following. Lemma 3.25.
For a cut-off function c : X → R ≥ , we put c x ( g ) := µ ( g ) − / (cid:112) c ( g · x ) . Then, thecorresponding KK -element [ c X ] is given by the family of rank one projections x (cid:55)→ P √ c x . In thelanguage of modules, it is represented by (cid:0) C (cid:0) X × G,R L ( G ) ∗ (cid:1) , , (cid:1) .
26e would like to describe the descent homomorphism in the language of “generalized fixed-pointmodules” which is defined in an obvious way. Before that, we need to recall that Proposition 3.22follows from the following lemma and the fact that C ( G ) (cid:111) G is isomorphic to K ( L ( G )). Lemma 3.26 ([EE, Lemma 2.9 and Lemma 2.8]) . (1) Let G be a locally compact second countableHausdorff group and let K be a compact subgroup of G . Let A be a G - C ∗ -algebra, whose actionis denoted by α . Suppose that an action β : K → Aut( A ) commuting with α is given. Then, theinclusion ι : A K → A induces an isomorphism A K (cid:111) G → ( A (cid:111) G ) K , where B K for a K - C ∗ -algebra B is the subalgebra consisting of all K -invariant elements of B . (2) Let α and β be commuting actions of G on A . We define the following two C ∗ -algebras:Using the G -action on A (cid:111) α G given by [ (cid:101) β g F ]( h ) := β g [ F ( h )] , we define C ( X × G, (cid:101) β ( A (cid:111) α G )) ; Usingthe G -action τ (cid:98) ⊗ α on C ( X × G,β A ) given by [( τ (cid:98) ⊗ α ) g F ]( x ) := α g [ F ( g − x )] , we define the crossedproduct C ( X × G,β A ) (cid:111) τ (cid:98) ⊗ α G . Then, they are isomorphic: C ( X × G,β A ) (cid:111) τ (cid:98) ⊗ α G ∼ = C ( X × G, (cid:101) β ( A (cid:111) α G )) . Remark . (2) follows from (1) by the following argument: The fiber of the left hand side at x ∈ X is A G x (cid:111) α G ; The fiber of the right hand side at x is [ A (cid:111) α G ] G x ; They are isomorphic thanksto (1). Proposition 3.22 follows from (2) by a formal argument.Let us describe the descent homomorphism. For simplicity, we deal with only actually equivari-ant Kasparov modules. Note that every R KK G -element has such a representative because we canuse the averaging procedure thanks to the properness of the G -action on X . See [Kas3, Section 5].See also Corollary 4.40. Proposition 3.28. (1)
Let G be a locally compact second countable Hausdorff group and let X bea σ -compact locally compact Hausdorff space equipped with a proper G -action. We assume that G is amenable, for simplicity. Let A and B be X (cid:111) G - C ∗ -algebras, and let ( E, π, F ) be an X (cid:111) G -equivariant Kasparov ( A, B ) -module. The G -action on E is denoted by α E . We assume that F isactually equivariant: g ( F ) = F for every g ∈ G . Then, the descent homomorphism is given by j G ( E, π, F ) = (cid:16) [ K ( L ( G )) (cid:98) ⊗ E ] G, Ad R (cid:98) ⊗ α E , [id (cid:98) ⊗ π ] G , [id (cid:98) ⊗ F ] G (cid:17) , where [id (cid:98) ⊗ π ] G ( a ) and [id (cid:98) ⊗ F ] G are induced maps from id (cid:98) ⊗ π ( a ) and id (cid:98) ⊗ F to the fixed-point module,respectively. (2) In the language of u.s.c. fields, j G ( E, π, F ) can be written as follows. Let A := ( { A x } x ∈ X , Γ A ) , B := ( { B x } x ∈ X , Γ B ) , E := ( { E x } x ∈ X , Γ E ) be the u.s.c. fields associated to A , B and E , re-spectively. Let π = { π x } x ∈ X and F = { F x } x ∈ X be the field description of π and F . Then, j G ( E , { π x } x ∈ X , { F x } x ∈ X ) is given by (cid:16) C (cid:16) X × G,α E (cid:98) ⊗ Ad R (cid:8) E x (cid:98) ⊗ K ( L ( G )) (cid:9) x ∈ X (cid:17) , { π x (cid:98) ⊗ id } x ∈ X , { F x (cid:98) ⊗ id } x ∈ X (cid:17) . Proof. (2) is obvious from (1).We first prove that E (cid:111) G is isomorphic to the fixed-point module [ K ( L ( G )) (cid:98) ⊗ E ] G, Ad R (cid:98) ⊗ α E as bimodules. By the “same” argument of the proof of Proposition 3.22, we have an isometric27omomorphism from the left hand side to the right hand side as bimodules. This is possiblebecause all the algebraic operations on E (cid:111) G (right B (cid:111) G -action, left A (cid:111) G -action and the B (cid:111) G -valued inner product) are given by the parallel formulas of the corresponding operations oncrossed products. For example, the right action of b ∈ C c ( G, B ) on e ∈ C c ( G, E ) looks like the“multiplication of e and b in a crossed product algebra”, although e and b live in different places.This argument does not guarantee that the isometry is surjective. Thus, what we need to prove isthe analogue of Lemma 3.26 (1) for Hilbert modules: Let K be a compact subgroup of G . We supposethat K acts on B and E , whose actions are denoted by β B : K → Aut( B ) and β E : K → Aut( E ) ,and we also suppose that they are compatible in the following sense β Ek ( eb ) = β Ek ( e ) β Bk ( b ) and β Bk (( e | e )) = (cid:0) β Ek ( e ) (cid:12)(cid:12) β Ek ( e ) (cid:1) . We assume that the K -actions β ’s commute with the G -actions α ’s. Then, the inclusion ι : E K → E induces an isomorphism E K,β E (cid:111) α G → ( E (cid:111) α G ) K,β E as Hilbert modules. We borrow an idea from [LG, Lemma 4.1]. We prove that this statement is derived from Lemma3.26 (1). Let us consider the C ∗ -algebra K B ( E ⊕ B ). It is decomposed into four components: K B ( E ⊕ B ) = (cid:18) K B ( E ) EE ∗ B (cid:19) . Note that the product on the matrix algebra K B ( E ⊕ B ) contains all the information on the Hilbertmodule E as follows: For k ∈ K B ( E ), e, e , e ∈ E and b ∈ B , (cid:18) e (cid:19) (cid:18) b (cid:19) = (cid:18) eb (cid:19) ; (cid:18) k
00 0 (cid:19) (cid:18) e (cid:19) = (cid:18) ke (cid:19) ; and (cid:18) e ∗ (cid:19) (cid:18) e (cid:19) = (cid:18) e | e ) B (cid:19) , where e ∗ is the map x (cid:55)→ ( e | x ) B .Moreover, the group action preserves the decomposition into 2 × g · (cid:18) k ef b (cid:19) = (cid:18) g ( k ) g ( e ) g ( f ) g ( b ) (cid:19) for g ∈ G , k ∈ K B ( E ), e ∈ E , f ∈ E ∗ and b ∈ B , and similarly for the K -action. Thus, we have K B ( E ⊕ B ) K = (cid:18) K B ( E ) K E K [ E ∗ ] K B K (cid:19) and K B ( E ⊕ B ) (cid:111) G = (cid:18) K B ( E ) (cid:111) G E (cid:111) GE ∗ (cid:111) G B (cid:111) G (cid:19) . Applying Lemma 3.26 (1) to K B ( E ⊕ B ), we obtain the isomorphism (cid:18) K B ( E ) K (cid:111) G E K (cid:111) G [ E ∗ ] K (cid:111) G B K (cid:111) G (cid:19) ∼ = (cid:18) [ K B ( E ) (cid:111) G ] K [ E (cid:111) G ] K [ E ∗ (cid:111) G ] K [ B (cid:111) G ] K (cid:19) as C ∗ -algebras . Therefore, we have an isomorphism between E K (cid:111) G and [ E (cid:111) G ] K as Hilbertmodules . 28inally, we prove the property on (cid:101) F . We use the language of fields and integral kernels. Let F = { F x } x ∈ X . For e ∈ C c ( G, E ), the integral kernel of the corresponding element of [ K ( L ( G )) (cid:98) ⊗ E ] G is given by k e ( g , g ; x ) = µ ( g ) − g − [ e ( g g − )( g x )] . Thus, k (cid:101) F e ( g , g ; x ) = µ ( g ) − g − (cid:104)(cid:110) (cid:101) F e (cid:111) ( g g − )( g x ) (cid:105) = µ ( g ) − g − (cid:2) F g x (cid:8) e ( g g − )( g x ) (cid:9)(cid:3) = F x (cid:8) µ ( g ) − g − [ e ( g g − )( g x )] (cid:9) = F x [ k e ( g , g ; x )] , where we have used the actual G -equivariance of F at the third equality.The final tool to define the original topological assembly map is the “descent of the Diracelement” j G ([ d X ]). According to Definition 3.14, for the reformulated one, we need σ S ε ◦ j G (fgt[ S ] (cid:98) ⊗ [ d X ]) ∈ KK ( S ε (cid:98) ⊗ ( C ( X ) (cid:111) G ) , S ε (cid:98) ⊗ ( C (cid:111) G )) . Thus, we compute j G (fgt[ S ] (cid:98) ⊗ [ d X ]).The KK -element fgt[ S ] (cid:98) ⊗ [ d X ] is given by the index element [ D ] of the following Spin c -Diracoperator D . In fact, since Cliff + ( T X ) ∼ = S ∗ (cid:98) ⊗ S , we have C ( X, S ) (cid:98) ⊗ L ( X, Cliff + ( T X )) = L ( X, S ).Put D := (cid:80) n c ( v n ) ◦ ∇ Sv n for an orthonormal base { v n } . π denotes left multiplication of C ( X ) on L ( X, S ). Then, fgt[ S ] (cid:98) ⊗ [ d X ] can be represented by (cid:0) L ( X, S ) , π, D (cid:1) .We explicitly compute j G ([ D ]) with the same spirit of Proposition 3.28. We do not need toassume that S is a Spinor bundle, and so we compute it in the following (slightly more general)situation: On a Clifford bundle E equipped with a G -equivariant Clifford multiplication c : T X → End( E ) and a G -equivariant Clifford connection ∇ E , we have an equivariant Dirac operator D = (cid:80) n c ( v n ) ◦ ∇ Ev n . Note that G has a unitary representation on L ( G ) by R g φ ( h ) := (cid:112) µ ( g ) φ ( hg ), and G acts on ( C(cid:111) G ) by rt g ( b )( h ) := b ( hg ). The latter action gives Hilbert C(cid:111) G -module automorphismswith respect to the following Hilbert module structure: b · a := a ∨ ∗ b and ( b | b ) C(cid:111) G := ( b ∗ b ∗ ) ∨ for a, b, b , b ∈ C (cid:111) G , where b ∨ ( g ) := (cid:112) µ ( g ) − b ( g − ). See Definition 3.1 for the origin of thisHilbert module structure.The following definition will be justified in the following proposition. Definition 3.29.
Let X be a complete Riemannian manifold equipped with an isometric properaction of a locally compact second countable Hausdorff group G . For simplicity, we suppose that G is amenable. Let E be a G -equivariant Clifford bundle over X equipped with a G -equivariant Diracoperator D .(0) Let C c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) be the set of compactly supported continuoussections k : X × G × G → E satisfying the equivariance condition k ( g , g ; x ) = (cid:112) µ ( g ) α Eg k ( g g, g g ; g − x ) , where “compactly supported” means that the closure of the set { x ∈ X | k ( • , • , x ) (cid:54) = 0 } /G is com-pact. It has a pre-Hilbert C c ( G )-module structure by the following operations: For k, k , k ∈ C c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) , b ∈ C c ( G ), x ∈ X and g, g , g ∈ G .( k | k ) ( C(cid:111) G ) ( g ) := (cid:112) µ ( g ) − (cid:90) X c ( x ) (cid:90) G (cid:90) G (cid:0) k ( η, ξ ; x ) (cid:12)(cid:12) k ( η, g − ξ ; x ) (cid:1) E dηdξdx ;29 · b ( g , g ; x ) := (cid:90) G k ( g , η − g ; x ) b ( η − ) (cid:112) µ ( η ) − dη. (1) We define a Hilbert ( C (cid:111) G )-module L (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) by the completion of C c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) with respect to the above innerproduct.(2) We define a ∗ -homomorphism π (cid:111) lt : C ( X ) (cid:111) G → L C(cid:111) G (cid:16) L (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17)(cid:17) under the identification C ( X ) (cid:111) G ∼ = C ( X × G K ( L ( G ))), by the formula[ π (cid:111) lt( a ) k ]( g , g ; x ) := (cid:90) G k a ( g , η ; x ) k ( η, g ; x ) dη. (3) For k ∈ C c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) , we define an element k ( g , g ) ∈ C c ( X, E )by k ( g , g )( x ) := k ( g , g ; x ). We define C ∞ c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) by the sub-set consisting of all smooth sections. Associated to D , we define an unbounded operator D on C ∞ c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) by D ( k )( g , g ; x ) := D [ k ( g , g )]( x ) . Since D ( k ) is smooth and G -invariant, D is well-defined as a map on C ∞ c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) .Its extension to an appropriate domain is denoted by the same symbol. Proposition 3.30. (0) C c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) is a pre-Hilbert C c ( G ) -moduleand π (cid:111) lt is actually a ∗ -homomorphism. (1) This ( C ( X ) (cid:111) G, C (cid:111) G ) -bimodule is isomorphic to L ( X, E ) (cid:111) G by the following corre-spondence: For e ∈ C c ( G, C c ( X, E )) ⊆ L ( X, E ) (cid:111) G , we define k e by k e ( x )( g , g ) := (cid:112) µ ( g ) − g − (cid:2) e ( g g − , g x ) (cid:3) . We regard it as “a function on G × G (cid:98) ⊗ an element of E x ”. This family defines an element of C c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) . (2) For e ∈ C c ( G, C c ( X, E )) , Dk e = k (cid:101) De . Thus, we denote D by (cid:101) D from now on. Consequently, j G ([ D ]) is represented by (cid:16) L (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) , π (cid:111) lt , (cid:101) D (cid:17) . roof. (0) Simple computations show the statement. We leave it to the reader.(1) We denote the correspondence e (cid:55)→ k e by Ψ. We need to check the following things: ( a ) Ψ isa left module homomorphism; ( b ) Ψ is a right module homomorphism; ( c ) Ψ is isometric; and ( d )The image of Ψ is dense.( a ) For e ∈ C c (cid:0) X × G { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:1) and a ∈ C c ( X × G K ( L ( G ))), k π (cid:111) lt( a )( e ) ( g , g ; x )= (cid:112) µ ( g ) − α Eg − (cid:2) π (cid:111) lt( a )( e )( g g − ; g x ) (cid:3) = (cid:112) µ ( g ) − α Eg − (cid:20)(cid:90) G a ( η ; g x ) α Eη [ e ( η − g g − ; η − g x )] (cid:21) dη = (cid:112) µ ( g ) − α Eg − (cid:20)(cid:90) G µ ( η − g ) α Eg [ k a ( g , η − g ; x )] α Eη (cid:16)(cid:112) µ ( g ) α Eη − α Eg [ k e ( η − g , g ; x )] (cid:17) dη (cid:21) = (cid:90) G k a ( g , η − g ; x ) k e ( η − g , g ; x ) µ ( η − g ) dη = (cid:90) G k a ( g , η − ; x ) k e ( η − , g ; x ) µ ( η − ) dη = (cid:90) G k a ( g , η ; x ) k e ( η, g ; x ) dη = [ π (cid:111) lt( a ) k ]( x )( g , g ) , where we used the definitions of k a and k e at the second equality, the left invariance of the measureat the fifth equality, and the property of the modular function at the sixth equality.( b ) It is obtained by a similar calculation of ( a ).( c ) For e , e ∈ C c ( G, L ( X, E )),( k e | k e ) ( C(cid:111) G ) ( g )= (cid:112) µ ( g ) − (cid:90) X c ( x ) (cid:90) G (cid:90) G (cid:0) k e ( g , g ; x ) (cid:12)(cid:12) k e ( g , g − g ; x ) (cid:1) E x dg dg dx = (cid:112) µ ( g ) − (cid:90) X c ( x ) (cid:90) G (cid:90) G (cid:16)(cid:112) µ ( g ) − α Eg − e ( g g − ; g x ) (cid:12)(cid:12)(cid:12) (cid:112) µ ( g − g ) − α Eg − e ( g g − g ; g x ) (cid:17) E x dg dg dx = (cid:90) G (cid:90) G (cid:90) X c ( x ) (cid:0) e ( g g − ; g x ) (cid:12)(cid:12) e ( g g − g ; g x ) (cid:1) E g x µ ( g ) − dxdg dg = (cid:90) G (cid:90) G (cid:90) X c ( g − x ) ( e ( g ; x ) | e ( g g ; x )) E x dxdg dg = (cid:90) G ( e ( g ) | e ( g g )) L ( X,E ) dg = ( e | e ) ( C(cid:111) G ) ( g ) , where we have used the definition of k e at the second equality, Fubini’s theorem at the third one,the G -invariance of the measure on X and the left invariance of the measure on G at the fourthone, and the definition of the cut-off function at the fifth one.( d ) For k ∈ C c (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:17) , we prove that there exists e ∈ C c ( G, C c ( X, E )) ⊆ ( X, E ) (cid:111) G such that k e = k . In fact, if we put e ( η ; x ) := (cid:112) µ ( η ) − k (1 , η − ; x ), we obtain k e ( g , g ; x ) = (cid:112) µ ( g ) − α Eg − e ( g g − ; g x )= (cid:112) µ ( g ) − α Eg − (cid:20)(cid:113) µ ( g g − ) k (1 , g g − ; g x ) (cid:21) = (cid:112) µ ( g ) − α Eg − (cid:20)(cid:113) µ ( g g − ) (cid:112) µ ( g ) α Eg k (1 · g , g g − g ; g − g x ) (cid:21) = k ( g , g ; x ) , where we used the G -invariance of k at the third equality.(2) One can easily prove this statement by using the definition of k e and the G -invariance of D .We leave it to the reader.Moreover we can define descent homomorphism in terms of only unbounded Kasparov modulesfor actually equivariant unbounded Kasparov modules [T4, Definition-Proposition 2.10], by thesame formula. Thus, this result immediately follows from (1) and (2). Remarks . (1) On the fiber E x (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C(cid:111) G ) at x ∈ X , we can define a Hilbert C(cid:111) G -modulestructure by the following operations:( e (cid:98) ⊗ φ (cid:98) ⊗ b (cid:48) ) · b := e (cid:98) ⊗ φ (cid:98) ⊗ [ b ∨ ∗ b (cid:48) ] , (cid:0) e (cid:98) ⊗ φ (cid:98) ⊗ b (cid:12)(cid:12) e (cid:98) ⊗ φ (cid:98) ⊗ b (cid:1) C(cid:111) G := ( e | e ) E x ( φ | φ ) L ( G ) ( b ∗ b ∗ ) ∨ , where b ∨ ( g ) := (cid:112) µ ( g ) − b ( g − ). Moreover, ( π (cid:111) lt) x ( a x ) is given by id (cid:98) ⊗ a x (cid:98) ⊗ id for a x ∈ K ( L ( G )).These operations vary continuously on X , and hence we obtain a locally trivial bundle of Hilbert C (cid:111) G -module. This structure naturally induces the above bimodule structure to the section space.(2) This Hilbert module bundle structure is given by the tensor product of the locally trivialHilbert bundle E (cid:98) ⊗ L ( G ) and the trivial Hilbert C (cid:111) G -module bundle X × C (cid:111) G . Thus, in orderto use the formulas, we will often denote symbolically a section of this bundle as φ (cid:98) ⊗ ψ , where φ isregarded as a map X → E (cid:98) ⊗ L ( G ) and ψ is regarded as a map X → C (cid:111) G . In this notation, theabove bimodule structure can be described as follows:[ π (cid:111) lt( a ) φ (cid:98) ⊗ ψ ]( x ) = [id E x (cid:98) ⊗ a ( x )] φ ( x ) (cid:98) ⊗ ψ ( x ) , [ φ (cid:98) ⊗ ψ · b ]( x ) = φ ( x ) (cid:98) ⊗ [ b ∨ ∗ ψ ( x )] , (cid:0) φ (cid:98) ⊗ ψ (cid:12)(cid:12) φ (cid:98) ⊗ ψ (cid:1) ( C(cid:111) G ) = (cid:90) X c ( x ) ( φ ( x ) | φ ( x )) E x (cid:98) ⊗ L ( G ) [ ψ ( x ) ∗ ψ ( x ) ∗ ] ∨ dx, where b ∨ ( g ) := (cid:112) µ ( g ) − b ( g − ). These formulas will be adopted as the definition of the descenthomomorphism for proper LT -spaces.We can further rewrite these simple formulas in much more algebraic way, under the followingassumptions. Assumption 3.32.
Until the next subsection, we suppose the following conditions on X and G : • G is a finite-dimensional unimodular Lie group, and H is a closed Lie subgroup of the centerof G ; 32 The H -action given by the restriction of that of G , is free and smooth; and • The orbit map H (cid:51) h (cid:55)→ h · x ∈ X is isometric for each x ∈ X .For v ∈ h = Lie( H ), we denote the infinitesimal action of v on L ( X, E ) by dα Ev : dα Ev s ( x ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 α E exp( tv ) s (exp( − tv ) · x ) . The associated operator on L ( X, E ) (cid:111) G is denoted by (cid:103) dα Ev as usual. Lemma 3.33.
For e ∈ C ∞ c ( G, C ∞ c ( X, E )) and v ∈ h , k (cid:103) dα Ev e is computed as follows: k (cid:103) dα Ev e ( x ) = dR − v k e ( x ) + d rt − v k e ( x ) . Proof.
The following computation shows it: k (cid:103) dα Ev e ( g , g ; x ) = α Eg − dα Ev e ( g g − ; g x )= α Eg − ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 α E exp( tv ) e ( g g − ; exp( − tv ) g x )= α Eg − ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 α E exp( tv ) e ( g exp( − tv )[ g exp( − tv )] − ; [ g exp( − tv )] x )= α Eg − ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 α E exp( tv ) α Eg exp( − tv ) k e ( g exp( − tv ) , g exp( − tv ); x )= ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 k e ( g exp( − tv ) , g exp( − tv ); x )= dR − v k e ( g , g ; x ) + d rt − v k e ( g , g ; x ) . Example . Suppose that H = G = R , X = R and E = X × C . Let us consider the G -actionon X given by g : ( x, y ) (cid:55)→ ( x + g, y ), and its lift on E given by g : (( x, y ) , z ) (cid:55)→ (( x + g, y ) , z ). Wedenote the infinitesimal generator 1 ∈ g of G by v . Then, dα Ev = − ∂∂x .Let e : G → C ∞ c ( X, E ) be a smooth function. Then, k e is a smooth function on G × G × X = R × R × R . The coordinate is denoted by ( g , g ; x, y ). Then, dR v = ∂∂g , d rt v = ∂∂g . Since k e ( g , g ; x, y ) = e ( g − g ; x + g , y ), we have k (cid:103) dα Ev e ( g , g ; x, y ) = − ∂∂x e ( g − g ; x + g , y )= − ∂∂g e ( g − g ; x + g , y ) − ∂∂g e ( g − g ; x + g , y )= dR − v k e ( g , g ; x, y ) + d rt − v k e ( g , g ; x, y ) . This is the most fundamental case of the above formula.
Notations 3.35.
In order to emphasize how (cid:103) dα Ev acts on each fiber, we denote the above formulaby k (cid:103) dα Ev e = id E (cid:98) ⊗ dR − v (cid:98) ⊗ id( k e ) + id E (cid:98) ⊗ id (cid:98) ⊗ d rt − v ( k e ).33et D be the Dirac operator given by (cid:88) i c ( e i ) dα E − e i + D base , where c ( e i ) is the Clifford multiplication of the vector field induced by e i ∈ h , and D base = (cid:80) n c ( v n ) ◦∇ Ev n is the “Dirac operator for the base direction”, where { v n } is an orthonomal base of the normalbundle of the H -orbit.The following is fundamental. Lemma 3.36.
Let V → X be a G -equivariant vector bundle (it can be a C ∗ -algebra bundle, aHilbert space bundle, or a Hilbert module bundle). We say two elements v, w ∈ V are H -equivalentif w = h · v for some h ∈ H . Consequently, if v ∈ V x , w = h · v ∈ V hx . Then, the quotientspace under the H -equivalence admits a G/H -equivariant vector bundle structure over
X/H . A G -invariant section on V defines a G/H -invariant section on
V /H and vice versa.
Notations 3.37.
By regarding
V /H as the “restriction of V to local slices”, and taking into accountthe above lemma, we often denote C c ( X × G V ) by C c ( X/H × G/H V ). Following this notation, thefunction spaces appearing in Definition 3.29 are also denoted by C (cid:0) X/H × G/H K ( L ( G )) (cid:1) , C (cid:0) X/H × G/H { E (cid:98) ⊗ K ( L ( G )) } (cid:1) and L (cid:0) X/H × G/H { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:1) . Let us rewrite j G ([ D ]) in an algebraic way. A cut-off function c : X → R ≥ induces a cut-offfunction c on X/H by the orbit integral c ( x ) := (cid:82) H c ( h − (cid:101) x ) dh , where x ∈ X/H and (cid:101) x is a lift of x .The following is obvious by the arguments so far. Proposition 3.38. (1) L (cid:0) X/H × G/H { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:1) admits a Hilbert ( C (cid:111) G ) -modulestructure given by the following operations: For φ (cid:98) ⊗ ψ, φ (cid:98) ⊗ ψ , φ (cid:98) ⊗ ψ , ∈ L (cid:0) X/H × G/H { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:1) (for the remark on this notation, see Remark 3.31 (2)) and b ∈ C (cid:111) G , • [ φ (cid:98) ⊗ ψ ] · b ( x ) = φ ( x ) (cid:98) ⊗ [ b ∨ ∗ ψ ( x )] ; and • (cid:0) φ (cid:98) ⊗ ψ (cid:12)(cid:12) φ (cid:98) ⊗ ψ (cid:1) ( C(cid:111) G ) = (cid:82) X/H c ( x ) ( φ ( x ) | φ ( x )) E x (cid:98) ⊗ L ( G ) [ ψ ( x ) ∗ ψ ( x ) ∗ ] ∨ dx . (2) This Hilbert module admits a left module structure π (cid:111) lt : C (cid:0) X/H × G/H K ( L ( G )) (cid:1) → L C(cid:111) G (cid:0) L (cid:0) X/G × G/H { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:1)(cid:1) given by [ π (cid:111) lt( a ) φ (cid:98) ⊗ ψ ]( x ) = [id E x (cid:98) ⊗ a ( x ) φ ( x )] (cid:98) ⊗ ψ ( x ) . (3) D denotes the operator (cid:88) i [id E (cid:98) ⊗ dR v i (cid:98) ⊗ id + id E (cid:98) ⊗ id (cid:98) ⊗ d rt v i ] (cid:98) ⊗ c ( v i ) + D base . Then, j G ([ D ]) is represented by (cid:0) L (cid:0) X/H × G/H { E (cid:98) ⊗ L ( G ) (cid:98) ⊗ ( C (cid:111) G ) } (cid:1) , π (cid:111) lt , D (cid:1) . .4 Twisted equivariant version In the previous three subsections, we supposed that the G -action on X lifts to the Clifford bundle.In this subsection, we will study the case of a Clifford bundle over X which admits an action of a U (1) -central extension of G which is compatible with the G -action on X . It is a natural situation forHamiltonian loop group spaces. The formulas proved this subsection will be adopted as definitionsin Section 6.More concretely, we will describe parallel results for the following setting. For a continuous U (1)-central extension of G → U (1) i −→ G τ p −→ G → , we suppose that G (not G τ ) acts on a complete Riemannian manifold X in an isometric, properand cocompact way, and that X has a G -equivariant Spinor bundle ( S, c ). We consider a Hermitianvector bundle π : F → X equipped with a G τ -action satisfying α Fi ( z ) = z id F and π ( g · f ) = p ( g ) · π ( f )for z ∈ U (1), g ∈ (cid:101) G and f ∈ F . Such a vector bundle is said to be τ -twisted G -equivariant . Wewould like to study the τ -twisted G -equivariant index of S (cid:98) ⊗ F . Needless to say, it is possible to dealwith this problem as a G τ -twisted problem. However, we need more economical formulas in orderto generalize them to infinite-dimensional manifolds. We refer to [T3, Section 3.3] for the detailedarguments. Notations 3.39.
Let 1 → U (1) i −→ G τ p −→ G → U (1)-central extension of G .(1) A function f from G τ to a vector space (Hilbert spaces, C ∗ -algebras, Hilbert modules, andso on) is said to be at level q if f ( i ( z ) g ) = z q f ( g ).(2) The set of compactly supported continuous function on G τ at level q , is denoted by C c ( G, qτ ).Other types of function spaces are denoted in the same way: C ∞ c ( G, qτ ), L ( G, qτ ), and so on.(3) Let A be a G - C ∗ -algebra. It is automatically a G τ - C ∗ -algebra through the homomorphism p : G τ → G . The subalgebra A (cid:111) qτ G of A (cid:111) G τ is defined by the completion of C c ( G, qτ ) in A (cid:111) G τ .It is called the qτ -twisted crossed product of A by G . For a Hilbert A -module E , we define E (cid:111) qτ G in the same way. Definition 3.40.
We suppose the same conditions.(1) Let A and B be separable G - C ∗ -algebras. They are automatically equipped with the G τ -actions. A G τ -equivariant Kasparov module ( E, π, F ) satisfying the following is said to be qτ -twisted G -equivariant : α Ei ( z ) ( e ) = z q e for all z ∈ U (1).We define KK qτG ( A, B ) by the set of homotopy classes of qτ -twisted G -equivariant Kasparov ( A, B )-modules for q ∈ Z . These are direct summands of KK G τ ( A, B ), that is to say, KK G τ ( A, B ) = (cid:76) q ∈ Z KK qτG ( A, B ) . (2) Let ( E, π, F ) be a qτ -twisted G -equivariant Kasparov ( A, B )-module. Then, we can definea Kasparov ( A (cid:111) ( p − q ) τ G, B (cid:111) pτ G )-module ( E (cid:111) pτ G, π (cid:111) lt , (cid:101) F | E (cid:111) pτ G ). By the correspondence( E, π, F ) (cid:55)→ ( E (cid:111) pτ G, π (cid:111) lt , (cid:101) F | E (cid:111) pτ G ), we define a homomorphism j pτG : KK qτG ( A, B ) → KK ( A (cid:111) ( p − q ) τ G, B (cid:111) pτ G ) . It is called the partial descent homomorphism . When we should emphasize “ q ” in the partialdescent homomorphism, j pτG is denoted by j pτG,q . 353) In the same situation, we define the analytic assembly map µ pτG : KK pτG ( C ( X ) , C ) → KK ( C , C (cid:111) pτ G )by µ pτG ( x ) := [ c X ] (cid:98) ⊗ C ( X ) (cid:111) G j pτG,p ( x ) (see the following remark).(4) We do the parallel constructions for R KK -theory: R KK qτG ( X ; A, B ) ⊆ R KK G τ ( X ; A, B ) ,j pτG : R KK qτG ( X ; A, B ) → R KK ( X ; A (cid:111) ( p − q ) τ G, B (cid:111) pτ G ) and ν pτG : R KK pτG ( X ; C ( X ) , Cl τ ( X )) → KK ( C , C (cid:111) pτ G ) . Remarks . (1) We use the notation of the twisted K -theory, because a central extension of agroup acting on a space is a special example of a twisting. See [FHT1] for the details.(2) The Kasparov product takes the following form: KK pτG ( A, A ) × KK qτG ( A , B ) → KK ( p + q ) τG ( A, B ) . In particular, we can still define the Poincar´e duality homomorphism by the same formula:PD : KK pτG ( C ( X ) , C ) ∼ = R KK pτG ( X ; C ( X ) , Cl τ ( X )) . Moreover, µ pτG = ν pτG ◦ PD. The reformulated versionPD : KK pτG ( A ( X ) , S ε ) ∼ = R KK pτG ( X ; S ε (cid:98) ⊗ C ( X ) , S ε (cid:98) ⊗ C ( X )) ,µ pτG = ν pτG ◦ PD : KK pτG ( A ( X ) , S ε ) → KK ( S ε , S ε (cid:111) pτ G )can be easily proved.(3) Let us briefly explain the reason why j pτG,p (which is defined on KK pτG ( C ( X ) , C )) takes valuesin KK ( C ( X ) (cid:111) G, C (cid:111) pτ G ). Let a : G τ → C ( X ) be at level q , and let e : G τ → E be at level p .Then π (cid:111) lt( a )( e )( g ) = (cid:90) G τ π ( a ( h )) α Eh ( e ( h − g )) dh = (cid:90) G (cid:90) U (1) π ( a ( zh )) α Ezh ( e ([ zh ] − g )) dh = (cid:90) G (cid:90) U (1) z q π ( a ( h )) z p (cid:110) α Eh ( z − p e ( h − g )) (cid:111) dh. This integral vanishes unless q = 0.(4) By definition, [ c X ] is an element of KK ( C , C ( X ) (cid:111) G τ ). However, since i ( U (1)) acts on C ( X ) trivially, the projection c ∈ C ( X ) (cid:111) G τ is at level 0. Therefore, [ c X ] belongs to the directsummand KK ( C , C ( X ) (cid:111) G ). We used this fact to define µ pτG .The above assembly maps contain all the information of the assembly maps for G τ in thefollowing sense. Note that C (cid:111) pτ G is contained in C (cid:111) G τ as a direct summand. Thus, we have aninjection KK ( C , C (cid:111) pτ G ) (cid:44) → KK ( C , C (cid:111) G τ ). 36 roposition 3.42. The following two diagrams commute: KK pτG ( C ( X ) , C ) (cid:31) (cid:127) (cid:47) (cid:47) µ pτG (cid:15) (cid:15) KK G τ ( C ( X ) , C ) µ Gτ (cid:15) (cid:15) KK ( C , C (cid:111) pτ G ) (cid:31) (cid:127) (cid:47) (cid:47) KK ( C , C (cid:111) G τ ) R KK pτG ( X ; C ( X ) , Cl τ ( X )) (cid:31) (cid:127) (cid:47) (cid:47) ν pτG (cid:15) (cid:15) R KK G τ ( X ; C ( X ) , Cl τ ( X )) ν Gτ (cid:15) (cid:15) KK ( C , C (cid:111) pτ G ) (cid:31) (cid:127) (cid:47) (cid:47) KK ( C , C (cid:111) G τ ) . Let us describe the twisted versions of Proposition 3.23, Proposition 3.28 and Proposition 3.38.Note that G acts on K (cid:0) L ( G, qτ ) (cid:1) by the formulaAd R g ( k ) := R (cid:101) g ◦ k ◦ R (cid:101) g − , where (cid:101) g is a chosen lift of g ∈ G . The ambiguity of the choice of a lift is cancelled out. The sameargument of [Loi, Proposition 4.8] shows the following. Proposition 3.43.
Let A be an X (cid:111) G - C ∗ -algebra and let A := ( { A x } x ∈ X , Γ A ) be the u.s.c. fieldassociated to A . Then, we have an isomorphism A (cid:111) qτ G ∼ = C (cid:16) X × G, Ad R (cid:98) ⊗ α A (cid:8) K ( L ( G, qτ )) (cid:98) ⊗ A x (cid:9) x ∈ X (cid:17) . Let a ∈ C c ( G, A ) . The integral kernel of the corresponding equivariant section is given by x (cid:55)→ (cid:104) ( g, h ) (cid:55)→ µ ( h ) − α Ag − [ a ( gh − )( gx )] (cid:105) , where a ( gh − )( gx ) is the evaluation of a ( gh − ) ∈ A at gx . The above integral kernel is also denotedby k a ( g, h ; x ) . Note that K (cid:0) L ( G, pτ ) , L ( G, ( p − q ) τ ) (cid:1) is a Hilbert K (cid:0) L ( G, pτ ) (cid:1) -module by the followingoperations: For k, k , k ∈ K (cid:0) L ( G, pτ ) , L ( G, ( p − q ) τ ) (cid:1) and b ∈ K (cid:0) L ( G, pτ ) (cid:1) , k · b := k ◦ b , and ( k | k ) K ( L ( G,pτ )) := k ∗ ◦ k .This Hilbert module admits a left K (cid:0) L ( G, ( p − q ) τ ) (cid:1) -module structure π : K (cid:0) L ( G, ( p − q ) τ ) (cid:1) → L K ( L ( G,pτ )) (cid:0) K (cid:0) L ( G, pτ ) , L ( G, ( p − q ) τ ) (cid:1)(cid:1) given by π ( a )( k ) := a ◦ k .Let { V x } be a qτ -twisted X (cid:111) G -equivariant u.s.c. field of vector spaces. Then, G acts on neither K ( L ( G, pτ ) , L ( G, ( p − q ) τ )) nor { V x } x ∈ X , but it does act on { K ( L ( G, pτ ) , L ( G, ( p − q ) τ )) (cid:98) ⊗ V x } x ∈ X by the following: For F ∈ K ( L ( G, pτ ) , L ( G, ( p − q ) τ )), v ∈ V x , (cid:101) g ∈ G τ and z ∈ U (1), R z (cid:101) g ◦ F ◦ R ( z (cid:101) g ) − (cid:98) ⊗ ( z (cid:101) g ) · v = [ z p − q R (cid:101) g ] ◦ F ◦ [ z − p R (cid:101) g − ] (cid:98) ⊗ z q ( (cid:101) g · v ) = R (cid:101) g ◦ F ◦ R (cid:101) g − (cid:98) ⊗ (cid:101) g · v. With this observation, we can describe the twisted descent homomorphism for R KK -theory in thelanguage of fields. Proposition 3.44.
Let A and B be X (cid:111) G - C ∗ -algebras and let ( E, π, F ) be a qτ -twisted X (cid:111) G -equivariant Kasparov ( A, B ) -module. We suppose that F is actually equivariant. Let A :=37 { A x } x ∈ X , Γ A ) , B := ( { B x } x ∈ X , Γ B ) and E := ( { E x } x ∈ X , Γ E ) be the u.s.c. fields associated to A , B and E , respectively. Then, we have an isomorphism E (cid:111) pτ G ∼ = C (cid:16) X × G,α E (cid:98) ⊗ Ad R (cid:8) E x (cid:98) ⊗ K ( L ( G, pτ ) , L ( G, ( p − q ) τ )) (cid:9) x ∈ X (cid:17) as bimodules. Moreover, π (cid:111) ( p − q ) τ lt corresponds to { π x (cid:98) ⊗ id } x ∈ X and F corresponds to { F x (cid:98) ⊗ id } x ∈ X .Therefore, j pτG ( E , { π x } x ∈ X , { F x } x ∈ X ) is represented by (cid:16) C (cid:16) X × G,α E (cid:98) ⊗ Ad R (cid:8) E x (cid:98) ⊗ K ( L ( G, pτ ) , L ( G, ( p − q ) τ )) (cid:9) x ∈ X (cid:17) , { π x (cid:98) ⊗ id } x ∈ X , { F x (cid:98) ⊗ id } x ∈ X (cid:17) . The following is the twisted version of Proposition 3.30. Let [ D ] = ( L ( X, E ) , π, D ) be an indexelement of a G -equivariant Dirac operator. Proposition 3.45.
We define a ( C ( X ) (cid:111) pτ G, C (cid:111) pτ G ) -bimodule (cid:16) L (cid:16) X × G,α E (cid:98) ⊗ R (cid:98) ⊗ rt { E (cid:98) ⊗ L ( G, pτ ) (cid:98) ⊗ ( C (cid:111) − pτ G ) } (cid:17) , π (cid:111) lt , (cid:101) D (cid:17) in the same way of Definition 3.29. Then, it is isomorphic to j pτG, ([ D ]) by the correspondence e (cid:55)→ k e given by k e ( g , g ; x ) := (cid:112) µ ( g ) − g − (cid:2) e ( g g − , g x ) (cid:3) . Remarks . (1) Since e ∈ E (cid:111) qτ G is at level q , k e is at level q with respect to g , and it is atlevel − q with respect to g .(2) The Hilbert ( C (cid:111) G )-module structure on C (cid:111) G used in Definition 3.29 is defined by e · b := b ∨ ∗ e and ( e | e ) C(cid:111) G = [ e ∗ e ∗ ] ∨ for e, e , e , b ∈ C (cid:111) G . The correspondence b (cid:55)→ b ∨ exchanges C (cid:111) pτ G and C (cid:111) − pτ G . Thus, C (cid:111) − pτ G has a Hilbert C (cid:111) pτ G -module structure by theseformulas.Let us give the twisted version of Proposition 3.38. We work on the same situation. We cancompute k (cid:103) dα Ev e by the same argument of Lemma 3.33. Take a linear splitting s : g (cid:44) → g τ . For v ∈ h and e ∈ C ∞ c ( G τ , C ∞ c ( X, E )) at level p , k (cid:103) dα Ev e ( x ) = dR − s ( v ) k e ( x ) + d rt − s ( v ) k e ( x ) . Note that the right hand side is independent of the choice of s , because the infinitesimal generatorof i ( U (1)) acts on L ( G, pτ ) as p √− C (cid:111) − pτ G as − p √− Proposition 3.47. (1) L (cid:0) X/H × G/H { E (cid:98) ⊗ L ( G, pτ ) (cid:98) ⊗ ( C (cid:111) − pτ G ) } (cid:1) admits a Hilbert C (cid:111) pτ G -module structure, and it admits a ∗ -homomorphism π (cid:111) lt : C (cid:0) X/H × G/H K ( L ( G )) (cid:1) → L C(cid:111) pτ G (cid:0) L (cid:0) X/H × G/H { E (cid:98) ⊗ L ( G, pτ ) (cid:98) ⊗ ( C (cid:111) − pτ G ) } (cid:1)(cid:1) by the same formulas of the untwisted cases Proposition 3.38. (2) We define an operator D on L (cid:0) X/H × G/H { E (cid:98) ⊗ L ( G, pτ ) (cid:98) ⊗ ( C (cid:111) − pτ G ) } (cid:1) by (cid:88) i [id E (cid:98) ⊗ dR s ( v i ) (cid:98) ⊗ id + id E (cid:98) ⊗ id (cid:98) ⊗ d rt s ( v i ) ] (cid:98) ⊗ c ( v i ) + D base . Then, j pτG, ([ D ]) is represented by (cid:0) L (cid:0) X/H × G/H { E (cid:98) ⊗ L ( G, pτ ) (cid:98) ⊗ ( C (cid:111) − pτ G ) } (cid:1) , π (cid:111) lt , D (cid:1) . Index problem on proper LT -spaces From this section, we start the study of infinite-dimensional spaces. The aim of this section is toexplain the precise setting of the problem and formulate the main result. First, we will define proper LT -spaces and set up the concrete problem. We will clarify what is necessary to formulate. Second,we will review representation theory of LT from [FHT2]. In addition, we will recall the substitutefor the “ τ -twisted group C ∗ -algebra of LT ” from [T3]. Third, we will introduce R KK -theory fornon-locally compact spaces. The detailed study on this subject will be done in [NT]. Finally, wewill study LT -equivariant KK -theory, which was introduced in [T4] but the study on general theorywas left. In this subsection, we will prove that the Kasparov product is well-defined and associativefor reformulated LT -equivariant KK -theory. Then, we will formulate the main results. We define the loop group of the circle group T = S as a Hilbert Lie group as follows. In thissetting, the paper is full of circle groups, and hence we must distinguish all of them: The target ofthe loop group is denoted by T ; The source of loops is denoted by T rot ; When we consider a centralextension by a circle group, we denote it by U (1). Definition 4.1. (1) Let U L m be the completion of (cid:26) f ∈ C ∞ ( T rot , R ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) f ( θ ) dθ = 0 (cid:27) with respect to the “ L m -metric” (cid:107) f (cid:107) L m := 1 π (cid:90) π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddθ (cid:12)(cid:12)(cid:12)(cid:12) m f ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) dθ for m ≥ , where (cid:12)(cid:12) ddθ (cid:12)(cid:12) m is given by the functional calculus of the self-adjoint operator ddθ on C ∞ ( T rot , R ).(2) Let Π T ⊆ t be the kernel of exp : t → T , which can be identified with the set of homotopyclasses of free loops on T : Π T ∼ = π ( T ) ∼ = Z . We define the Hilbert Lie group LT L m by LT L m := T × Π T × U L m . It is canonically identified with the set of “ L m -loops” by the following correspondence: For an L m -loop l , there is an element n ∈ Π T and an L m -map f : S → R satisfying l ( θ ) = exp( θn ) exp( f ( θ ));The T -component is given by exp (cid:0)(cid:82) S f ( θ ) dθ (cid:1) ; The U L m -component is given by u ( θ ) := f ( θ ) − (cid:82) T rot f ( θ ) dθ .(3) Each of U L m and LT L m has a rotation symmetry given by the same formula θ · u ( θ ) := u ( θ + θ ).(4) The Hilbert space L t ∗ L m − := Ω L m − ( T rot , t ) consisting of t -valued 1-forms over T rot can beregarded as the set of connections on the trivial T -bundle over T rot . Thus, it admits an LT L m -actiondefined by the gauge transformation: l · A ( θ ) := A ( θ ) + dl ( θ ) l ( θ ) − . We denote the holonomy mapby hol : L t ∗ L m − → T .(5) If m (cid:48) > m , there exist continuous homomorphisms U L m (cid:48) → U L m and LT L m (cid:48) → LT L m , andthese make two inverse systems. The inverse limits of them are denoted by U and LT , and they39re identified with the set of C ∞ -loops, thanks to the Sobolev embedding theorem . We can do thesame thing of (4) for this case: We define L t ∗ := Ω C ∞ ( T rot , t ); It is equipped with the holonomymap hol : L t ∗ → T ; It is equipped with an LT -action given by the gauge transformation. Remarks . (1) Although the standard norm of U L m is given by (cid:107) f (cid:107) L m := π (cid:82) π (cid:12)(cid:12) (1 + (cid:52) ) m/ f ( θ ) (cid:12)(cid:12) dθ ,we use the above, in order to simplify the formulas. The same kind of simplification is mentionedin [Fre].(2) The gauge action given in (4) is isometric, proper and cocompact. Thus, the LT L m -manifold L t ∗ L m − satisfies the setting of the index theorem of [Kas3], except that the manifold and the groupare infinite-dimensional. In fact, manifolds which we study in the present paper, are different fromit by compact sets.(3) The Lie algebra Lie( U L m ) of U L m can be identified with itself. That of LT L m is isomorphicto t ⊕ Lie( U L m ).(4) The holonomy map is given by the following composition: L t ∗ L m − (cid:82) −→ t exp −−→ T. Definition 4.3 (See [T3]) . Let M be an even-dimensional compact Riemannian T -equivariant Spin c -manifold equipped with a T -invariant smooth map φ : M → T . We define an LT L m -manifold M L m for 1 / ≤ m by the fiber product M L m Φ −−−−→ L t ∗ L m − Hol (cid:121) (cid:121) hol M φ −−−−→ T and we call a proper LT -space . The induced maps are denoted by Hol and Φ as above. Remarks . (1) A proper LT -space is a Hilbert manifold equipped with an isometric, proper andcocompact LT L m -action, and it has a proper smooth equivariant map M L m → L t ∗ L m − induced by φ . Apply this construction for m = ∞ , we obtain an ILH-manifold M , where we omit the subscript L ∞ .(2) We can also say that φ is T -equivariant with respect to the adjoint action on T : φ ( t .x ) = t φ ( x ) t − , since T is commutative. If one wants to replace T with some noncommutative group, oneneeds to assume that φ : M → G is equivariant with respect to the adjoint action. See [AMM, Mei1]for details. Example . Let M be an S -symplectic manifold. Then, by perturbing the symplectic form, M admits an S -valued moment map φ [McD]. Then, the induced proper LT -space is a Hamiltonian LT -space [AMM]. In this case, the induced map M L m → L t ∗ L m − is called the moment map . Inthis sense, proper LT -spaces are obvious generalizations of Hamiltonian LT -spaces.A proper LT -space has the following simple topological type. Although there are no directlycorresponding results for noncommutative loop groups, the following is related to the abelianizationof [LMS]. Lemma 4.6 ([T3]) . Let M L k = M φ × hol L t ∗ L k − be a proper LT -space. Let t ⊆ L t ∗ L k − be thesubspace consisting of constant sections. Then, (cid:102) M := Φ − ( t ) ⊆ M L k is a smooth manifold and it The resulting Lie groups are ILH-Lie groups. “ILH” stands for “Inverse Limit of Hilbert”. See [Omo] for details. s a principal Π T -bundle over M . Moreover, it is a global slice of the t × U L k -action, and hence wehave M L k ∼ = (cid:102) M × U L k . The goal of our project is to formulate and prove an LT -equivariant index theorem for proper LT -spaces. We investigate the topological aspects of this project in the present paper. In order toexplain the problem precisely, we need to introduce a U (1)-central extension and a Spinor of LT .In the following, we identify t and R to simplify the notations. Inner products are represented likeproducts of real numbers. Definition 4.7 (See [PS, FHT2, T3]) . (1) For f , f ∈ U L m , we define a two-cocycle τ ( f , f ) := exp (cid:18) iπ (cid:90) T rot f ( θ ) f (cid:48) ( θ ) dθ (cid:19) ∈ U (1) . The central extension U τL m is given by U L m × U (1) equipped with the multiplication given by( f , z ) · ( f , z ) := ( f + f , z z τ ( f , f )) for f , f ∈ U L m . The lift of the T rot -action is given by θ · ( f, z ) = ( θ · f, z ),(2) Let κ τ : Π T → Hom(Π T , Z ) be an injective homomorphism such that κ τ ( n, m ) ∈ Z . Then,we can define a 1-dimensional representation κ τ ( n/
2) : T → U (1) for n ∈ Π T in an obviousway. This representation is also denoted by κ τn/ . We define a U (1)-central extension ( T × Π T ) τ of T × Π T by ( T × Π T ) × U (1) whose multiplication is defined by (( t , n ) , z ) · (( t , n ) , z ) =(( t t , n + n ) , z z κ τn / ( t ) κ τn / ( t ) − ) for ( t , n ) , ( t , n ) ∈ T × Π T .(3) The U (1)-central extension of LT L m is defined by the exterior tensor product: LT τL m :=( T × Π T ) τ (cid:2) U τL m .In order to define a Spinor space, we need to specify a base of L t L m . Notations 4.8. (1) On Lie( U L m ), we take a complete orthonomal systemcos θ, sin θ, cos 2 θ m , sin 2 θ m , cos 3 θ m , sin 3 θ m , · · · . The corresponding tangent vectors are denoted by ( e , f , e , f , · · · ) and the coordinate with respectto this base is denoted by ( x , y , x , y , · · · ).(2) We define an unbounded operator d on Lie( U L m ) by du := du/dθ , which is the infinitesimalgenerator of the T rot -action. We introduce the complex structure J there by d/ | d | : Concretely, J ( e n ) = − f n and J ( f n ) = e n . We introduce the following complex orthonormal base: z n := 1 √ (cid:0) e n + √− f n (cid:1) and z n := 1 √ (cid:0) e n − √− f n (cid:1) . (3) The finite-dimensional approximation U L m ,N is the subgroup whose Lie algebra is given bythe linear span of e , f , e , f , · · · , e N , f N . The algebraic inductive limit with respect to the natural inclusion U L m ,N (cid:44) → U L m ,N +1 is denotedby U fin . We define d , J and the complex base on U L m ,N in the same way of (1) and (2).For convenience of the reader, we explicitly describe the Lie bracket on Lie( U L m ). One can provethe following by a simple calculation. 41 emma 4.9. We denote the infinitesimal generator of i ( U (1)) by K . Then, [ e n , f n ] = n − m K and [ z n , z n ] = −√− n − m K . Let us introduce the Spinor space. In addition, we introduce its dual and other two Cliffordmultiplications on the tensor product of the Spinor and its dual, which will be used to define a KK -element substituting for the index element in the next section. Definition 4.10. (1) The Spinor space S U of Lie( U L m ) is defined by the exterior algebra of thenegative part of the complexfication of Lie( U L m ): • S U, fin := (cid:86) alg (cid:76) n> C z n . • S U is the completion of S U, fin with respect to the metric given by( z i ∧ z i ∧ · · · ∧ z i n | z j ∧ z j ∧ · · · ∧ z j m ) S U = (cid:40) n = m and i = j , i = j , · · · , i n = j m )0 (otherwise) , where we have assumed that i < i < · · · < i n and j < j < · · · < j m . • The Clifford multiplication is defined by γ ( z n ) := √ z n ∧ γ ( z n ) := −√ z n (cid:99) , where (cid:99) is the interior product ( z n ∧ ) ∗ .The unit vector corresponding to “1” in the exterior algebra S U is denoted by f , where “ f ” comesfrom “fermion”.(2) The dual of ( S U , γ ) is naturally a left Cliff + (Lie( U L m ))-module. We explicitly construct itas follows: • S ∗ U, fin := (cid:86) alg (cid:76) n> C z n . • S ∗ U is the completion of S ∗ U, fin with respect to the parallel way of S U . • S U and S ∗ U are mutually dual by the bilinear pairing (cid:104) z i ∧ z i ∧ · · · ∧ z i n , z j ∧ z j ∧ · · · ∧ z j m (cid:105) := (cid:40) n = m and i = j , i = j , · · · , i n = j m )0 (otherwise) , where we have assumed that i < i < · · · < i n and j < j < · · · < j m . • The Clifford multiplication is given by γ ∗ ( z n ) = −√ z n (cid:99) ◦ (cid:15) S ∗ U γ ∗ ( z n ) = −√ z n ∧ ◦ (cid:15) S ∗ U . S ∗ U is denoted by ∗ f .(3) We define two other Clifford multiplications on S ∗ U (cid:98) ⊗ S U by c ( v ) := 1 √ (cid:0) id (cid:98) ⊗ γ ( v ) − √− γ ∗ ( v ) (cid:98) ⊗ id (cid:1) c ∗ ( v ) := √− √ (cid:0) id (cid:98) ⊗ γ ( v ) + √− γ ∗ ( v ) (cid:98) ⊗ id (cid:1) for v ∈ Lie( U fin ). Note that c ( v ) = −(cid:107) v (cid:107) id and { c ∗ ( v ) } = (cid:107) v (cid:107) id. See [FHT2, T4] for details.The geometrical situation of the problem of the present paper is the following. Problem 4.11.
Let M be an even-dimensional compact Riemannian T -equivariant Spin c -manifoldequipped with a T -invariant smooth map φ : M → T and let M = M φ × hol L t ∗ be the correspondingproper LT -space. Suppose that a T -equivariant Spinor bundle S M and a τ -twisted LT L k -equivariantline bundle L over M , are given. The pullback of S M to (cid:102) M is denoted by S (cid:102) M . We define an LT -equivariant Spinor bundle S M over M by the exterior tensor product of S (cid:102) M and the trivial bundle U × S U . Then, formulate and prove the index theory for the τ -twisted LT -equivariant Clifford modulebundle L (cid:98) ⊗ S M .Remark . The assumption that M is even-dimensional and Spin c is not essential. One canremove this assumption by considering “ K -theoretical orientation sheaf” Cliff + (Hol ∗ T M ).The main result of the present paper is the topological side of the above problem: a constructionof two homomorphisms substituting for the Poincar´e duality homomorphism and the topologicalassembly map . In order to explain them, we need to introduce the substitute for the τ -twistedgroup C ∗ -algebra of LT , a non-locally compact version of R KK -theory, and LT -equivariant KK -theory. The following three subsections are devoted to them. LT In this subsection, we give a review of the representation theory of LT and recall the constructionof a substitute for the τ -twisted group C ∗ -algebra of LT .We have defined the central extension1 → U (1) i −→ LT τL m p −→ LT L m → m ≥ / T rot -action of LT L m lifts to the central extension LT τL m . With this lift, we define the concept of positive energy representation. The unitary group U ( V ) of a Hilbert space V is equipped with the compact-open topology Definition 4.13 ([PS]) . A positive energy representation (PER for short) of LT L m at level τ on a separable Hilbert space H , is a continuous homomorphism ρ : LT τL m → U ( H ) satisfying thefollowing: • ρ is at level 1, that is to say, ρ ( e LT , z ) = z id H for ( e LT , z ) ∈ i ( U (1)) ⊆ LT τL m ; • ρ lifts to LT τL m (cid:111) T rot ; and 43 The orthogonal decomposition H = ⊕ H n given by the weight of the circle action defined bythe restriction of ρ to T rot , satisfies the following: dim H n < ∞ for all n and H n = 0 forall sufficiently small n ’s. We may impose that H n = 0 for all n <
0, by retaking the lift to LT τL m (cid:111) T rot .We introduce several standard notions of representation theory (the irreducibility, the directsum and so on) in an obvious way. In particular, the dual representation of a PER is at level − T × Π T , U L m and its subgroups in an obvious way. Itis known that U τL m has the unique PER up to isomorphism as an infinite-dimensional version of theStone-von Neumann theorem (See [Kir, Theorem 2.4]). See also [PS, FHT2, T1] for details. We willconstruct it by the following recipe: First,we define a PER of the finite-dimensional approximation U τL m ,N ; Second, we define a homomorphism from the PER of U τL m ,N to that of U τL m ,M for N < M ;Finally, we take the Hilbert space inductive limit of this system and it is the desired PER. We defineit at the infinitesimal level.
Definition 4.14. (1) On L ( R N ), we define an infinitesimal representation dρ of U τL m ,N and anaction of the infinitesimal generator d of T rot , by the following operators: dρ ( e n ) := n − m ∂∂x n ; dρ ( f n ) := √− n − m x n × ; dρ ( d ) := i N (cid:88) n =1 n (cid:20)(cid:18) − ∂ ∂x n + x n − (cid:19)(cid:21) . (2) Let L ( R N ) fin be the subspace which is algebraically spanned by functions of the form“polynomial × e − (cid:107) x (cid:107) ”. Obviously, the operators dρ ( e n )’s, dρ ( f n )’s and dρ ( d ) preserve it. Remark . By using the complex base of the Lie algebra, we can rewrite dρ ( d ) as dρ ( d ) = −√− (cid:88) n n m dρ ( z n ) dρ ( z n ) . This expression is more appropriate than the above for the infinite-dimensional case.With the above representation of U L m ,N , we define a PER of U L m as follows: Definition 4.16. (1) We define an isometric embedding I N : L ( R N ) (cid:51) f (cid:55)→ f (cid:98) ⊗ π / e − x N +12 ∈ L ( R N +1 )for each N . Note that I N ’s are equivariant: dρ ( d ) ◦ I N = I N ◦ dρ ( d ) and I N ◦ dρ ( v ) = dρ ( v ) ◦ I N for v ∈ U L m ,N . The former equivariance is because π / e − x N +12 belongs to the kernel of dρ ( z N +1 ) dρ ( z N +1 ).Note that I N ’s preserve L ( R N ) fin ’s.(2) We define L ( R ∞ ) by the Hilbert space inductive limit lim −→ L ( R N ), and L ( R ∞ ) fin by the algebraic inductive limit lim −→ alg L ( R N ) fin . On L ( R ∞ ) fin , we can define the operators dρ ( e n ) , dρ ( f n ) and dρ ( d )44ince the embeddings I N ’s are equivariant. The extension of these operators are denoted by thesame symbols. Notations 4.17.
The “infinite tensor product” π / e − x (cid:98) ⊗ π / e − x (cid:98) ⊗ · · · defines a unit vector de-noted by b , where “ b ” comes from “boson”.We can do the same things for the dual representation, that is to say, we can define a continuoushomomorphism ρ ∗ : U τL m → U ( L ( R ∞ ) ∗ ) in the standard way, which is at level −
1, and L ( R ∞ ) ∗ hasthe “highest weight vector” ∗ b . Then, ρ ∗ induces a ∗ -isomorphism Op : C(cid:111) τ U L m ,N → K ( L ( R N ) ∗ ). On the right hand side, we have a natural connecting homomorphism K ( L ( R N ) ∗ ) → K ( L ( R N +1 ) ∗ )given by k (cid:55)→ k (cid:98) ⊗ P , where P is the one-dimensional projection onto C e − x N +1 . With this idea, asubstituting C ∗ -algebra for the τ -twisted group C ∗ -algebra of U L m has been defined in [T1]. Definition 4.18. (1) We define a C ∗ -algebra C (cid:111) τ U L m by the C ∗ -algebra inductive limit: C (cid:111) τ U L m := lim −→ N →∞ K ( L ( R N ) ∗ ) . It is naturally isomorphic to K (cid:16) L ( R ∞ ) ∗ (cid:17) . When we regard an element b ∈ C (cid:111) τ U L m as anelement of K (cid:16) L ( R ∞ ) ∗ (cid:17) by the natural identification, it is denoted by Op( b ). This symbol comesfrom “Operator”. The rank one projection onto C ∗ b is denoted by P C ∗ b .(2) We define a C ∗ -algebra C (cid:111) τ LT L m by C (cid:111) τ LT L m := C (cid:111) τ ( T × Π T ) (cid:98) ⊗ C (cid:111) τ U L m . (3) For every N ∈ N , we define [ C (cid:111) τ U L m ,N ] fin ⊆ K ( L ( R N ) ∗ ) by the set of finite-rank operatorspreserving L ( R N ) ∗ fin . These subspaces are preserved by the connecting homomorphisms. Hencethe algebraic inductive limit lim −→ alg [ C (cid:111) τ U L m ,N ] fin makes sense, and it is denoted by [ C (cid:111) τ U L m ] fin .We define a dense subalgebra [ C (cid:111) τ LT L m ] fin of C (cid:111) τ LT L m by the algebraic tensor product[ C (cid:111) τ LT L m ] fin := C ∞ c ( T × Π T , τ ) (cid:98) ⊗ alg [ C (cid:111) τ U L m ] fin . Notations 4.19.
A PER of LT L m is given by the tensor product of those of T × Π T ’s and U L m ’s.Thus, we can define a ∗ -homomorphism C (cid:111) τ LT L m → L ( H ) for a PER H . This homomorphism isdenoted by “Op”, following the notation of Definition 4.18 (1). Remarks . (1) The rank one operator appearing in the definition of the connecting homomor-phism K ( L ( R N ) ∗ ) → K ( L ( R N +1 ) ∗ ) is given by the Gaussianvac( a N +1 , b N +1 ) := N − m π e − ( N +1)1 − m ( a N +1 + b N +1 ) A function f on U τL m ,N at level p defines the trivial operator on a representation space ( V, σ ) at level q , unless p + q = 0. This is because the integral (cid:90) U τL m,N f ( g ) σ ( g ) dg = (cid:90) U L m,N (cid:90) U (1) z p f ( g ) z q σ ( g ) dzdg vanishes unless p + q = 0. U L m ,N +1 (cid:9) U L m ,N . This is proved by a simple calculation using the definition of ρ and π ρ , andwe leave it to the reader.(2) A finite rank operator preserving L ( R N ) fin is given by a finite linear combination of[ dρ ( z )] α [ dρ ( z )] α · · · [ dρ ( z N )] α N ◦ P C ∗ b ◦ [ dρ ( z )] β [ dρ ( z )] β · · · [ dρ ( z N )] β N . One can prove the following formulas by integration by parts and the Leibniz rule: dρ ( z k ) ◦ Op( f ) = Op (cid:18)(cid:20) − k − + m (cid:18) ∂∂a k + 12 k − m a k (cid:19) + √− k − + m (cid:18) ∂∂b k + 12 k − m b k (cid:19)(cid:21) f (cid:19) , Op( f ) ◦ dρ ( z k ) = Op (cid:18)(cid:20) − k − + m (cid:18) ∂∂a k + 12 k − m a k (cid:19) − √− k − + m (cid:18) ∂∂b k + 12 k − m b k (cid:19)(cid:21) f (cid:19) . By this formula, one finds that [
C (cid:111) U L m ,N ] fin is the set of functions of the form “polynomial × Gaussian”. Therefore, [
C (cid:111) U L m ] fin is regarded as the set of “functions” of the form “polynomial × Gaussian”.We have used the G -action “rt” on C (cid:111) G to compute the descent homomorphism in Definition3.29, which is given by rt g f ( h ) := f ( hg ) for f ∈ C c ( G ) and g, h ∈ G . In addition, we define anotheraction “lt” by lt g f ( h ) := f ( g − h ). For a unitary representation ρ : G → U ( V ), we can define a ∗ -homomorphism π ρ : C (cid:111) G → L ( V ). By simple calculations, we have π ρ (lt g f ) = ρ g ◦ π ρ ( f ) and π ρ (rt g f ) = π ρ ( f ) ◦ ρ g − . If G is a Lie group, we can also define “ d lt” and “ d rt” and they satisfy π ρ ( d lt X f ) = dρ X ◦ π ρ ( f ) and π ρ ( d rt X f ) = π ρ ( f ) ◦ dρ − X for X ∈ g .Since C (cid:111) τ LT L m is not defined by the completion of C c ( LT L m , τ ), the original definitions of“lt” and “rt” do not work. However, the right hand sides of π ρ (lt g f ) = ρ g ◦ π ρ ( f ) and π ρ (rt g f ) = π ρ ( f ) ◦ ρ g − still make sense. With this observation, we introduce the LT -counterparts of “lt” and“rt”. Definition 4.21. (1) For b ∈ C (cid:111) τ U L m and g ∈ U τL m , we define lt g b := Op − ( ρ g ◦ Op( b )) andrt g b := Op − (cid:0) Op( b ) ◦ ρ g − (cid:1) . We define the infinitesimal versions of “lt” and “rt” in an obviousway.(2) They and the corresponding actions on T × Π T , induce two LT τL m -actions on C (cid:111) τ LT L m .We use the same symbols “lt” and “rt” to denote these new actions. R KK -theory for non-locally compact action groupoids The goal of this subsection is to define “ R KK -theory for non-locally compact action groupoids”based on the idea which we have explained in the last several paragraphs of Section 2.2. In thepresent paper, we deal with only action groupoids. We do not explain the whole story and we willmerely define necessary concepts and prove several easy results. For more general cases and details,see [NT]. I emphasize that the primitive idea of this theory is due to Shintaro Nishikawa. I thankhim for allowing me to introduce this theory before the collaboration paper [NT].For simplicity, in this subsection, we work under the following assumption. It is satisfied forproper LT -spaces. Assumption 4.22.
Throughout this subsection, X is assumed to be a metrizable space. In par-ticular, we have Urysohn’s lemma: For any closed subsets
C, C (cid:48) ⊆ X with C ∩ C (cid:48) = ∅ , there existsa continuous function f : X → [0 , such that f | C = 0 and f | C (cid:48) = 1 . We also suppose that G is atopological metrizable group and it acts on X continuously. Then, G × X is also metrizable.46et us introduce the substitute for the concept of “ X (cid:111) G - C ∗ -algebras”. We begin with thedefinition of upper semi-continuous fields (u.s.c. fields, for short) of Banach spaces. Definition 4.23 ([Dix, Definition 10.1.2]) . (1) A u.s.c. field of Banach spaces over X is a pair E = ( { E x } x ∈X , Γ E ) of a family of Banach spaces { E x } x ∈X parameterized by X , and a subset Γ E of (cid:8) s : X → (cid:96) x ∈X E x | s ( x ) ∈ E x (cid:9) satisfying the following conditions: • Γ E is a complex vector space by the pointwise operations; • The evaluation homomorphism Γ E (cid:51) s (cid:55)→ s ( x ) ∈ E x is surjective; • The function
X (cid:51) x (cid:55)→ (cid:107) s ( x ) (cid:107) ∈ R ≥ is upper semi-continuous for each s ∈ Γ E ; and • A section s : X → (cid:96) x ∈X E x is an element of Γ E if the following is satisfied: For every x ∈ X and every (cid:15) >
0, these exists a neighborhood U x,(cid:15) of x and an s (cid:48) ∈ Γ E such that (cid:107) s ( y ) − s (cid:48) ( y ) (cid:107) ≤ (cid:15) for y ∈ U x,(cid:15) .(2) A u.s.c. field E = ( { E x } x ∈X , Γ E ) is Z -graded if every E x is Z -graded: E x = E ,x (cid:98) ⊕ E ,x ,and both of E i = ( { E i,x } x ∈X , Γ E i ) ( i = 0 ,
1) are u.s.c. fields, whereΓ E i := { s ∈ Γ E | s ( x ) ∈ E i,x for every x ∈ X } . Remark . If the function
X (cid:51) x (cid:55)→ (cid:107) s ( x ) (cid:107) is continuous for every s ∈ Γ E , the field is called a continuous field of Banach spaces . Definition 4.25.
Let E = ( { E ,x } x ∈X , Γ E ) and E = ( { E ,x } x ∈X , Γ E ) be two u.s.c. fields ofBanach spaces over X .(1) A family of bounded linear maps φ = { φ x : E ,x → E ,x } is called a homomorphismbetween E and E over X if it satisfies the following: For every s ∈ Γ E , the resulting section φ ( s ) : x (cid:55)→ φ x ( s x ) also belongs to Γ E . It is said to be injective/surjective/isomorphic/isometric ifevery φ x is injective/surjective/isomorphic/isometric.(2) A homomorphism φ : E → E over X between Z -graded u.s.c. fields of Banach spaces is even if it preserves the Z -grading, and is odd if it reveres the Z -grading. Remark . A Z -graded u.s.c. field of Banach space E admits an isometric isomorphism (cid:15) = { (cid:15) x } : E → E such that (cid:15) x ( e ,x + e ,x ) = e ,x − e ,x , where e i,x ∈ E i,x for i ∈ { , } and x ∈ X . Itis called the grading homomorphism. A homomorphism is even and odd if and only if it commutesand anti-commutes with (cid:15) , respectively.As mentioned in [TXLG, Appendix], the “total space” (cid:101) E := (cid:96) x ∈X E x admits the unique topol-ogy so that Γ E is the set of all continuous sections of the bundle (cid:101) E → X . Then, the abovehomomorphism { φ x } defines a continuous bundle map φ : (cid:101) E → (cid:101) E . We can construct the pullbackof a u.s.c. field of Banach spaces by a continuous map F : Y → X . This field over Y is denoted by F ∗ E .Using this construction, we can define the concept of continuous group actions. See also [LG,Definition 3.5]. By the G -action on X , we define two continuous maps s, r : G × X → X by s ( g, x ) := x and r ( g, x ) := g · x . 47 efinition 4.27. (1) Let E = ( { E x } x ∈X , Γ E ) be a u.s.c. field of Banach spaces over X . A continuous G -action on E is an isomorphism α : s ∗ E → r ∗ E over G × X satisfying the followingcommutative diagram for every x ∈ X and g ∈ G : E x α ( g,x ) (cid:47) (cid:47) α ( hg,x ) (cid:34) (cid:34) E g · xα ( h,g · x ) (cid:123) (cid:123) E ( hg ) · x An isomorphism α ( g,x ) is also denoted by ( α g ) x , by regarding α g as an automorphism on E . If everyisomorphism is isometric, this action is said to be isometric. A u.s.c. field of Banach spaces over X equipped with a G -action is said to be X (cid:111) G -equivariant .(2) For X (cid:111) G -equivariant u.s.c. fields of Banach spaces E and E , a homomorphism φ : E → E over X is said to be X (cid:111) G -equivariant if it satisfies ( α g ) x ◦ φ x = φ g · x ◦ ( α g ) x .More interesting types of fields, including fields of C ∗ -algebras or those of Hilbert modules, canbe defined. Definition 4.28. (1) Let A = ( { A x } , Γ A ) be a u.s.c. field of Banach spaces. It is a u.s.c. fieldof C ∗ -algebras over X if the following conditions are satisfied: • Each fiber A x is a C ∗ -algebra; • Γ A is closed under the multiplication and the adjoint; and • The norm satisfies the C ∗ -condition (cid:107) a ∗ a (cid:107) = (cid:107) a (cid:107) for each x ∈ X and a ∈ A x .For u.s.c. fields of C ∗ -algebras A and A over X , a homomorphism φ : A → A over X is called a ∗ -homomorphism if it preserves the multiplication and the adjoint. We define theconcepts of Z -graded u.s.c. fields of C ∗ -algebras, even/odd ∗ -homomorphisms, in the same way forfields of Banach spaces.(2) A u.s.c. field of C ∗ -algebras A = ( { A x } , Γ A ) over X is said to be locally separable ifthe following condition is satisfied: For every x ∈ X , there exists an open neighborhood U x and acountable set of continuous sections { a n } ⊆ Γ A such that { a n ( y ) } ∈ A y is dense for every y ∈ U x .(3) Let A = ( { A x } , Γ A ) be a u.s.c. field of C ∗ -algebras. A u.s.c. field of Banach spaces E = ( { E x } , Γ E ) is a u.s.c. field of Hilbert A -modules over X if the following conditions aresatisfied: • Each fiber E x is a Hilbert A x -module; • Γ E is closed under the multiplication by Γ A , and the inner product of two elements e , e ∈ Γ E defines an element of Γ A ; and • The Banach space norm of e ∈ E x is given by (cid:107) e (cid:107) E x = (cid:113) (cid:107) ( e | e ) E x (cid:107) A x .(4) A u.s.c. field E of Hilbert A -modules over X is said to be locally countably generated if the following condition is satisfied: For any x ∈ X , there exist a countable set { s n } n ∈ N ⊆ Γ E andan open neighborhood U x such that the A y -linear span of { s n ( y ) } n ∈ N ⊆ E y is dense in E y for every y ∈ U x . 485) A u.s.c. field of C ∗ -algebras A over X is said to be X (cid:111) G -equivariant if an isometric G -action on A is given and each ( α g ) x preserves the multiplication and the adjoint. We define theconcept of X (cid:111) G -equivariant Hilbert A -modules in a parallel way. Remark . Let E be a locally countably generated u.s.c. field of Hilbert A -modules over X .Then, thanks to the proof of [MS, Lemma 5.9] and a partition of unity, we can find a countable set { s n } ⊆ Γ E so that { s n ( x ) } spans E x over A x for every x ∈ X . Moreover, E can be written as anorthogonally complementable submodule of the trivial bundle l ( N ) (cid:98) ⊗ A . This result will be provedin [NT].Let us give typical examples. Example . (1) Let C ( X ) be the ∗ -algebra consisting of C -valued continuous functions on X .Then, C ( X ) = ( { C x } x ∈X , C ( X ))is an X (cid:111) G -equivariant continuous field of C ∗ -algebras by the “trivial” action ( α g ) x : C x (cid:51) z (cid:55)→ z ∈ C g · x . Note that the single C ∗ -algebra C ( X ) is trivial if X is not locally compact. In this sense, theconcept of u.s.c. fields of C ∗ -algebras is more general than single C ∗ -algebras.(2) Let E = (cid:96) x ∈X E x → X be a G -equivariant locally trivial Hilbert space bundle whose fibersare separable, and let C ( X , E ) be the set of continuous sections. Then, E = ( { E x } x ∈X , C ( X , E ))is an X (cid:111) G -equivariant continuous field of Hilbert C ( X )-modules.(3) For a G - C ∗ -algebra A and a G -equivariant Hilbert A -module E , A (cid:98) ⊗ C ( X ) = ( { A } , C ( X , A ))is an X (cid:111) G -equivariant continuous field of C ∗ -algebras and E (cid:98) ⊗ C ( X ) = ( { E } , C ( X , E )) is an X (cid:111) G -equivariant continuous field of Hilbert A (cid:98) ⊗ C ( X )-module by the diagonal action. These fields aresaid to be trivial .Let us define the filed of adjointable operators and that of compact operators for a u.s.c. fieldof C ∗ -algebras B and a u.s.c. field of Hilbert B -modules E . Definition 4.31.
Let B be an X (cid:111) G -equivariant u.s.c. field of C ∗ -algebras and let E be a locallycountably generated X (cid:111) G -equivariant u.s.c. field of Hilbert B -modules.(1) The filed of adjointable operators L ( E ) = L B ( E ) over E is defined by the family of C ∗ -algebras { L ( E x ) } x ∈X and the set of sections Γ L ( E ) given by the following: A section x (cid:55)→ T x iscontinuous if and only if, for any continuous section ξ = { ξ x } x ∈X ∈ Γ E , the new sections x (cid:55)→ T x ξ x and x (cid:55)→ ( T x ) ∗ ξ x are again continuous.(2) We define a single C ∗ -algebra L B ( E ) by the set of bounded sections (cid:8) { T x } ∈ Γ L ( E ) | (cid:107) T x (cid:107) is bounded (cid:9) by the pointwise operations and the norm (cid:107){ T x }(cid:107) := sup x ∈X (cid:107) T x (cid:107) .(3) A continuous section { F x } ∈ Γ L ( E ) is a locally finite rank operator if the followingcondition is satisfied: For each point x ∈ X , there exists an open neighborhood U x and contin-uous sections e , f , · · · , e n , f n of E such that F y = (cid:80) e i ( y ) (cid:98) ⊗ [ f i ( y )] ∗ on U x , where the operator e i ( y ) (cid:98) ⊗ [ f i ( y )] ∗ is defined by E y (cid:51) v (cid:55)→ e i ( y ) ( f i ( y ) | v ) B y ∈ E y .(4) The filed of B -compact operators K ( E ) = K B ( E ) over E is defined by the family of C ∗ -algebras { K ( E x ) } x ∈X and the set of sections Γ K ( E ) given by the following. A section x (cid:55)→ T x ∈ ( E x ) is continuous if and only if there exists a net of locally finite rank operators {{ ( F λ ) x } x ∈X } λ ∈ Λ satisfying the following condition: For any (cid:15) > x ∈ X , there exists an open neighborhood U (cid:15),x and λ such that λ ≥ λ implies (cid:107) T y − ( F λ ) y (cid:107) < (cid:15) on y ∈ U (cid:15),x .The field K ( E ) is an X (cid:111) G -equivariant u.s.c. field of C ∗ -algebras, where the G -action is definedby ( α K ( E ) g ) x ( k x ) := ( α E g ) x ◦ k x ◦ ( α E g − ) g · x . This will be proved in [NT] Definition 4.32.
Let A = ( { A x } , Γ A ) and B = ( { B x } , Γ B ) be X (cid:111) G -equivariant u.s.c. fields of C ∗ -algebras.(1) Let E = ( { E ,x } , Γ E ) be an X (cid:111) G -equivariant u.s.c. field of Hilbert A -modules, andlet E = ( { E ,x } , Γ E ) be an X (cid:111) G -equivariant u.s.c. field of Hilbert B -modules. Suppose thatan X (cid:111) G -equivariant ∗ -homomorphism π = { π x } x ∈X : A → L B ( E ) is given. Then, we definean X (cid:111) G -equivariant u.s.c. field of Hilbert B -modules E (cid:98) ⊗ A E by ( { E ,x (cid:98) ⊗ A x E ,x } x ∈X , Γ E (cid:98) ⊗ A E ),where a section s is an element of Γ E (cid:98) ⊗ A E , if for every x ∈ X and for every (cid:15) > U x,(cid:15) of x and finitely many sections { s ,i } ⊆ Γ E and { s ,i } ⊆ Γ E such that (cid:107) s ( y ) − (cid:80) i s ,i ( y ) (cid:98) ⊗ s ,i ( y ) (cid:107) < (cid:15) on y ∈ U x,(cid:15) . For an adjointable operator T on E , we can define T (cid:98) ⊗ id in an obvious way.(2) In particular, we can define the pushout as follows. Let σ : A → B be an X (cid:111) G -equivariant ∗ -homomorphism. It gives a Hilbert B -module B equipped with a ∗ -homomorphism σ : A → B ⊆ L B ( B ). We define the pushout σ ∗ ( E ) by E (cid:98) ⊗ A B . Remark . Let I be the closed interval [0 , X (cid:111) G -equivariant u.s.c. field of C ∗ -algebras A , we define another equivariant u.s.c. field A I by ( { A x (cid:98) ⊗ C ( I ) } x ∈X , Γ A I ), where Γ A I is definedby the same way of (1) of the above definition. Then, we have a ∗ -homomorphismev t = { ev t,x : A x I = A x (cid:98) ⊗ C ( I ) (cid:51) a (cid:98) ⊗ f (cid:55)→ a · f ( t ) ∈ A x } . Therefore, for an X (cid:111) G -equivariant u.s.c. field of Hilbert A I -module E , we can define the “evalu-ation at t of E ” by ev t, ∗ E . We will use this construction to define the homotopy equivalence.For this tensor product construction and T ∈ L A I ( E ), T (cid:98) ⊗ id is denoted by ev t, ∗ ( T ) ∈ L A (ev t, ∗ ( E )).Now, we can extend R KK -theory to our infinite-dimensional setting. Definition 4.34.
Let A = ( { A x } , Γ A ) and B = ( { B x } , Γ B ) be locally separable X (cid:111) G -equivariantu.s.c. fields of C ∗ -algebras.(1) An X (cid:111) G -equivariant Kasparov ( A , B ) -module is a triple ( E , π, F ) satisfying the fol-lowing conditions: • E = ( { E x } , Γ E ) is a locally countably generated X (cid:111) G -equivariant u.s.c. field of Z -gradedHilbert B -modules. • π = { π x } is an X (cid:111) G -equivariant even ∗ -homomorphism A → L B ( E ). • F = { F x } ∈ L B ( E ) is an odd element and it satisfies the following conditions: For any a ∈ Γ A ,all the sections x (cid:55)→ [ π x ( a x ) , F x ], x (cid:55)→ π x ( a x )( T x − T ∗ x ) and x (cid:55)→ π x ( a x ) { − ( T x ) } belong to Γ K B ( E ) , and the section G × X (cid:51) ( g, x ) ∈ a g · x { F g · x − ( α g ) x ◦ F x ◦ ( α g − ) g · x } ∈ K B g · x ( E g · x )belongs to Γ r ∗ K B ( E ) . 502) The set of isomorphism classes of G -equivariant Kasparov ( A , B )-modules is denoted by E X (cid:111) G ( A , B ). It is an abelian semigroup, whose addition is defined by ( E , π , F ) + ( E , π , F ) :=( E ⊕ E , π ⊕ π , F ⊕ F ).(3) Two elements ( E , π , F ) , ( E , π , F ) ∈ E X (cid:111) G ( A , B ) are said to be homotopic if thereexists ( E , π, F ) ∈ E X (cid:111) G ( A , B I ) such that ( E i , π i , F i ) is isomorphic to ev i, ∗ ( E , π, F ) for i = 0 , E , π , F )and ( E , π , F ).(4) The set of homotopy classes of X (cid:111) G -equivariant Kasparov ( A , B )-modules is denoted by R KK G ( X ; A , B ). The homotopy class of ( E , π , F ) is denoted by [( E , π , F )] ∈ R KK G ( X ; A , B ).(5) If G is equipped with a U (1)-central extension, we define R KK τ G ( X ; A , B ) in the same wayof Definition 3.40. R KK G ( X ; A , B ) is an abelian group, whose addition is induced by the addition in E X (cid:111) G ( A , B ).In fact, the direct sum operation is homotopy invariant, because the direct sum of the homotopiesgives a homotopy between the direct sums.We can also define the concept of Kasparov products. Definition 4.35.
Let A , A and B be locally separable X (cid:111) G -equivariant u.s.c. fields of C ∗ -algebras. Let ( E , π , F ) ∈ E X (cid:111) G ( A , A ) and let ( E , π , F ) ∈ E X (cid:111) G ( A , B ).(1) For s ∈ Γ E , we define T s : E → E (cid:98) ⊗ A E by Γ E (cid:51) e (cid:55)→ [ x (cid:55)→ s ,x (cid:98) ⊗ s ,x ∈ E ,x (cid:98) ⊗ A ,x E ,x ] ∈ Γ E (cid:98) ⊗ A E . An element F ∈ L B ( E (cid:98) ⊗ E ) is a F -connection if (cid:20)(cid:18) F F (cid:19) , (cid:18) T s T ∗ s (cid:19)(cid:21) ∈ Γ K B ( E ⊕ E ) for any s ∈ Γ E (2) ( E (cid:98) ⊗ E , π, F ) ∈ E X (cid:111) G ( A , B ) is a Kasparov product of ( E , π , F ) and ( E , π , F )if F is an F -connection satisfying the following: For arbitrary a ∈ Γ A , there exists a con-tinuous section P ∈ L B ( E ) whose value at each point is positive, such that the section x (cid:55)→ π x ( a x )[ F x , F ,x (cid:98) ⊗ id] π x ( a x ) ∗ − P x belongs to Γ K B ( E (cid:98) ⊗ E ) . Theorem 4.36 ([NT]) . We suppose that the G -action on X is proper and there exists a localslice at every point of X . We further suppose that A , A and B are locally separable. Then,for arbitrary ( E , π , F ) ∈ E X (cid:111) G ( A , A ) and ( E , π , F ) ∈ E X (cid:111) G ( A , B ) , there exists a Kasparovproduct ( E (cid:98) ⊗ E , π, F ) ∈ E X (cid:111) G ( A , B ) . It is well-defined at the R KK -level, and it is associative. The Kasparov product of x ∈ R KK G ( X ; A , A ) and y ∈ R KK G ( X ; A , B ) is denoted by x (cid:98) ⊗ A y ∈ R KK G ( X ; A , B ).We have used the fact that an equivariant R KK -element is represented by an actually equivari-ant R KK -cycle in Proposition 3.28. An analogous result for non-locally compact groupoids whichsatisfies a certain condition, will be proved in [NT]. For this time, we prove it for a special case,which will be necessary for the main theorem.We begin with a useful criterion to be operator homotopic. Lemma 4.37 (See [JT, Lemma 2.1.18]) . Let A and B be X (cid:111) G -equivariant u.s.c. fields of C ∗ -algebras. Suppose that ( E , π, F ) , ( E , π, F (cid:48) ) ∈ E X (cid:111) G ( A , B ) satisfy the following: For any a ∈ Γ A ,there exists a continuous section P ∈ L B ( E ) whose value at each point is positive such that thesection x (cid:55)→ π x ( a x )[ F x , F (cid:48) x ] π x ( a x ) ∗ − P x belongs to Γ K B ( E ) . Then, ( E , π, F ) is homotopic to ( E , π, F (cid:48) ) as equivariant Kasparov modules. roof. We define two C ∗ -subalgebras Q A ( E ) and I A ( E ) of L B ( E ) by Q A ( E ) := (cid:8) T ∈ L B ( E ) | The section x (cid:55)→ [ π x ( a ( x )) , T x ] belongs to Γ K ( E ) for any a ∈ Γ A (cid:9) I A ( E ) := (cid:8) T ∈ Q A ( E ) | The section x (cid:55)→ π x ( a ( x )) T x belongs to Γ K ( E ) for any a ∈ Γ A (cid:9) Note that I A ( E ) is a two-sided ideal of Q A ( E ). Since F and F (cid:48) belongs to Q A ( E ), so does [ F, F (cid:48) ].We check that [
F, F (cid:48) ] ≥ I A ( E ). For each state ω of L B x ( E x ) / K B x ( E x ) and each a ∈ A x for x ∈ X , let us consider the state of Q A ( E ) /I A ( E ) defined by [ T ] (cid:55)→ ω ( q x ( π x ( a ) T x π x ( a ) ∗ )), where[ T ] is the equivalence class of T in Q A ( E ) /I A ( E ) and q x is the natural projection L B x ( E x ) → L B x ( E x ) / K B x ( E x ). This new state is denoted by µ ( ω, a, x ). Then, the set of all such states { µ ( ω, a, x ) } is a faithful family of states of Q A ( E ) /I A ( E ). This is because (cid:81) x ∈X ev x : Q A ( E ) /I A ( E ) → (cid:81) x ∈X Q A x ( E x ) /I A x ( E x ) is injective and the set { µ ( ω, a, x ) } ω,a is a faithful family of Q A x ( E x ) /I A x ( E x )for each x ∈ X . Since µ ( ω, a, x )([ F, F (cid:48) ]) = ω ( π x ( a )[ F, F (cid:48) ] π x ( a )) = ω ( q x ( P x )) ≥ a, ω, x ,the commutator [ F, F (cid:48) ] ≥ I A ( E ). The remainder of the proof is the same with that of [JT,Lemma 2.1.18] and left to the reader. Definition 4.38.
Let A , B be locally separable X (cid:111) G -equivariant u.s.c. fields of C ∗ -algebras andlet C and D be separable G - C ∗ -algebras. Suppose that all of C (cid:98) ⊗ A , D (cid:98) ⊗ A , C (cid:98) ⊗ B and D (cid:98) ⊗ B are X (cid:111) G -equivariant u.s.c. fields of C ∗ -algebras (see the following remark). We define the followingtwo operations: σ X , A : KK G ( C, D ) → R KK G ( X ; C (cid:98) ⊗ A , D (cid:98) ⊗ A )by ( E, π, F ) (cid:55)→ ( { E (cid:98) ⊗ C A x } x ∈X , { π (cid:98) ⊗ id A x } x ∈X , { T (cid:98) ⊗ id A x } x ∈X ), and σ C : R KK G ( X ; A , B ) → R KK G ( X ; C (cid:98) ⊗ A , C (cid:98) ⊗ B )by ( { E x } , { π x } , { F x } ) (cid:55)→ ( { C (cid:98) ⊗ C E x } , { id C (cid:98) ⊗ π x } , { id C (cid:98) ⊗ F x } ). Remark . Even for locally compact cases, it is non-trivial whether tensor product of a u.s.c.field of C ∗ -algebras and a single C ∗ -algebra is again a u.s.c. field [KW]. However, even for infinite-dimensional cases, it is obvious that the tensor product of a single nuclear C ∗ -algebra and a locallytrivial field of C ∗ -algebras is again u.s.c. This is the case which we will deal with in the followingsections. Corollary 4.40.
Let C and D be separable G - C ∗ -algebras, and let ( E, π, F ) be a G -equivariant Kas-parov ( C, D ) -module. Then, σ G , C ( G ) ( E, π, F ) is operator homotopic to the family ( { E } g ∈G , { π } g ∈G , { g ( F ) } g ∈G ) .Proof. Let c ∈ Γ C (cid:98) ⊗ C ( G ) . By an easy computation, π ( c g )[ F, g ( F )] π ( c g ) ∗ is given by2 π ( c g ) F π ( c g ) ∗ + π ( c g ) F · { π ( c ∗ g )( g ( F ) − F ) } ∗ + π ( c g )( g ( F ) − F ) · F π ( c g ) ∗ . The first term is a positive element of L C (cid:98) ⊗ C ( G ) ( E (cid:98) ⊗ C ( G )). The second and the third ones arecontinuous sections of K ( C (cid:98) ⊗ C ( G )). By Lemma 4.37, we obtain the result.52 .4 Loop group equivariant KK -theory and the main result Loop groups are non-locally compact. Although the LT -equivariant Kasparov modules and LT -equivariant KK -groups make sense, and although the phrase “a Kasparov module is a Kasparovproduct of other two Kasparov modules” makes sense, we encounter a serious trouble when we tryto study the Kasparov product at the level of KK -theory. In the previous paper [T4], such problemsare put off.In this subsection, we re-introduce “ LT -equivariant KK -theory” using the inductive limit of LT L m -equivariant KK -theory, and we will prove the desired properties on the Kasparov product.This definition shares the same spirit with the concept of ILH-Lie groups of [Omo].Let us begin with a fundamental construction on equivariant KK -theory. Let ≤ m ≤ m (cid:48) .Suppose that A and B are LT L m - C ∗ -algebras. They are at the same time LT L m (cid:48) - C ∗ -algebras bythe inclusion i m (cid:48) ,m : LT L m (cid:48) (cid:44) → LT L m . Thus, we can define a homomorphism i ∗ m (cid:48) ,m : KK LT L m ( A, B ) → KK LT L m (cid:48) ( A, B )by the pullback of the group action by i m (cid:48) ,m . The family of homomorphisms { i m (cid:48) ,m } satisfies thecondition to be a directed system i ∗ m (cid:48) ,m ◦ i ∗ m (cid:48)(cid:48) ,m (cid:48) = i ∗ m (cid:48)(cid:48) ,m for m (cid:48)(cid:48) ≥ m (cid:48) ≥ m . Using it, we re-define“ LT -equivariant KK -theory” as follows. Definition 4.41. (1) Let A and B be LT - C ∗ -algebras. Suppose that the LT -action on themextends to LT L m for some m ≥ /
2. By the inductive limit of this system, we define KK LT ( A, B ) := lim −→ m (cid:48) →∞ KK LT L m (cid:48) ( A, B ) . We call it the LT -equivariant KK -theory .(2) We define M (cid:111) LT -equivariant KK -group by the same construction. Let A and B be M (cid:111) LT -equivariant u.s.c. fields of C ∗ -algebras, which are obtained by M L m (cid:111) LT L m -equivariantu.s.c. fields of C ∗ -algebras for some m ≥ /
2. Then, we define R KK LT ( A , B ) := lim −→ m (cid:48) →∞ R KK LT L m (cid:48) ( A , B ) . We can define the concept of Kasparov products on KK LT -theory and on R KK LT ( M )-theory.For the latter one, we can do that at the level of R KK LT L m ( M L m )-theory [NT]. For the formerone, we need an appropriate version of the Kasparov technical theorem.The following statement is the copy of Theorem 1.4 of [Kas2] except that the two non-locallycompact groups LT L m (cid:48) and LT L m appear. Note that the assumption of the statement is about the LT L m -equivariance and the conclusion is about the LT L m (cid:48) -equivariance which is weaker than the LT L m -equivariance since m (cid:48) > m . This result is sufficient for KK LT -theory. Theorem 4.42.
Let m (cid:48) > m ≥ / .Let J be a σ -unital LT L m -algebra, A and A σ -unital subalgebras in M ( J ) such that A is an LT L m -algebra. Let ∆ be a subset in M ( J ) which is separable in the norm topology and derives A and φ : LT m (cid:48) → M ( J ) a bounded function. Assume that A · A ⊆ J , A · φ ( LT L m ) ⊆ J , On a Z -graded algebra B , a subset S derives a subalgebra B (cid:48) if the graded commutator [ d, b ] ∈ B (cid:48) for every d ∈ ∆ and b ∈ B (cid:48) . ( LT L m ) · A ⊆ J , and the functions g (cid:55)→ aφ ( g ) and g (cid:55)→ φ ( g ) a are norm-continuous on LT L m forany a ∈ A + J . Then, there are LT L m (cid:48) -continuous positive even elements M , M ∈ M ( J ) such that M + M = 1 , all elements M i a i , [ M i , d ] , M φ ( g ) , φ ( g ) M , g ( M i ) − M i belong to J for any a i ∈ A i , d ∈ ∆ , g ∈ LT L m (cid:48) ( i = 1 , , and the functions g (cid:55)→ M φ ( g ) and g (cid:55)→ φ ( g ) M are norm-continuouson LT L m (cid:48) .Proof. In order to prove it, we follow the argument of [Kas2, Theorem 1.4]. We explain only thenecessary changes from it. See also [JT, Blac] for the detailed expositions on the technical theorem.The proof of [Kas2, Theorem 1.4] is outlined as follows: (1) A lemma related to the quasi-central and quasi-invariant approximate unit is verified; (2) Using this result, the technical theoremis verified.The corresponding result to (1) is the following (it is again the copy of Lemma of Theorem 1.4 of[Kas2] except that the two groups LT L m (cid:48) and LT L m appear): Let B be a C ∗ -algebra with an LT L m -action, A a σ -unital LT L m - C ∗ -algebra which is an LT L m -subalgebra of B , Y a σ -compact, locallycompact space, φ : Y → B a function satisfying the following condition: [ φ ( y ) , a ] ∈ A for all a ∈ A and y ∈ Y , and all the functions y (cid:55)→ [ φ ( y ) , a ] are norm-continuous for all a ∈ A . Then, there is acountable approximate unit { u i } for A which has the following properties: lim i →∞ (cid:107) [ u i , φ ( y )] (cid:107) = 0 for all y ∈ Y and lim i →∞ (cid:107) g ( u i ) − u i (cid:107) = 0 for all g ∈ LT L m (cid:48) . Both limits are uniform on compactsubsets of Y and bounded subsets of LT L m (cid:48) . Note that A is automatically LT L m (cid:48) -subalgebra becausethe inclusion LT L m (cid:48) → LT L m is continuous.In the proof of the lemma in [Kas2], an exhaustive sequence consisting of relatively compactopen subsets of G , was chosen. Then, the set Z in the proof of the lemma is compact, and hence theproof works. In our case, the same argument does not work, because LT L m is non-locally compact.However, there is an exhaustive sequence X ⊆ X ⊆ · · · LT L m (cid:48) consisting of bounded sets . Then,thanks to the Rellich lemma, i m (cid:48) ,m ( X i )’s are relatively compact. Take an exhaustive sequence Y ⊆ Y ⊆ · · · ⊆ Y such that Y i is compact and (cid:83) Y i = Y , just like the proof of [Kas2, Lemma 1.4].Using these sequences and the same argument in that proof, we obtain an approximate unit { u i } of A such that (cid:107) [ u i , φ ( y )] (cid:107) < i − for all y ∈ Y i and (cid:107) g ( u i ) − u i (cid:107) < i − for all g ∈ i m (cid:48) ,m ( X i ). Note thatthe closure i m (cid:48) ,m ( X i ) is taken in the L m -topology. Since i m (cid:48) ,m ( X i ) ⊆ i m (cid:48) ,m ( X i ), the approximateunit { u i } satisfies the desired properties.The argument to prove (2) from (1) still works for our case. We leave the details to the reader. Remark . The above argument works for a topological group G which has an “approximationfrom the outside by the compact homomorphism”. This concept is defined as follows: (1) For each i ∈ N , suppose that a topological group G i with a metric function which contains G as a densesubgroup, is given; (2) Suppose that a continuous homomorphism j i (cid:48) ,i : G i (cid:48) → G i such that allbounded sets are mapped to relatively compact sets, is given for every i < i (cid:48) ∈ N ; (3) Suppose that j i (cid:48) ,i ◦ j i (cid:48)(cid:48) ,i (cid:48) = j i (cid:48)(cid:48) ,i for all i < i (cid:48) < i (cid:48)(cid:48) ∈ N ; and (4) G = ∩ i ∈ N G i . For such a group, we can define KK G -theory by the inductive limit of KK G i -theory, and this theory has the Kasparov product. Needlessto say, we can replace N with a more general directed system including R ≥ / .Thanks to the Rellich lemma, any mapping group C ∞ ( M, G ) for a compact manifold M anda compact Lie group G satisfies the above conditions. In particular, KK C ∞ ( M,G ) -theory can bedefined.By using this theorem, we can prove the desired properties on the Kasparov product in anappropriate sense. For example, for x ∈ KK LT L m ( A, A ) and y ∈ KK LT L m ( A , B ), there exists a54 K -element z ∈ KK LT L m (cid:48) ( A, B ) which is a Kasparov product of i ∗ m (cid:48) ,m ( x ) and i ∗ m (cid:48) ,m ( y ) for m (cid:48) > m .Such desired properties and standard arguments on inductive limits guarantee the following. Corollary 4.44.
Let
A, B, A and A be separable LT - C ∗ -algebras whose actions continuouslyextend to LT L m for some m < ∞ . Then, the Kasparov product KK LT ( A, A ) × KK LT ( A , B ) → KK LT ( A, B ) is well-defined. This product is denoted by ( x, y ) (cid:55)→ x (cid:98) ⊗ A y . This Kasparov product is associa-tive, that is to say, for x ∈ KK LT ( A, A ) , y ∈ KK LT ( A , A ) and z ∈ KK LT ( A , B ) , we have ( x (cid:98) ⊗ A y ) (cid:98) ⊗ A z = x (cid:98) ⊗ A ( y (cid:98) ⊗ A z ) . Since LT τ satisfies the condition explained in Remark 4.43, the same arguments work for LT τ -equivariant theory, and hence τ -twisted LT -equivariant KK -theory can be defined. Corollary 4.45.
Let
A, B and A be separable LT - C ∗ -algebras whose actions continuously extendto LT L m for some m < ∞ . We can define KK pτLT -theory for p ∈ Z , and we can prove that theKasparov product KK pτLT ( A, A ) × KK qτLT ( A , B ) → KK ( p + q ) τLT ( A, B ) is well-defined and associative. With this theory, we can state the main result of the present paper. A ( M L k ) is a C ∗ -algebraplaying the role of “ S ε (cid:98) ⊗ Cl τ ( M L k )”, which will be defined in Section 5. Main-Theorem.
Let M be a proper LT -space. (1) We can define a homomorphism substituting for the Poincar´e duality homomorphism
PD : KK τLT ( A ( M L k ) , S ε ) → R KK τLT ( M ; S ε (cid:98) ⊗ C ( M ) , S ε (cid:98) ⊗ C ( M )) . This homomorphism assigns σ S ε ([ L ]) to the “index element [ (cid:101) D ] of the Dirac operator twisted by a τ -twisted LT -equivariant line bundle L ” constructed in [T4]. (2) We can define a homomorphism substituting for the topological assembly map ν τLT L m : R KK τLT L m ( M L m ; C ( M L m ) , C ( M L m )) → KK ( C , C (cid:111) τ LT L m ) . This homomorphism assigns to [ L ] the “analytic index of the Dirac operator twisted by L ” con-structed in [T3], which is denoted by ind( D L ) . (3) Consequently, we have ν τLT L m (PD([ (cid:101) D ])) = ind( D L ) .Remark . In fact, what we will define for (1) is a homomorphism KK τLT L m ( A ( M L k ) , S ) → R KK τLT L m ( M L m ; S (cid:98) ⊗ C ( M L m ) , S (cid:98) ⊗ C ( M L m ))for k ≥ / m which is sufficiently larger than k . We will reformulate the “index element of theDirac operator twisted by a τ -twisted LT -equivariant line bundle L ” in the next section in order tofit the new construction of A ( M L k ). 55 The Poincar´e duality homomorphism for infinite-dimensionalmanifolds
In this section, we formulate the Poincar´e duality homomorphism for infinite-dimensional Hilbertmanifolds, and we compute it for proper LT -spaces. Strictly speaking, a modification of the Poincar´eduality homomorphism for the case of proper LT -spaces will be formulated and computed (seeSection 6.4 for this modification).For this aim, we begin with an explanation of the concept of “ C ∗ -algebras of Hilbert manifolds”which was announced in [Yu]. Since we think this C ∗ -algebra is quite important, we will study itfor proper LT -spaces in details. C ∗ -algebra of a Hilbert manifold and Poincar´e duality homomorphism The goal of this subsection is to formulate an infinite-dimensional version of the reformulated versionof the Poincar´e duality (see also Definition 3.14 and Proposition 3.19). In the present paper, we donot prove that it is isomorphic. Instead, we will show that the homomorphism constructed in thissection, gives an appropriate result for proper LT -spaces in the sense that the answer is parallel toExample 3.20. It will be done in the following two subsections.Let us begin with the concept of “ C ∗ -algebras of Hilbert manifolds” following [GWY, Yu]. Definition 5.1 (See [GWY]) . Let X be a Hilbert manifold and let ε >
0. First, we consider thespace Π( X ) := (cid:89) ( x,t ) ∈X × [0 ,ε ) Cliff + ( T x X ⊕ t R ) , where t R := (cid:40) R ( t (cid:54) = 0)0 ( t = 0) . This is a space of possibly non-continuous Clifford algebra-valued functions. Then, we consider the C ∗ -algebra Π b ( X ) := (cid:110) s ∈ (cid:89) ( X ) (cid:12)(cid:12)(cid:12) (cid:107) s ( x, t ) (cid:107) is bounded. (cid:111) equipped with pointwise algebraic operations (addition, multiplications and the adjoint) and theuniform norm. Definition 5.2 ([GWY, Yu]) . (1) Let X be a Hilbert manifold. Suppose that its injectivity radiusis greater than ε everywhere for some < ε ≤ ∞ . Let x , x ∈ X , and suppose that d ( x, x ) < ε .Then, x is contained in the image of exp x : B ε ( T x X ) → X , and hence it is contained in the domainof log x : exp x ( B ε ( T x X )) → T x X . The local Clifford operator at x is defined by C x ( x, t ) := ( − log x ( x ) , t ) ∈ T x X ⊕ t R , or equivalently C x ( x, t ) = (cid:0) ( d exp x ) x (log x ( x )) , t (cid:1) =“( −−→ x x, t )”.(2) The Bott homomorphism β x : S ε → Π b ( X ) centered at x ∈ X is defined by β x ( f )( x, t ) := (cid:40) f ( C x ( x, t )) ( d ( x, x ) < ε )0 ( d ( x, x ) ≥ ε ) , for f ∈ S ε , where f ( C x ( x, t )) is the functional calculus in the C ∗ -algebra Cliff + ( T x X ⊕ t R ).56 efinition 5.3 ([Yu]) . The C ∗ -algebra A ( X ) is defined by the C ∗ -subalgebra of Π b ( X ) generatedby the image of the Bott homomorphisms: A ( X ) := C ∗ ( { β x ( f ) | x ∈ X , f ∈ S ε } ) . Remark . This C ∗ -algebra depends on the choice of ε . If X is at the same time a Hilbert-Hadamard space, and if we chose ∞ as ε , we obtain the same C ∗ -algebra constructed in [GWY].Since this definition is parallel to [GWY, Definition 5.14], A ( X ) has similar properties. Thesame arguments work, except for [GWY, Lemma 5.8]. It is due to the fact that Hilbert-Hadamardspaces are “non-positively curved”. We prove a corresponding result by the infinite-dimensionalversion of the Rauch comparison theorem [Bil, Theorem 19]. See also [KN2, Sak]. Lemma 5.5.
Let Y and (cid:101) Y be Hilbert manifolds. Fix p ∈ Y and (cid:101) p ∈ (cid:101) Y , and take an isometricembedding I : T (cid:101) p (cid:101) Y (cid:44) → T p Y . Suppose that both injectivity radii of Y and (cid:101) Y are greater than ε , andsuppose that all the sectional curvatures of (cid:101) Y are not greater than those of Y . For a smooth curve (cid:101) c : [0 , → B ε ( (cid:101) p ) ⊆ (cid:101) Y , we define c := exp p ◦ I ◦ exp − (cid:101) p ◦ (cid:101) c : [0 , → B ε ( p ) ⊆ Y . Then, we have aninequality (cid:90) (cid:107) ˙ c ( t ) (cid:107) dt ≤ (cid:90) (cid:107) ˙ (cid:101) c ( t ) (cid:107) dt. For simplicity, we impose the following “bounded geometry type” assumption. Note that it isautomatically satisfied if X admits an isometric cocompact group action. In particular, a proper LT -space satisfies this assumption. Assumption 5.6.
We suppose that X is a Hilbert manifold whose injectivity radius is boundedbelow by 2 ε . We also suppose that all the sectional curvatures of X are bounded above by δ . When δ >
0, by retaking ε if necessary, we suppose that ε < π/ √ δ from the beginning. Lemma 5.7 (Compare with [GWY, Lemma 5.8]) . Let X be a Hilbert manifold satisfying Assump-tion 5.6. For x, x , x ∈ X satisfying d ( x, x ) , d ( x, x ) < ε , we have an inequality (cid:107) C x ( x, t ) − C x ( x, t ) (cid:107) ≤ d ( x , x ) . Proof.
We would like to apply the Rauch comparison theorem to the case when (cid:102) M = X and M = S ∞ ( δ − / ), where S ∞ ( δ − / ) is the infinite-dimensional sphere with radius δ − / . Pick up y ∈ S ∞ ( δ − / ) and take an isometric embedding I : T x X → T y S ∞ ( δ − / ). We may assume that y is the north pole. By the composition exp y ◦ I ◦ log x , we can define a local embedding from the ε -neighborhood of x in X to that of y in S ∞ ( δ − / ). Note that this neighborhood of y in S ∞ ( δ − / ) iscontained in the northern hemisphere, thanks to the condition ε < π √ δ . Let y i := exp y ◦ I ◦ log x ( x i )for i = 0 ,
1. Let (cid:101) c be the unique minimal geodesic connecting x and x , and let c := exp y ◦ I ◦ log x ◦ (cid:101) c .Then, thanks to the comparison theorem, we have the inequality L ( c ) ≤ L ( (cid:101) c ) . Since (cid:101) c is a geodesic, L ( (cid:101) c ) = d X ( x , x ). Clearly, d S ∞ ( δ − / ) ( y , y ) ≤ L ( c ). Combining them, we obtain the inequality d S ∞ ( δ − / ) ( y , y ) ≤ d X ( x , x ).By definition of the Clifford operator and y i ’s, exp y I [ − C x i ( x, t )] = y i . Since I is isometric, wehave (cid:107) C x ( x, t ) − C x ( x, t ) (cid:107) = (cid:107) I [ C x ( x, t )] − I [ C x ( x, t )] (cid:107) . Therefore, in order to obtain the result,it is sufficient to prove that d S ∞ ( δ − / ) ( y , y ) ≥ (cid:107) I [ C x ( x, t )] − I [ C x ( x, t )] (cid:107) .Take the minimal geodesic c connecting y and y in S ∞ ( δ − / ). Put c := log y ◦ c . Then, L ( c ) = d S ∞ ( δ − / ) ( y , y ) and L ( c ) ≥ (cid:107) I [ C x ( x, t )] − I [ C x ( x, t )] (cid:107) . Thus, it is sufficient to prove57hat L ( c ) ≥ L ( c ). By definition of c and c , we have ˙ c ( t ) = ( d exp y ) c ( t ) [ ˙ c ( t )]. Since L ( c i ) = (cid:82) (cid:107) ˙ c i ( t ) (cid:107) dt , it suffices to prove that (cid:107) ( d exp y ) z ( v ) (cid:107) ≥ (cid:107) v (cid:107) for all z which belongs to the northernhemisphere and v ∈ T z S ∞ ( δ − / ). Now, this is clear by the following arguments: Consider the twodimensional sphere S ( δ − / ) ⊆ S ∞ ( δ − / ) which contains y, z and whose tangent space contains v ;Then, calculate the differential of exp y at x using the polar coordinate system. We leave the detailsto the reader.In the same way of Corollary 5.12 of [GWY], we can prove the following. Corollary 5.8.
If a net { x i } i ∈ I ⊆ X converges to x ∈ X , we have lim i →∞ (cid:107) β x ( f ) − β x i ( f ) (cid:107) = 0 for every f ∈ S ε . By this corollary, we can prove several important properties. Let Γ be a Hausdorff (possiblynon-locally compact) group. Suppose that Γ acts on X in an isometric and continuous way. Then,Γ acts on (cid:81) b ( X ) as follows: For an isometry φ : X → X and f ∈ (cid:81) b ( X ), we define φ ∗ ( f )( x, t ) := Cliff + { ( dφ ) φ − ( x ) ⊕ id t R } f ( φ − ( x ) , t ) ∈ Cliff + ( T x X ⊕ t R ) , where Cliff + { F } for a linear map F on a Hilbert space V is the induced homomorphism on Cliff + ( V ). This Γ-action on (cid:81) b ( X ) gives a continuous action on A ( X ) as follows. Proposition 5.9. (1)
For an isometry φ : X → X and x ∈ X , φ ∗ ◦ β x = β φ ( x ) . Consequently, φ ∗ preserves A ( X ) . (2) Therefore, Γ acts on A ( X ) . This action is continuous in the point-norm topology.Proof. See [GWY, Section 6].We summarize other necessary properties of A ( X ). Thanks to Lemma 5.7 , one can prove themin the same way of [GWY]. Proposition 5.10.
Let X be a Hilbert manifold satisfying Assumption 5.6. (1) A ( X ) is separable whenever X is separable. (2) When X is finite-dimensional, A ( X ) coincides with the set of continuous sections vanishingat infinity (cid:8) f ∈ C ( X × [0 , ε ) , Cliff + ( T X ⊕ t R )) (cid:9) . This C ∗ -algebra is isomorphic to C ([0 , ε ) , Cliff + ( t R )) (cid:98) ⊗ Cl τ ( X ) . (3) For a subset Y of X , we define A ( X , Y ) by the C ∗ -subalgebra of A ( X ) generated by { β x ( f ) | x ∈ Y , f ∈ S ε } , following Lemma 7.2 of [GWY]. Then, • A ( X , X ) = A ( X ) ; If Y ⊆ Z , we have A ( X , Y ) ⊆ A ( X , Z ) ; • A ( X , Y ) = A ( X , Y ) ; and • If Y ⊆ Y ⊆ · · · , we have A ( X , ∪ i Y i ) = lim −→ i A ( X , Y i ) .Consequently, if ∪ i Y i = X , we have A ( X ) = lim −→ i A ( X , Y i ) .Remarks . (1) S ε is isomorphic to C ([0 , ε ) , Cliff + ( t R ))by the following homomorphism. For f ∈ S ε , we have the even-odd decomposition f = f + f ,where f ( t ) = f ( t )+ f ( − t )2 and f = f − f . Then, we can define an element of C ([0 , ε ) , Cliff + ( t R )) by s (cid:55)→ f ( s ) f + f ( s ) v , where f is “1 ∈ Cliff + ( t R ))” and v is the unit vector of “ t R ” for s >
0. Notethat f (0) = 0, and hence f ( s ) v makes sense for any s ≥
0. One can prove that the correspondence f (cid:55)→ [ s (cid:55)→ f ( s ) f + f ( s ) v ] is a ∗ -isomorphism. By this description, we have a short exact sequence0 → Cl τ (0 , ε ) → S ε → C → . We have used a similar exact sequence in the proof of 2.18.(2) Therefore, (2) above gives an isomorphism A ( X ) ∼ = S ε (cid:98) ⊗ Cl τ ( X ) for finite-dimensional X .This isomorphism has been used in Section 3.2 as a convention. Moreover, by (3), we have A ( l ( N )) ∼ = lim −→ N S (cid:98) ⊗ Cl τ ( R N ), where we choose ε = ∞ . The right hand side is the C ∗ -algebradefined in [HKT].(3) Since all of Cliff + ( R N ), C ( R N ), Cl τ ( R ) and C are nuclear, so is S (cid:98) ⊗ Cl τ ( R N ) (see [Mur,Section 6.3–6.5]). Thus, its limit A ( l ( N )) is again nuclear. This fact will be used in Section 5.3.We can formulate an infinite-dimensional version of the Poincar´e duality homomorphism, byimitating Proposition 3.19. Definition 5.12.
Let X be a Hilbert manifold satisfying Assumption 5.6. Suppose that a Hausdorffgroup Γ acts on X in a proper and isometric way. Then, the reformulated local Bott element is defined by[ (cid:93) Θ X , ] := ( {A ( X ) } x ∈X , { β x } x ∈X , { } x ∈X ) ∈ R KK Γ ( X , S ε (cid:98) ⊗ C ( X ) , A ( X ) (cid:98) ⊗ C ( X )) , where the Γ-action on the filed A ( X ) (cid:98) ⊗ C ( X ) is given by the diagonal action A ( X ) x (cid:51) a (cid:55)→ γ ∗ ( a ) ∈ A ( X ) γ.x . Definition 5.13.
We define the Poincar´e duality homomorphism by the Kasparov productPD : KK Γ ( A ( X ) , S ε ) (cid:51) [ D ] (cid:55)→ [ (cid:93) Θ X , ] (cid:98) ⊗ σ X , C ( X ) ([ D ]) ∈ R KK Γ ( X , S ε (cid:98) ⊗ C ( X ) , S ε (cid:98) ⊗ C ( X )) . We call the image of this map PD([ D ]) the symbol element of [ D ].59 .2 The index element of [T4] and a study on C ∗ -algebras of Hilbert manifolds We would like to apply the above construction for proper LT -spaces. The value of “PD” at theindex element of the “Dirac operator twisted by the τ -twisted LT -equivariant line bundle L ” shouldbe σ S ε ([ L ]), according to Example 3.20.The task of this subsection is to define a KK τLT -element substituting for the reformulated indexelement studied in Section 3.2. Although we have defined a similar KK τLT -element in [T4], we willreconstruct it for the following two reasons. First, we adopted the C ∗ -algebra of a Hilbert space asa substitute for the function algebra in [T4], which does not coincides with A ( M L k ) in the presentpaper. Second, the definition of the A ( M )-module structure of the previous paper was not quitenatural, although the Hilbert space is natural. We would like to reconstruct it so that the modulestructure looks natural. Instead, we must modify the Hilbert space.Let us outline this subsection. Fix k > / l > m ≥ k + l . We first study the C ∗ -algebra A ( M L k ). We will introduce a possibly bigger C ∗ -algebra A HKT ( M L k ) which is more convenientto study the K -homological element. Thanks to it, we can deal with the U L k -part and the (cid:102) M -part separately. After that, we will reconstruct an equivariant unbounded Kasparov module whichplays a role of the reformulated index element, by the following steps: (1) We will define a Hilbertspace substituting for “ L ( U L k , L )”; (2) We will prove that it admits a continuous U L m -action L which looks like the left regular representation; (3) We will define the A HKT ( U L k )-module structure π on the Hilbert S -module S (cid:98) ⊗H substituting for “ S (cid:98) ⊗ L ( U L k , L (cid:98) ⊗ S ∗ U (cid:98) ⊗ S U )”; (4) We will define anoperator substituting for the Dirac operator (cid:19) ∂ on H ; (5) We will prove that the triple ( S (cid:98) ⊗H , π, id (cid:98) ⊗ (cid:19) ∂ )is an unbounded U L m -equivariant Kasparov ( A HKT ( U L k ) , S )-module; (6) We will prove that thisunbounded Kasparov module can be restricted to A ( U L k ); and (7) We will finally prove that theconstructed element is independent of l as an element of KK τLT ( A ( M L k ) , S ε ). (1)–(5) have beenessentially done in [T4], but we will clarify the proofs. We will also give several different argumentsin order to investigate the C ∗ -algebra of a Hilbert space from the viewpoint of global analysis more.When we use the same arguments or estimates of [T4], we will refer to the corresponding resultsand we will omit the details.At the KK -theory level, the index element we are going to construct is, roughly speaking, thepullback of the one constructed in [T4] via the inclusion A ( M L k ) → A HKT ( M L k ). However, thefact that A ( M L k ) has a natural unbounded KK -element, is itself interesting. In addition, it isimportant to know “how an element of A ( M L k ) can be seen as a function on M L k ”. Our result isuseful to consider this problem.Let us begin with the construction of A HKT ( M L k ) inspired by [HKT]. See also [Tro]. Wewill also introduce the alternative description of A ( M L k ) related to A HKT ( M L k ) with the idea ofProposition 5.10 (3) and [GWY, Remark 7.7].Recall that each C ∗ -algebra S and S ε has an (un)bounded multiplier X given by Xf ( t ) := tf ( t ). They correspond to one another under the zero-extension ι : S ε (cid:44) → S . We denote thecoordinate of U L k ,N by ( x , y , · · · , x N , y N ), and we denote the corresponding base of the Lie algebraby { e , f , · · · , e N , f N } . The Clifford multiplication by v ∈ Lie( U L k ,N ) from the left is denoted by v : Cliff + (Lie( U L k ,N )) → Cliff + (Lie( U L k ,N )). Definition 5.14. (1) We define an unbounded multiplier on Cl τ ( U L k ,M (cid:9) U L k ,N ) by C MN +1 := (cid:80) Mj = N +1 ( x j (cid:98) ⊗ e j + y j (cid:98) ⊗ f j ). Using it, we define a Bott-homomorphism β MN : S (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) → (cid:98) ⊗ Cl τ ( U L k ,M × (cid:102) M ) for N ≤ M by β MN : f (cid:98) ⊗ h (cid:55)→ f ( X (cid:98) ⊗ (cid:98) ⊗ C MN +1 ) (cid:98) ⊗ h ∈S (cid:98) ⊗ Cl τ ( U L k ,M (cid:9) U L k ,N ) (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) ∼ = S (cid:98) ⊗ Cl τ ( U L k ,M × (cid:102) M ) . These make a directed system, namely β MN (cid:48) ◦ β N (cid:48) N = β MN for N ≤ N (cid:48) ≤ M . We define a C ∗ -algebra A HKT ( M L k ) by the inductive limit of this system: A HKT ( M L k ) := lim −→ N S (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) . The canonical homomorphisms are denoted by β ∞ N : S (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) → A HKT ( M L k ).(2) In the same way, we define A HKT ( U L k ) := lim −→ N S (cid:98) ⊗ Cl τ ( U L k ,N ) . (3) Let S fin be the subalgebra generated by the following two functions: f ev ( t ) := 1 t + 1 and f od ( t ) := tt + 1 . Note that S fin is a dense subalgebra of S thanks to the Stone-Weierstrass theorem. We define thedense subalgebra A HKT ( M L k ) fin by the subalgebra generated by ∪ N β ∞ N (cid:16) S fin (cid:98) ⊗ alg Cl τ, S ( U L k ,N × (cid:102) M ) (cid:17) , where Cl τ, S stands for the set of Clifford algebra-valued Schwartz class functions.Let us introduce an alternative description of A ( M L k ) using Proposition 5.10 (3). Let β MN : S ε (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) → S ε (cid:98) ⊗ Cl τ ( U L k ,M × (cid:102) M ) be the homomorphism given by β MN : f (cid:98) ⊗ h (cid:55)→ f ( X (cid:98) ⊗ (cid:98) ⊗ C MN +1 ) (cid:98) ⊗ h for N ≤ M . These make a directed system and its limitlim −→ N S ε (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M )is isomorphic to A ( M L k ) thanks to Proposition 5.10 (3). The canonical homomorphisms are denotedby β ∞ N : S ε (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) → A ( M L k ).This alternative description has the following advantages compared to the original one. Lemma 5.15. (1)
Thanks to the canonical isomorphism Cl τ ( U L k ,N × (cid:102) M ) ∼ = Cl τ ( U L k ,N ) (cid:98) ⊗ Cl τ ( (cid:102) M ) ,we have a ∗ -isomorphism A ( M L k ) ∼ = A ( U L k ) (cid:98) ⊗ Cl τ ( (cid:102) M ) . For the same reason, we have A HKT ( M L k ) ∼ = A HKT ( U L k ) (cid:98) ⊗ Cl τ ( (cid:102) M ) . (2) By the zero-extension ι : S ε (cid:44) → S , the unbounded multiplier X on S corresponds to thebounded multiplier X on S ε . Consequently, we have the following commutative diagram S ε (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) β MN −−−−→ S ε (cid:98) ⊗ Cl τ ( U L k ,M × (cid:102) M ) ι (cid:98) ⊗ id (cid:121) (cid:121) ι (cid:98) ⊗ id S (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) β MN −−−−→ S (cid:98) ⊗ Cl τ ( U L k ,M × (cid:102) M ) . herefore, we have a ∗ -homomorphism A ( M L k ) → A HKT ( M L k ) defined by the limit of ι (cid:98) ⊗ id : S ε (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) → S ε (cid:98) ⊗ Cl τ ( U L k ,M × (cid:102) M ) . This homomorphism preserves the tensor product decomposition given in (1) . The constructedhomomorphism is denoted by ι (cid:98) ⊗ id . These two C ∗ -algebras admit LT L k = U L k × ( T × Π T )-actions which look like left translations. Lemma 5.16. (1)
The isometric LT L k -action on M L k induces a continuous action denoted by “ lt ”on A ( M L k ) thanks to Proposition 5.9. It is given by lt g ◦ β x ( f ) = β g · x ( f ) . (2) The dense subgroup LT fin := U fin × ( T × Π T ) acts on A HKT ( M L k ) by the following: For g ∈ U L k ,M × ( T × Π T ) , we define lt g ∈ Aut( A ( U L k ,M )) by left translation; For g ∈ LT fin and a ∈ S (cid:98) ⊗ Cl τ ( U L k ,N × (cid:102) M ) , we define lt g [ β ∞ N ( a )] := β ∞ N (cid:48) (lt g ( β N (cid:48) N ( a ))) , where N (cid:48) ≥ N is chosen so that g ∈ U L k ,N (cid:48) × ( T × Π T ) . This action extends to LT L k as a continuous action. (3) The action given above can be written as the tensor product of two actions on A HKT ( U L k ) (cid:98) ⊗ Cl τ ( (cid:102) M ) ,that is to say, for g = ( u, γ ) ∈ U L k × ( T × Π T ) , a ∈ A HKT ( U L k ) and f ∈ Cl τ ( (cid:102) M ) , lt g ( a (cid:98) ⊗ f ) = lt u ( a ) (cid:98) ⊗ lt γ ( f ) . (4) The ∗ -homomorphism ι (cid:98) ⊗ id : A ( M L k ) → A HKT ( M L k ) is LT L k -equivariant. Consequently,the LT L k -action on A ( M L k ) is written as the tensor product of the actions of U L k and T × Π T .Proof. (1) is automatic from Proposition 5.9. For (2), see [T4, Theorem 4.18]. (3) is obvious bydefinition. (4) is also obvious by definition. Thanks to the continuity of the group actions provedin (2), the ∗ -homomorphism is LT L k -equivariant.Therefore, we can deal with the U L k -part and the (cid:102) M -part separately. We concentrate on theformer one until Definition 5.35.We re-define a Hilbert space substituting for the “ L -space of the line bundle L on U L k ”. In[T4], we used the infinitely many copies of a Gaussian in order to define the “ L -space” by theinductive limit. In the present paper, however, we use the infinite sequence of different Gaussianswhich tends to the Dirac δ -function. This is because we would like to define the A ( U L k )-modulestructure on a Hilbert S ε -module in a natural way.Since the restriction of L to U L k ,N is topologically trivial, we can fix a trivialization and wedescribe L ( U L k ,N , L ) as the set of scalar-valued functions with a τ -twisted U L k ,N -action. Definition 5.17.
Let l > L ( U L k ,N , L ) fin of L ( U L k ,N , L ) by the set ofpolynomial × ( N !) l/ π N/ e − (cid:80) Nn =1 n l ( x n + y n ) . (2) We define an isometric embedding J N : L ( U L k ,N , L ) fin → L ( U L k ,N +1 , L ) fin ∼ = L ( U L k ,N , L ) fin (cid:98) ⊗ L ( R , L ) fin
62y the following: φ (cid:55)→ φ (cid:98) ⊗ (cid:114) ( N + 1) l π e − ( N +1) l ( x N +1 + y N +1 ) . Note that the Gaussian (cid:113) ( N +1) l π e − ( N +1) l ( x N +1 + y N +1 ) is an L -unit vector, and hence J N is anisometric embedding. We denote the composition of these maps by J MN := J M − ◦ J M − ◦ · · · ◦ J N : L ( U L k ,N , L ) fin → L ( U L k ,M , L ) fin . These make a directed system.(3) The algebraic inductive limit of this system is denoted by L ( U L k , L ) fin . We define a Hilbert space L ( U L k , L ) by the completion of L ( U L k , L ) fin . Remarks . (1) L ( U L k , L ) can be given by the Hilbert space inductive limit of the directedsystem · · · J N − −−−→ L ( U L k ,N , L ) J N −−→ L ( U L k ,N +1 , L ) J N +1 −−−→ · · · , where J N : L ( U L k ,N , L ) → L ( U L k ,N +1 , L ) is the canonical extension of J N : L ( U L k ,N , L ) fin → L ( U L k ,N +1 , L ) fin . The canonical homomorphism is denoted by J ∞ N : L ( U L k ,N , L ) (cid:44) → L ( U L k , L ).(2) We will construct an unbounded Kasparov module using this Hilbert space, which dependson l . However, the resulting KK τLT -element is independent of l as proved in Proposition 5.36.(3) Incidentally, an element of L ( U L k , L ) is an “asymptotically Dirac δ -function”. We willexplain the reason why we should adopt such a strange Hilbert space in Section 6.4. Notations 5.19.
We use the following symbols about the “Gaussians”:vac NN ( x N , y N ) := (cid:114) N l π e − Nl ( x N + y N ) , vac MN := J MN (vac NN ) = vac NN (cid:98) ⊗ vac N +1 N +1 (cid:98) ⊗ · · · (cid:98) ⊗ vac MM , vac ∞ N := J ∞ M (vac MN ) = vac NN (cid:98) ⊗ vac N +1 N +1 (cid:98) ⊗ · · · , vac := vac ∞ . The Hilbert space L ( U L k ,N , L ) admits a τ -twisted continuous U L k ,N -action L defined by thefollowing: For φ ∈ L ( U L k ,N , L ) and ( g, z ) ∈ U τL k ,N , we define[ L ( g,z ) φ ]( x ) := zφ ( x − g ) τ ( g, x ) . Recall that τ ( g, x ) is given by τ ( g, x ) = e √− (cid:80) Nn =1 n − k ( a n y n − b n x n ) for x = ( x , y , · · · , x N , y N ) , g = ( a , b , · · · , a N , b N ) ∈ U L k ,N .We need to prove that the action L extends to U L m for m ≥ k + l . We begin with an elementaryexercise of the Lebesgue integration. 63 emma 5.20. On R N , consider the Gaussian function vac( x ) := π − N/ e −(cid:107) x (cid:107) / and an element a ∈ R N . For any (cid:15) > , there exists ∆ > such that (cid:107) a (cid:107) < ∆ implies that (cid:90) R N | vac( x − a ) − vac( x ) | dx < (cid:15). ∆ can be chosen independently of N .Proof. This is clear from the dominated convergence theorem, but we describe the complete proofwhich will be useful in the following discussions.We may assume that a is of the form ( a , , , · · · ,
0) for a = (cid:107) a (cid:107) , because the Gaussian isrotation invariant. Then, the integral we are estimating becomes (cid:90) R π − / (cid:20) e − ( x − a )22 − e − x (cid:21) dx. We can choose a large number
K > (cid:90) | x |≥ K π − / e − x dx < (cid:15) . We may assume that a < K , and hence we have inequalities (cid:90) x ≥ K π − / (cid:20) e − ( x − a − e − x (cid:21) dx ≤ (cid:90) x ≥ K π − / e − ( x − a ) dx < (cid:15) (cid:90) x ≤− K π − / (cid:20) e − ( x − a − e − x (cid:21) dx ≤ (cid:90) x ≤− K π − / e − x dx < (cid:15) . Finally, choose ∆ small enough so that the following holds:4 K · π − / (cid:20) e − ( x − ∆)22 − e − x (cid:21) < (cid:15) − K ≤ x ≤ K . This is possible because the Gaussian is uniformly continuous. If | a | < ∆,we have an inequality 4 K · π − / (cid:20) e − ( x − a )22 − e − x (cid:21) < (cid:15) − K ≤ x ≤ K , and we obtain the result. Proposition 5.21 (See also Proposition 4.7 of [T4]) . Let m ≥ k + l . The Hilbert space L ( U L k , L ) admits a continuous U τL m -action L such that the restriction of it to U τL m ,N on the image of L ( U L k ,N , L ) in L ( U L k , L ) coincides with the left regular representation: [ L ( g,z ) φ ]( x ) = zφ ( x − g ) τ ( g, x ) . Proof.
We have defined the U L m ,N -action L on L ( U L k ,N , L ) for each N . These are compatible inthe following sense: Let i N,N +1 : U L m ,N (cid:44) → U L m ,N +1 be the canonical embedding; For g ∈ U L m ,N and φ ∈ L ( U L k ,N , L ), L ( i N,N +1 ( g ) ,z ) J N ( φ ) = J N ( L ( g,z ) φ ).64his means that U fin acts on L ( U L k , L ) fin . Since this action is unitary, it extends to a U fin -action on L ( U L k , L ). We would like to extend this action to the whole group U L m by L ( g,z ) φ := lim L ( g i ,z ) φ for φ ∈ L ( U L k , L ) and g = lim g i in U L m .For this aim, it suffices to check that the U fin -action is continuous in the L m -topology. Concretely,we need to prove the following: For any φ ∈ L ( U L k , L ) and a net { ( g i , z i ) } ⊆ U τ fin convergingto ( g, z ) ∈ U τ fin in the L m -topology, the net { L ( g i ,z i ) φ } converges to L ( g,z ) φ in norm. We wouldlike to find, for a positive real number (cid:15) >
0, a large element i such that i ≥ i implies that (cid:107) L ( g i ,z ) φ − L ( g,z ) φ (cid:107) < (cid:15) . Since L ( U L k , L ) fin is dense in L ( U L k , L ), we can find φ (cid:48) ∈ L ( U L k , L ) fin satisfying (cid:107) φ − φ (cid:48) (cid:107) < (cid:15) . Then, we have an inequality (cid:107) L ( g i ,z i ) φ − L ( g,z ) φ (cid:107) ≤ (cid:107) L ( g i ,z i ) φ − L ( g i ,z i ) φ (cid:48) (cid:107) + (cid:107) L ( g i ,z i ) φ (cid:48) − L ( g,z ) φ (cid:48) (cid:107) + (cid:107) L ( g,z ) φ (cid:48) − L ( g,z ) φ (cid:107) < (cid:15) + (cid:107) L ( g i ,z i ) φ (cid:48) − L ( g,z ) φ (cid:48) (cid:107) . Thus, we may assume that φ ∈ L ( U L k , L ) fin . By definition, φ is of the form φ (cid:98) ⊗ vac ∞ N , for some φ ∈ L ( U L k ,N − , L ) for sufficiently large N .We may focus on vac ∞ N and we may assume that z = z i = 1 for each i . This is because L ( g i ,z i ) ( φ (cid:98) ⊗ vac ∞ N ) = L (( a i, ,b i, , ··· ,a i,N − ,b i,N − ) ,z i )) φ (cid:98) ⊗ L (( a i,N ,b i,N , ··· ) , (vac ∞ N )for g i = ( a i, , b i, , · · · ) ∈ U fin and g = ( a , b , · · · ) ∈ U fin ; The first component converges to L (( a ,b , ··· ,a N − ,b N − ) ,z )) φ in norm as i → ∞ , thanks to a standard argument of Lebesgue inte-gration. This also means that we may ignore the z -component. Moreover, by a standard argumentof group actions, we may assume that the limit g is the identity element.Combining the arguments so far, we notice that we need to prove that (cid:107) L ( g i , vac − vac (cid:107) → (cid:90) U L k,Mi (cid:12)(cid:12)(cid:12) vac M i ( x − g i ) τ ( g, x ) − vac M i ( x ) (cid:12)(cid:12)(cid:12) dx dy · · · dx M i dy M i → i → ∞ , where M i is chosen so that g i ∈ U L k ,M i .This integral converges to 0 by the following argument. First, we notice that (cid:90) U L k,Mi (cid:12)(cid:12)(cid:12) vac M i ( x − g i ) τ ( g i , x ) − vac M i ( x ) (cid:12)(cid:12)(cid:12) dx dy · · · dx M i dy M i = (cid:90) U L k,Mi (cid:12)(cid:12)(cid:12) vac M i ( x − g i ) τ ( g i , x ) − vac M i ( x ) τ ( g i , x ) + vac M i ( x ) τ ( g i , x ) − vac M i ( x ) (cid:12)(cid:12)(cid:12) dx dy · · · dx M i dy M i ≤ (cid:90) U L k,Mi (cid:12)(cid:12)(cid:12) vac M i ( x − g i ) − vac M i ( x ) (cid:12)(cid:12)(cid:12) dx dy · · · dx M i dy M i + 2 (cid:90) U L k,Mi vac M i ( x ) | τ ( g i , x ) − | dx dy · · · dx M i dy M i . (cid:90) U L k,Mi (cid:12)(cid:12)(cid:12) vac M i ( x − g i ) − vac M i ( x ) (cid:12)(cid:12)(cid:12) dx dy · · · dx M i dy M i = (cid:90) U L k,Mi ( M i !) l π N (cid:12)(cid:12)(cid:12)(cid:12) e − (cid:80) Mij =1 j l [( x j − a i,j ) +( y j − b i,j ) ] − e − (cid:80) Mij =1 j l [ x j + y j ] (cid:12)(cid:12)(cid:12)(cid:12) dx dy · · · dx M i dy M i = (cid:90) U L k,Mi π N (cid:12)(cid:12)(cid:12)(cid:12) e − (cid:80) Mij =1 [( X j − A i,j ) +( Y j − B i,j ) ] − e − (cid:80) Mij =1 [ X j + Y j ] (cid:12)(cid:12)(cid:12)(cid:12) dX dY · · · dX M i dY M i , where X j := (cid:112) j l x j and Y j := (cid:112) j l y j , and we introduced the notations A i,j := (cid:112) j l a i,j and B i,j := (cid:112) j l b i,j . Then, thanks to Lemma 5.20, this integral is arbitrary small when the norm (cid:110)(cid:80) j ( A i,j + B i,j ) (cid:111) / is sufficiently small. This quantity is the L k + l -norm of g i , and it is notgreater than the L m -norm of g i since m ≥ k + l .For the latter integral, we follow the same story of Lemma 5.20. Let (cid:15) >
0. Note that thefunction x (cid:55)→ | τ ( g i , x ) − | is bounded. We can find K > (cid:90) (cid:80)
Mij =1 ( x j + y j ) ≥ K vac M i ( x ) | τ ( g i , x ) − | dx dy · · · dx M i dy M i < (cid:15) . Thus, it suffices to prove that | τ ( g i , x ) − | < (cid:15) on (cid:110) x (cid:12)(cid:12)(cid:12) (cid:80) M i j =1 ( x j + y j ) ≤ K (cid:111) . For this aim, itsuffices to check that (cid:80) M i j =1 j − k ( a i,j y j − b i,j x j ) is uniformly small there. This quantity is boundedabove by (cid:118)(cid:117)(cid:117)(cid:116) M i (cid:88) j =1 ( x j + y j ) (cid:118)(cid:117)(cid:117)(cid:116) M i (cid:88) j =1 j − k ( a i,j + b i,j )thanks to the Cauchy-Schwarz inequality. (cid:113)(cid:80) Mj =1 j − k ( a i,j + b i,j ) is the L − k -norm and it is notgreater than the L m -norm of g i .In order to define a substitute for the index element associated to the line bundle L as anunbounded U L m -equivariant Kasparov ( A ( U L k ) , S ε )-module, we set H := L ( U L k , L ) (cid:98) ⊗ S ∗ U (cid:98) ⊗ S U ; and H fin := L ( U L k , L ) fin (cid:98) ⊗ alg S ∗ U, fin (cid:98) ⊗ alg S U, fin and we consider the Hilbert S ε -module S ε (cid:98) ⊗H . The next task is to define a left module structure π : A ( M L k ) → L S ε ( S ε (cid:98) ⊗H ) . We will construct it with the help of A HKT ( U L k ) and the ∗ -homomorphism ι (cid:98) ⊗ id : A ( U L k ) →A HKT ( U L k ). As essentially proved in [T4], A HKT ( U L k ) admits a U L m -equivariant ∗ -homomorphism π : A HKT ( U L k ) → L S ( S (cid:98) ⊗H ) as outlined in the following proposition.Before that, we notice that we can define the “multiplication operators by { x n , y n } n ∈ N ” on L ( U L k , L ) fin . Strictly speaking, for φ ∈ J ∞ N ( L ( U L k ,N , L )) ⊆ L ( U L k , L ) fin , x n φ is defined by J ∞ N (cid:48) ( x n J N (cid:48) N ( ψ )), where N (cid:48) is chosen so that n ≤ N (cid:48) .66 roposition 5.22. Let π ( C ∞ N ) be the infinite sum of unbounded operators on L ( U L k , L ) (cid:98) ⊗ S ∗ U (cid:98) ⊗ S U π ( C ∞ N ) := ∞ (cid:88) k = N (cid:0) x k (cid:98) ⊗ c ∗ ( e k ) + y k (cid:98) ⊗ c ∗ ( f k ) (cid:1) . This infinite sum strongly converges to an unbounded self-adjoint operator on L ( U L k , L ) (cid:98) ⊗ S ∗ U (cid:98) ⊗ S U .Proof. It is proved by the same argument of [T4, Proposition 4.24]. The necessary change is onlythe following: In the previous paper, we used the property that“the L -norms of n − l x n (cid:114) π e − ( x n + y n ) and n − l y n (cid:114) π e − ( x n + y n ) are small”,but, in the present paper, we need to use the property that“the L -norms of x n (cid:114) n l π e − nl ( x n + y n ) and y n (cid:114) n l π e − nl ( x n + y n ) are small”instead. This property is guaranteed because the Gaussian (cid:113) n l π e − nl ( x n + y n ) is more localized as n becomes bigger. We leave the details to the reader.We have a ∗ -representation of Cl τ ( U L k ,N ) on L ( U L k ,N ) (cid:98) ⊗ S ∗ U N (cid:98) ⊗ S U N by the combination of themultiplication of scalar-valued functions and the Clifford multiplication c ∗ . We denote it by µ N : Cl τ ( U L k ,N ) → L C ( L ( U L k ,N ) (cid:98) ⊗ S ∗ U N (cid:98) ⊗ S U N ) . Definition-Proposition 5.23. (1) For every N ∈ N , we define a ∗ -homomorphism π N : S (cid:98) ⊗ Cl τ ( U L k ,N ) → L S ( S (cid:98) ⊗H ) by π N ( f (cid:98) ⊗ h ) := f ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C ∞ N +1 )) (cid:98) ⊗ µ N ( h )with the canonical isomorphism H ∼ = (cid:104) L ( U L k (cid:9) U L k ,N , L ) (cid:98) ⊗ S ∗ U (cid:9) U N (cid:98) ⊗ S U (cid:9) U N (cid:105) (cid:98) ⊗ (cid:104) L ( U L k ,N , L ) (cid:98) ⊗ S ∗ U N (cid:98) ⊗ S U N (cid:105) . Then, π N ’s satisfy the following commutative diagram: S (cid:98) ⊗ Cl τ ( U L k ,N ) β MN (cid:47) (cid:47) π N (cid:39) (cid:39) S (cid:98) ⊗ Cl τ ( U L k ,M ) π M (cid:119) (cid:119) L S ( S (cid:98) ⊗H )and hence it defines a ∗ -homomorphism π : A HKT ( U L k ) → L S ( S (cid:98) ⊗H ) . (2) π (cid:16) ι (cid:98) ⊗ id( A ( U L k )) (cid:17) ( S (cid:98) ⊗H ) ⊆ S ε (cid:98) ⊗H . Therefore, we can regard π ◦ ι (cid:98) ⊗ id as a map A ( U L k ) → L S ε ( S ε (cid:98) ⊗H ). We use the same symbol π to denote π ◦ ι (cid:98) ⊗ id, in order to simplify the notation.(3) These homomorphisms are U L m -equivariant in the following sense: For a ∈ A HKT ( U L k ), φ ∈ S (cid:98) ⊗H and g ∈ U L m , we have L ( g,z ) [ π ( a ) φ ] = π (lt g ( a ))( L ( g,z ) φ ), and similarly for A ( U L k ).67 roof. (1) and (3) are obvious from the definitions and the computation for U L k , fin . See [T4,Definition-Theorem 4.26 and Lemma 5.1].We check (2) in detail. First, we notice that S ∼ = C ( R ≥ , Cliff + ( t R )) ⊆ C ( R ≥ ) (cid:98) ⊗ Cliff + ( R ),just like Remark 5.11 (1). Thus, S (cid:98) ⊗H ∼ = C ([0 , ∞ ) , H (cid:98) ⊗ Cliff + ( t R )); S ε (cid:98) ⊗H ∼ = C ([0 , ε ) , H (cid:98) ⊗ Cliff + ( t R )) . We denote the base of R = T t R ≥ for t > e in the following.It is sufficient to prove that π ◦ ι (cid:98) ⊗ id( β ∞ N ( f (cid:98) ⊗ a ))( g (cid:98) ⊗ φ ) ∈ S ε (cid:98) ⊗H for f ∈ S ε , g ∈ S , a ∈ Cl τ ( U L k ,N ) and φ ∈ J ∞ M ( L ( U L k ,M , L )) (cid:98) ⊗ S ∗ U M (cid:98) ⊗ S U M , for the following reason: S ε (cid:98) ⊗H is closed in S (cid:98) ⊗H ; ∪ N β ∞ N ( S ε (cid:98) ⊗ alg Cl τ ( U L k ,N )) is dense in A ( U L k ); S ε (cid:98) ⊗ alg H fin is dense in S ε (cid:98) ⊗H ; and all the op-erations are continuous. We may assume that N ≤ M and f is smooth. By Lemma 5.26 (2) provedlater, we can prove that π ◦ ι (cid:98) ⊗ id( β ∞ N ( f (cid:98) ⊗ a ))( g (cid:98) ⊗ φ ) = lim N (cid:48) →∞ f ( X (cid:98) ⊗ id + id (cid:98) ⊗ C N (cid:48) N +1 ) (cid:98) ⊗ µ N ( a )( g (cid:98) ⊗ φ ) , and hence it is sufficient to prove that f ( X (cid:98) ⊗ id + id (cid:98) ⊗ C N (cid:48) N +1 ) (cid:98) ⊗ µ N ( a )( g (cid:98) ⊗ φ ) ∈ S ε (cid:98) ⊗H for all N (cid:48) .We may assume that φ is of the form φ (cid:98) ⊗ φ (cid:98) ⊗ vac ∞ M +1 , where φ ∈ L ( U L k ,N , L ) (cid:98) ⊗ S ∗ U N (cid:98) ⊗ S U N and φ ∈ L ( U L k ,M (cid:9) U L k ,N , L ) (cid:98) ⊗ S ∗ U M (cid:9) U N (cid:98) ⊗ S U M (cid:9) U N and N (cid:48) ≥ M . Then, f ( X (cid:98) ⊗ id + id (cid:98) ⊗ C N (cid:48) N +1 ) (cid:98) ⊗ µ N ( a )( g (cid:98) ⊗ φ ) = ± f ( X (cid:98) ⊗ id + id (cid:98) ⊗ C N (cid:48) N +1 )( g (cid:98) ⊗ φ (cid:98) ⊗ vac N (cid:48) M +1 ) (cid:98) ⊗ µ N ( a ) φ (cid:98) ⊗ vac ∞ N (cid:48) . In order to compute f ( X (cid:98) ⊗ id+id (cid:98) ⊗ C N (cid:48) N +1 )( g (cid:98) ⊗ φ (cid:98) ⊗ vac N (cid:48) M +1 ), we use the set-theoretical description. Weregard this element as a section on [0 , ∞ ) × ( U L k ,N (cid:48) (cid:9) U L k ,N ). Suppose that f ( t ) = f ( t )+ tf ( t ) (cid:98) ⊗ e for f , f ∈ C ([0 , ε )) (it is possible since f is smooth). Then, f ( X (cid:98) ⊗ id + id (cid:98) ⊗ C N (cid:48) N +1 )( g (cid:98) ⊗ φ (cid:98) ⊗ vac N (cid:48) M +1 )( t, x )= f ( t + (cid:107) x (cid:107) ) g ( t ) φ (cid:98) ⊗ vac N (cid:48) M +1 ( x ) + tf ( t + (cid:107) x (cid:107) ) g ( t ) φ (cid:98) ⊗ vac N (cid:48) M +1 ( x )+ ( − ∂g f ( t + (cid:107) x (cid:107) ) g ( t ) c ∗ ( x ) φ (cid:98) ⊗ vac N (cid:48) M +1 ( x ) . This function vanishes outside [0 , ε ) × ( U L k ,N (cid:48) (cid:9) U L k ,N ), and thus f ( X (cid:98) ⊗ id+id (cid:98) ⊗ C N (cid:48) N +1 )( g (cid:98) ⊗ φ (cid:98) ⊗ vac N (cid:48) M +1 ) ∈S ε (cid:98) ⊗ L ( U L k ,M (cid:9) U L k ,N , L ) (cid:98) ⊗ S ∗ U M (cid:9) U N (cid:98) ⊗ S U M (cid:9) U N . Consequently, π ◦ ι (cid:98) ⊗ id( β ∞ N ( f (cid:98) ⊗ a ))( g (cid:98) ⊗ φ ) ∈ S ε (cid:98) ⊗H .Let us define “the Dirac operator (cid:19) ∂ twisted by L ” acting on H . We define two operators R (cid:19) ∂ and L (cid:19) ∂ , and then we define (cid:19) ∂ by a linear combination of them following [T4, Definition 5.2]. Seealso [Kos, Was, FHT2, Mei2] for details on algebraic Dirac operators. Definition 5.24 (Definition 5.2 of [T4]) . (1) We define two linear maps from Lie( U fin ) to the setof unbounded operators on L ( U L k , L ) fin as follows: For φ ∈ L ( U L k , L ) fin , dR (cid:48) ( e n ) φ := ∂φ∂x n + in l y n φ, and dR (cid:48) ( f n ) φ := ∂φ∂y n − in l x n φ, L (cid:48) ( e n ) φ := − ∂φ∂x n + in l y n φ, and dL (cid:48) ( f n ) φ := − ∂φ∂y n − in l x n φ. The strict definition of ∂φ∂x n is the following: For φ = J ∞ N ( ψ ) ∈ J ∞ N [ L ( U L k ,N , L )], ∂φ∂x n := J ∞ N (cid:48) ( ∂J N (cid:48) N ( ψ ) ∂x n ),where N (cid:48) is chosen to be greater than N and n . We linearly extend the maps dL (cid:48) and dR (cid:48) toLie( U fin ) (cid:98) ⊗ C .(2) For φ ∈ H fin , we define L (cid:19) ∂φ := (cid:88) n n − l/ (cid:2) dL (cid:48) ( z n ) (cid:98) ⊗ γ ∗ ( z n ) (cid:98) ⊗ id + dL (cid:48) ( z n ) (cid:98) ⊗ γ ∗ ( z n ) (cid:98) ⊗ id (cid:3) φ, R (cid:19) ∂φ := (cid:88) n n − l/ (cid:2) dR (cid:48) ( z n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( z n ) + dR (cid:48) ( z n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( z n ) (cid:3) φ, (cid:19) ∂φ := 1 √ R (cid:19) ∂φ + i √ L (cid:19) ∂φ. Remarks . (1) The summand dR (cid:48) ( z n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( z n ) + dR (cid:48) ( z n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( z n ) of R (cid:19) ∂ can be rewrittenas dR (cid:48) ( e n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( e n ) + dR (cid:48) ( f n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( f n ), and similarly for L (cid:19) ∂ . Thus, they should be regarded asDirac operators.(2) By the two Clifford multiplications c ( v ) := 1 √ (cid:0) id (cid:98) ⊗ γ ( v ) − iγ ∗ ( v ) (cid:98) ⊗ id (cid:1) and c ∗ ( v ) := √− √ (cid:0) id (cid:98) ⊗ γ ( v ) + iγ ∗ ( v ) (cid:98) ⊗ id (cid:1) for v ∈ Lie( U L k ), we can rewrite the summand of (cid:19) ∂ √− √ (cid:2) dL (cid:48) ( z n ) (cid:98) ⊗ γ ∗ ( z n ) (cid:98) ⊗ id + dL (cid:48) ( z n ) (cid:98) ⊗ γ ∗ ( z n ) (cid:98) ⊗ id (cid:3) + 1 √ (cid:2) dR (cid:48) ( z n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( z n ) + dR (cid:48) ( z n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( z n ) (cid:3) can be rewritten as, by a direct computation, n − l/ (cid:26) ∂∂x n (cid:98) ⊗ c ( e n ) + ∂∂y n (cid:98) ⊗ c ( f n ) − n l x n (cid:98) ⊗ c ∗ ( J e n ) − n l y n (cid:98) ⊗ c ∗ ( J f n ) (cid:27) . This operator resembles the Bott-Dirac operator studied in [HK]. In fact, we can compute thespectrum of the square of this operator just like the Bott-Dirac case.(3) Each of L (cid:19) ∂φ , R (cid:19) ∂φ and (cid:19) ∂φ is a finite sum for every φ ∈ H fin . This is because the element“vac” is killed by dR (cid:48) ( z n ) and dL (cid:48) ( z n ) for all n , and ∗ f (cid:98) ⊗ f is killed by γ ∗ ( z n ) (cid:98) ⊗ id S U and id S ∗ U (cid:98) ⊗ γ ( z n ). H fin is the common core of these operators.(4) R (cid:19) ∂ is essentially self-adjoint and L (cid:19) ∂ is essentially skew-adjoint. They anti-commute withone another. Consequently, (cid:19) ∂ is essentially self-adjoint. The self-adjoint extension of (cid:19) ∂ is denotedby the same symbol.Since our operator “has a liner potential”, any element of the “Sobolev space defined by (cid:19) ∂ ”satisfies the following estimate on the “decay rate”. Lemma 5.26. (1)
The operator C : H fin → H extends to a bounded operator from dom( (cid:19) ∂ ) to H . (2) C N also extends to a bounded operator from dom( (cid:19) ∂ ) to H , and { C N } converges to C in L C (dom( (cid:19) ∂ ) , H ) . Note that [ γ ( z n )] ∗ = − γ ( z n ), [ γ ∗ ( z n )] ∗ = γ ∗ ( z n ), dL (cid:48) ( z n ) ∗ = − dL (cid:48) ( z n ) and dR (cid:48) ( z n ) ∗ = − dR (cid:48) ( z n ). roof. We prove both statements at the same time.We first take complete orthonomal systems (CONS for short) of H and dom( (cid:19) ∂ ). For −→ α =( α , α , · · · ), −→ β = ( β , β , · · · ), −→ ξ = ( ξ , ξ , · · · , ξ M ), and −→ η = ( η , η , · · · , η N ) such that α i , β i ∈ Z ≥ , α i = β i = 0 except for finitely many i ’s, ξ i , η i ∈ N , N, M ∈ Z ≥ , 0 < ξ < ξ < · · · < ξ M , and0 < η < η < · · · < η N , we put (cid:101) φ −→ α , −→ β , −→ ξ , −→ η := (cid:16) dR (cid:48) ( z ) α dL (cid:48) ( z ) β dR (cid:48) ( z ) α dL (cid:48) ( z ) β · · · (cid:17) vac (cid:98) ⊗ [ z ξ ∧ · · · ∧ z ξ M ] (cid:98) ⊗ [ z η ∧ · · · ∧ z η N ] , and we put φ −→ α , −→ β , −→ ξ , −→ η := (cid:107) (cid:101) φ −→ α , −→ β , −→ ξ , −→ η (cid:107) − (cid:101) φ −→ α , −→ β , −→ ξ , −→ η . Then, { φ −→ α , −→ β , −→ ξ , −→ η } is a CONS of H . We put λ −→ α , −→ β , −→ ξ , −→ η := (cid:80) (4 m l/ ) α m + (cid:80) (4 m l/ ) β m + 4 (cid:80) ξ l/ m + 4 (cid:80) η l/ m . Since (cid:19) ∂ φ −→ α , −→ β , −→ ξ , −→ η = (cid:16)(cid:88) (4 n l/ ) α n + (cid:88) (4 n l/ ) β n + 4 (cid:88) ξ l/ n + 4 (cid:88) η l/ n (cid:17) φ −→ α , −→ β , −→ ξ , −→ η = λ −→ α , −→ β , −→ ξ , −→ η φ −→ α , −→ β , −→ ξ , −→ η the following is a CONS of dom( (cid:19) ∂ ) with respect to the graph norm: (cid:40)(cid:18)(cid:113) λ −→ α , −→ β , −→ ξ , −→ η (cid:19) − φ −→ α , −→ β , −→ ξ , −→ η (cid:41) . In order to prove the statement, we will prove that each operator x n (cid:98) ⊗ c ∗ ( e n ) + y n (cid:98) ⊗ c ∗ ( f n ) = z n (cid:98) ⊗ c ∗ ( z n ) + z n (cid:98) ⊗ c ∗ ( z n ) is bounded, and that the infinite sum (cid:80) n (cid:107) z n (cid:98) ⊗ c ∗ ( z n ) + z n (cid:98) ⊗ c ∗ ( z n ) (cid:107) is finite.We consider ( z n (cid:98) ⊗ c ∗ ( z n ) + z n (cid:98) ⊗ c ∗ ( z n )) φ for φ = (cid:88) −→ α , −→ β , −→ ξ , −→ η c −→ α , −→ β , −→ ξ , −→ η (cid:113) λ −→ α , −→ β , −→ ξ , −→ η − φ −→ α , −→ β , −→ ξ , −→ η ∈ dom( (cid:19) ∂ ) . Note that the graph norm of φ is (cid:80) −→ α , −→ β , −→ ξ , −→ η | c −→ α , −→ β , −→ ξ , −→ η | .By a simple computation, we notice the following formulas: z n φ = 12 n l (cid:8) dR (cid:48) ( z n ) + dL (cid:48) ( z n ) (cid:9) φ and z n φ = − n l (cid:8) dR (cid:48) ( z n ) + dL (cid:48) ( z n ) (cid:9) φ. Thus, by the commutation relations on dR (cid:48) s and dL (cid:48) s, z n dR (cid:48) ( z n ) α n dL (cid:48) ( z n ) β n vac = 12 n l dR (cid:48) ( z n ) α n dL (cid:48) ( z n ) β n +1 vac − ndR (cid:48) ( z n ) α n − dL (cid:48) ( z n ) β n vac ,z n dR (cid:48) ( z n ) α n dL (cid:48) ( z n ) β n vac = − n l dR (cid:48) ( z n ) α n +1 dL (cid:48) ( z n ) β n vac − ndR (cid:48) ( z n ) α n dL (cid:48) ( z n ) β n − vac . By (cid:107) (cid:101) φ −→ α , −→ β , −→ ξ , −→ η (cid:107) = −→ α ! −→ β ! (cid:81) (2 n l ) α n + β n (where −→ γ ! := γ ! γ ! · · · for a multi-index −→ γ = ( γ , γ , · · · )), z n φ −→ α , −→ β , −→ ξ , −→ η = (cid:114) β n + 12 n l φ −→ α , −→ β + e n , −→ ξ , −→ η − n (cid:112) n l α n φ −→ α − e n , −→ β , −→ ξ , −→ η ,z n φ −→ α , −→ β , −→ ξ , −→ η = − (cid:114) α n + 12 n l φ −→ α + e n , −→ β , −→ ξ , −→ η − n (cid:112) n l α n φ −→ α , −→ β − e n , −→ ξ , −→ η , −→ γ = ( · · · , γ n − , γ n , γ n +1 , · · · ), we denote ( · · · , γ n − , γ n ± , γ n +1 , · · · ) by −→ γ ± e n . If α n = 0, we put n √ n l α n φ −→ α − e n , −→ β , −→ ξ , −→ η := 0, and similarly for n √ n l β n φ −→ α , −→ β − e n , −→ ξ , −→ η .Moreover, by definition, c ∗ ( z n ) φ −→ α , −→ β , −→ ξ , −→ η = (cid:0) z n (cid:99) ◦ (cid:15) S ∗ (cid:98) ⊗ id + id (cid:98) ⊗√− z n ∧ (cid:1) φ −→ α , −→ β , −→ ξ , −→ η =: ( − M (cid:110) φ −→ α , −→ β , −→ ξ \{ n } , −→ η + √− φ −→ α + e n , −→ β , −→ ξ , −→ η ∪{ n } (cid:111) . Similarly, c ∗ ( z n ) φ −→ α , −→ β , −→ ξ , −→ η = (cid:0) z n ∧ ◦ (cid:15) S ∗ (cid:98) ⊗ id − id (cid:98) ⊗√− z n (cid:99) (cid:1) φ −→ α , −→ β , −→ ξ , −→ η =: ( − M (cid:110) φ −→ α , −→ β , −→ ξ ∪{ n } , −→ η − √− φ −→ α , −→ β , −→ ξ , −→ η \{ n } (cid:111) . Therefore, (cid:8) z n (cid:98) ⊗ c ∗ ( z n ) + z n (cid:98) ⊗ c ∗ ( z n ) (cid:9) φ −→ α , −→ β , −→ ξ , −→ η is given by( − M (cid:32)(cid:114) β n + 12 n l φ −→ α , −→ β + e n , −→ ξ \{ n } , −→ η − n (cid:112) n l α n φ −→ α − e n , −→ β , −→ ξ \{ n } , −→ η (cid:33) + ( − M (cid:32)(cid:114) β n + 12 n l φ −→ α , −→ β + e n , −→ ξ , −→ η ∪{ n } − n (cid:112) n l α n φ −→ α − e n , −→ β , −→ ξ , −→ η ∪{ n } (cid:33) + ( − M (cid:32) − (cid:114) α n + 12 n l φ −→ α + e n , −→ β , −→ ξ ∪{ n } , −→ η − n (cid:112) n l α n φ −→ α , −→ β − e n , −→ ξ ∪{ n } , −→ η (cid:33) + ( − M (cid:32) − (cid:114) α n + 12 n l φ −→ α + e n , −→ β , −→ ξ , −→ η \{ n } − n (cid:112) n l α n φ −→ α , −→ β − e n , −→ ξ , −→ η \{ n } (cid:33) . Thus, (cid:8) z n (cid:98) ⊗ c ∗ ( z n ) + z n (cid:98) ⊗ c ∗ ( z n ) (cid:9) φ is given by (cid:88) −→ α , −→ β , −→ ξ , −→ η ( − M c −→ α , −→ β , −→ ξ , −→ η (cid:113) λ −→ α , −→ β , −→ ξ , −→ η (cid:32)(cid:114) β n + 12 n l φ −→ α , −→ β + e n , −→ ξ \{ n } , −→ η − n (cid:112) n l α n φ −→ α − e n , −→ β , −→ ξ \{ n } , −→ η (cid:33) + (cid:88) −→ α , −→ β , −→ ξ , −→ η ( − M c −→ α , −→ β , −→ ξ , −→ η (cid:113) λ −→ α , −→ β , −→ ξ , −→ η (cid:32)(cid:114) β n + 12 n l φ −→ α , −→ β + e n , −→ ξ , −→ η ∪{ n } − n (cid:112) n l α n φ −→ α − e n , −→ β , −→ ξ , −→ η ∪{ n } (cid:33) + (cid:88) −→ α , −→ β , −→ ξ , −→ η ( − M c −→ α , −→ β , −→ ξ , −→ η (cid:113) λ −→ α , −→ β , −→ ξ , −→ η (cid:32) − (cid:114) α n + 12 n l φ −→ α + e n , −→ β , −→ ξ ∪{ n } , −→ η − n (cid:112) n l α n φ −→ α , −→ β − e n , −→ ξ ∪{ n } , −→ η (cid:33) + (cid:88) −→ α , −→ β , −→ ξ , −→ η ( − M c −→ α , −→ β , −→ ξ , −→ η (cid:113) λ −→ α , −→ β , −→ ξ , −→ η (cid:32) − (cid:114) α n + 12 n l φ −→ α + e n , −→ β , −→ ξ , −→ η \{ n } − n (cid:112) n l α n φ −→ α , −→ β − e n , −→ ξ , −→ η \{ n } (cid:33) − (cid:88) −→ α , −→ β , −→ ξ , −→ η ( − M c −→ α , −→ β − e n , −→ ξ ∪{ n } , −→ η (cid:113) λ −→ α , −→ β − e n , −→ ξ ∪{ n } , −→ η (cid:114) β n n l − c −→ α + e n , −→ β , −→ ξ ∪{ n } , −→ η (cid:113) λ −→ α + e n , −→ β , −→ ξ ∪{ n } , −→ η n (cid:112) n l ( α n + 1) φ −→ α , −→ β , −→ ξ , −→ η + (cid:88) −→ α , −→ β , −→ ξ , −→ η ( − M c −→ α , −→ β − e n , −→ ξ , −→ η \{ n } (cid:113) λ −→ α , −→ β − e n , −→ ξ , −→ η \{ n } (cid:114) β n n l − c −→ α + e n , −→ β , −→ ξ , −→ η \{ n } (cid:113) λ −→ α + e n , −→ β , −→ ξ , −→ η \{ n } n (cid:112) n l ( α n + 1) φ −→ α , −→ β , −→ ξ , −→ η − (cid:88) −→ α , −→ β , −→ ξ , −→ η ( − M − c −→ α − e n , −→ β , −→ ξ ∪{ n } , −→ η (cid:113) λ −→ α − e n , −→ β , −→ ξ ∪{ n } , −→ η (cid:114) α n n l − c −→ α , −→ β + e n , −→ ξ ∪{ n } , −→ η (cid:113) λ −→ α , −→ β + e n , −→ ξ ∪{ n } , −→ η n (cid:112) n l ( β n + 1) φ −→ α , −→ β , −→ ξ , −→ η + (cid:88) −→ α , −→ β , −→ ξ , −→ η ( − M − c −→ α − e n , −→ β , −→ ξ , −→ η \{ n } (cid:113) λ −→ α − e n , −→ β , −→ ξ , −→ η \{ n } (cid:114) α n n l − c −→ α , −→ β + e n , −→ ξ , −→ η \{ n } (cid:113) λ −→ α , −→ β + e n , −→ ξ , −→ η \{ n } n (cid:112) n l ( β n + 1) φ −→ α , −→ β , −→ ξ , −→ η , where c −→ α − e n , −→ β , ··· := 0 if α n = 0, and c −→ α , −→ β − e n , ··· := 0 if β n = 0.Let (cid:63) −→ α , −→ β , −→ ξ , −→ η be (cid:12)(cid:12)(cid:12) c −→ α , −→ β − e n , −→ ξ , −→ η \{ n } (cid:12)(cid:12)(cid:12) λ −→ α , −→ β − e n , −→ ξ , −→ η \{ n } β n n l + (cid:12)(cid:12)(cid:12) c −→ α + e n , −→ β , −→ ξ , −→ η \{ n } (cid:12)(cid:12)(cid:12) λ −→ α + e n , −→ β , −→ ξ , −→ η \{ n } n n l ( α n + 1)+ (cid:12)(cid:12)(cid:12) c −→ α − e n , −→ β , −→ ξ ∪{ n } , −→ η (cid:12)(cid:12)(cid:12) λ −→ α − e n , −→ β , −→ ξ ∪{ n } , −→ η α n n l + (cid:12)(cid:12)(cid:12) c −→ α , −→ β + e n , −→ ξ ∪{ n } , −→ η (cid:12)(cid:12)(cid:12) λ −→ α , −→ β + e n , −→ ξ ∪{ n } , −→ η n n l ( β n + 1)+ (cid:12)(cid:12)(cid:12) c −→ α − e n , −→ β , −→ ξ ∪{ n } , −→ η (cid:12)(cid:12)(cid:12) λ −→ α − e n , −→ β , −→ ξ ∪{ n } , −→ η α n n l + (cid:12)(cid:12)(cid:12) c −→ α , −→ β + e n , −→ ξ ∪{ n } , −→ η (cid:12)(cid:12)(cid:12) λ −→ α , −→ β + e n , −→ ξ ∪{ n } , −→ η n n l ( β n + 1)+ (cid:12)(cid:12)(cid:12) c −→ α − e n , −→ β , −→ ξ , −→ η \{ n } (cid:12)(cid:12)(cid:12) λ −→ α − e n , −→ β , −→ ξ , −→ η \{ n } α n n l + (cid:12)(cid:12)(cid:12) c −→ α , −→ β + e n , −→ ξ , −→ η \{ n } (cid:12)(cid:12)(cid:12) λ −→ α , −→ β + e n , −→ ξ , −→ η \{ n } n n l ( β n + 1) . Then, (cid:107) (cid:8) z n (cid:98) ⊗ c ∗ ( z n ) + z n (cid:98) ⊗ c ∗ ( z n ) (cid:9) φ (cid:107) is bounded by8 (cid:88) −→ α , −→ β , −→ ξ , −→ η (cid:63) −→ α , −→ β , −→ ξ , −→ η . Since all of β n λ −→ α , −→ β − en, −→ ξ , −→ η \{ n } , λ −→ α + en, −→ β , −→ ξ , −→ η \{ n } )( α n +1) , · · · are bounded above by a constant (say K ) which is independent of −→ α , −→ β , −→ ξ , −→ η and n , (cid:63) −→ α , −→ β , −→ ξ , −→ η is bounded above by K (cid:18)(cid:12)(cid:12)(cid:12) c −→ α , −→ β − e n , −→ ξ , −→ η \{ n } (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c −→ α + e n , −→ β , −→ ξ , −→ η \{ n } (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c −→ α − e n , −→ β , −→ ξ ∪{ n } , −→ η (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c −→ α , −→ β + e n , −→ ξ ∪{ n } , −→ η (cid:12)(cid:12)(cid:12) (cid:19) n − l + K (cid:18)(cid:12)(cid:12)(cid:12) c −→ α − e n , −→ β , −→ ξ ∪{ n } , −→ η (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c −→ α , −→ β + e n , −→ ξ ∪{ n } , −→ η (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c −→ α − e n , −→ β , −→ ξ , −→ η \{ n } (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c −→ α , −→ β + e n , −→ ξ , −→ η \{ n } (cid:12)(cid:12)(cid:12) (cid:19) n − l . Therefore, the operator norm of z n (cid:98) ⊗ c ∗ ( z n ) + z n (cid:98) ⊗ c ∗ ( z n ) is bounded by 64 Kn − l . Since l >
4, theinfinite sum (cid:80) n (cid:107) z n (cid:98) ⊗ c ∗ ( z n ) + z n (cid:98) ⊗ c ∗ ( z n ) (cid:107) ≤ (cid:80) n n − l is finite.72ith this preliminaries, we can prove the following. Since we have proved almost the sameresult in [T4, Section 5.1], we just outline the proof here for the convenience of the reader, and inorder to prepare the proof of the next result. Lemma 5.27.
The triple ( S (cid:98) ⊗H , π, id (cid:98) ⊗ (cid:19) ∂ ) defines an unbounded U L m -equivariant Kasparov ( A HKT ( U L k ) , S ) -module.Proof. We need to check the following: ( A ) The pair is actually an unbounded Kasparov module;and ( B ) This Kasparov module is equivariant.( A ) can be divided into the following steps: (0) S (cid:98) ⊗H is a countably generated Hilbert S -module, and id (cid:98) ⊗ (cid:19) ∂ is a densely defined, regular, odd, and essentially self-adjoint operator; (1) Theset of a ∈ A HKT ( U L k ) preserving dom( (cid:19) ∂ ) and satisfying [ a, id (cid:98) ⊗ (cid:19) ∂ ] ∈ L S ( S (cid:98) ⊗H ) is dense; and (2) a (id (cid:98) ⊗ id + id (cid:98) ⊗ (cid:19) ∂ ) − belongs to K S ( S (cid:98) ⊗H ) for any a ∈ A HKT ( U L k ).(0) is clear from the fact that H is a separable Hilbert space, and the arguments about (cid:19) ∂ so far.(1) is proved like [T4, Proposition 5.10]. We give a slightly different and more concrete proof. We prove that π (cid:16) A HKT ( U L k ) fin (cid:17) preserves dom(id (cid:98) ⊗ (cid:19) ∂ ) and that the commutator is bounded. Letus check the simplest case: β ∞ ( f ) ∈ A HKT ( U L k ) fin , and the general cases are left to the reader. Wenotice that π ( β ∞ ( f )) φ = f ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C ∞ )) φ = lim N →∞ f ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) φ for φ ∈ S fin (cid:98) ⊗ alg H fin and f ∈ S , thanks to [T4, Lemma 5.8]. Since S fin (cid:98) ⊗ alg H fin is dense in S (cid:98) ⊗H , andsince (cid:107) π ( β ∞ ( f )) (cid:107) is bounded by (cid:107) f (cid:107) , we have π ( β ∞ ( f )) ψ = lim N →∞ f ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) (cid:98) ⊗ ψ forarbitrary ψ ∈ S (cid:98) ⊗H . By the fact that id (cid:98) ⊗ (cid:19) ∂ is a closed operator, it is sufficient to prove the following:( a ) f ( X (cid:98) ⊗ id+id (cid:98) ⊗ π ( C N )) φ ∈ dom(id (cid:98) ⊗ (cid:19) ∂ ) for every N and ( b ) { (id (cid:98) ⊗ (cid:19) ∂ ) ◦ f ( X (cid:98) ⊗ id+id (cid:98) ⊗ π ( C N ))( φ ) } N ∈ N converges. We will also prove that the commutator is bounded at the same time. It is enough toprove these properties for f = f ev , f od .We put f = f ev in this paragraph. ( a ) is obvious because f ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) is essentiallya multiplication operator by a smooth function vanishing at infinity on a finite-dimensional space.For ( b ), we compute (id (cid:98) ⊗ (cid:19) ∂ ) ◦ f ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))( φ ). Thanks to the formula on the gradedcommutator [ A, B − ] = − ( − ∂A∂B B − [ A, B ] B − , we have(id (cid:98) ⊗ (cid:19) ∂ ) ◦ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))( φ )= [id (cid:98) ⊗ (cid:19) ∂, f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))] φ + f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))[id (cid:98) ⊗ (cid:19) ∂ ( φ )]= − f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) ◦ [id (cid:98) ⊗ (cid:19) ∂, X (cid:98) ⊗ id + id (cid:98) ⊗ ( C N ) ] ◦ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))( φ )+ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))[id (cid:98) ⊗ (cid:19) ∂ ( φ )]= − f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) ◦ (cid:34) id (cid:98) ⊗ (cid:40) N (cid:88) n =1 n − l [2 x n (cid:98) ⊗ c ( e n ) + 2 y n (cid:98) ⊗ c ( f n )] (cid:41)(cid:35) ◦ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))( φ )+ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))[id (cid:98) ⊗ (cid:19) ∂ ( φ )]=: − C l,N ◦ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) ( φ ) + f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))[id (cid:98) ⊗ (cid:19) ∂ ( φ )] , In [T4], we have used { e − X , Xe − X } as a generating set of S . Then, π ( β ( e − X )) is given by “ e − C ”. However,it is difficult to say that it is given by (cid:80) n ( n !) − C n , because the infinite sum should be in the sense of uniformconvergence on compact sets, if we use similar arguments for finite-dimensional spaces. Instead, we use alternativegenerators in this proof, for the following reasons: The values at these generators are simple to define; We would liketo give as many computations on A ( X ) as possible. C l,N := id (cid:98) ⊗ (cid:40) N (cid:88) n =1 n − l [2 x n (cid:98) ⊗ c ( e n ) + 2 y n (cid:98) ⊗ c ( f n )] (cid:41) . By the same argument of the previous lemma, C l,N converges in L S (dom(id (cid:98) ⊗ (cid:19) ∂ ) , S (cid:98) ⊗H ). Thelimit is denoted by C l . Let us prove that − C l,N ◦ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) ( φ ) + f ev ( X (cid:98) ⊗ id +id (cid:98) ⊗ π ( C N ))[id (cid:98) ⊗ (cid:19) ∂ ( φ )] converges to − C l ◦ f ev ( X (cid:98) ⊗ id+id (cid:98) ⊗ π ( C ∞ )) ( φ )+ f ev ( X (cid:98) ⊗ id+id (cid:98) ⊗ π ( C ∞ ))[id (cid:98) ⊗ (cid:19) ∂ ( φ )].In fact, − C l,N ◦ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) ( φ ) + f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N ))[id (cid:98) ⊗ (cid:19) ∂ ( φ )]+ C l ◦ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C ∞ )) ( φ ) − f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C ∞ ))[id (cid:98) ⊗ (cid:19) ∂ ( φ )]= f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) ◦ ( C l − C l,N ) φ + (cid:8) f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C ∞ )) − f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) (cid:9) ◦ C l φ + (cid:8) f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) − f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C ∞ )) (cid:9) [id (cid:98) ⊗ (cid:19) ∂ ( φ )] . Since (cid:107) f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) (cid:107) ≤
1, the third line converges to 0. The fourth and fifth lines,converge to 0 thanks to the strong convergence f ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C N )) → π ( β ∞ ( f )) on S (cid:98) ⊗H .This means π ( β ∞ ( f ev )) φ ∈ dom(id (cid:98) ⊗ (cid:19) ∂ ). Moreover, by the above computation, the commutator[id (cid:98) ⊗ (cid:19) ∂, π ( β ∞ ( f ev ))] is given by − C l f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C ∞ )) , which is bounded. In fact, the operatorid (cid:98) ⊗ x n (cid:98) ⊗ c ( e n ) ◦ f ev ( X (cid:98) ⊗ id + id (cid:98) ⊗ π ( C ∞ )) is bounded and its operator norm is not greater than 1, and l/ f od = Xf ev , one can prove ( a ) and ( b ) by the following formulas:[id (cid:98) ⊗ (cid:19) ∂, π ( β ∞ ( Xf ev ))] = [id (cid:98) ⊗ (cid:19) ∂, X (cid:98) ⊗ id + id (cid:98) ⊗ C ] π ( β ∞ ( f ev )) − ( X (cid:98) ⊗ id + id (cid:98) ⊗ C )[id (cid:98) ⊗ (cid:19) ∂, π ( β ∞ ( f ev ))] . [id (cid:98) ⊗ (cid:19) ∂, X (cid:98) ⊗ id + id (cid:98) ⊗ C ] = id (cid:98) ⊗ id (cid:98) ⊗ (cid:88) n n − l/ { c ( e n ) c ∗ ( e n ) + c ( f n ) c ∗ ( f n ) } . We leave the details to the reader.(2) follows from the same argument of the proof of [T4, Proposition 5.11]. We have computed thespectrum of (cid:19) ∂ in Lemma 5.26. By this computation, the number of combinations of multi-indices −→ α , −→ β , −→ ξ , and −→ η such that the corresponding eigenvalue is less than K is finite, for every K ∈ R .Thus, (1 + (cid:19) ∂ ) − is a compact operator on H . Therefore, π ( a ) ◦ (cid:8) id (cid:98) ⊗ (1 + (cid:19) ∂ ) − (cid:9) is S -compact.For ( B ), we need to check the following: The U L m -action preserves dom(id (cid:98) ⊗ (cid:19) ∂ ); g (id (cid:98) ⊗ (cid:19) ∂ ) − id (cid:98) ⊗ (cid:19) ∂ is bounded for each g ∈ U L m ; and the map g (cid:55)→ g (id (cid:98) ⊗ (cid:19) ∂ ) − id (cid:98) ⊗ (cid:19) ∂ is continuous.In order to prove them, we compute the difference g ( (cid:19) ∂ ) − (cid:19) ∂ . For this aim, we compute L ( g, ◦ dL (cid:48) ( v ) ◦ L ( − g, and L ( g, ◦ dR (cid:48) ( v ) ◦ L ( − g, for v = e n , f n . First, we suppose g ∈ U L m ,N and wecompute the difference on φ ∈ L ( U L k ,N , L ) for n ≤ N . For g = ( a , b , · · · , a N , b N ), L ( g, ◦ dL (cid:48) ( e n ) ◦ L ( − g, φ ( x )= dL (cid:48) ( e n ) ◦ L ( − g, φ ( x − g ) τ ( g, x )= (cid:26) − ∂L ( − g, φ∂x n ( x − g ) + in l ( y n − b n ) L ( − g, φ ( x − g )) (cid:27) τ ( g, x )= (cid:26) − ∂φ∂x n ( x ) τ ( − g, x − g ) − in − k b n φ ( x ) τ ( − g, x − g ) + in l ( y n − b n ) φ ( x ) τ ( − g, x − g ) (cid:27) τ ( g, x )= dL (cid:48) ( e n ) φ ( x ) + i (cid:16) − n − k − n l (cid:17) b n φ ( x ) .
74y similar computations, we obtain the following formulas: L ( g, ◦ dL (cid:48) ( f n ) ◦ L ( − g, = dL (cid:48) ( f n ) + i (cid:16) n − k + n l (cid:17) a n ; L ( g, ◦ dR (cid:48) ( e n ) ◦ L ( − g, = dR (cid:48) ( f n ) + i (cid:16) n − k − n l (cid:17) b n ; L ( g, ◦ dR (cid:48) ( f n ) ◦ L ( − g, = dR (cid:48) ( f n ) + i (cid:16) − n − k + n l (cid:17) a n . Let
J g be (cid:80) Nj =1 ( − b j e j + a j f j ) ∈ Lie( U L m ,N ). Then, g ( (cid:19) ∂ ) − (cid:19) ∂ = 1 √ (cid:98) ⊗ id (cid:98) ⊗ γ (cid:88) n ≤ N n − l/ (cid:104) − i (cid:16) − n − k + n l (cid:17) b n e n + i (cid:16) − n − k + n l (cid:17) a n f n (cid:105) + i √ (cid:98) ⊗ γ ∗ (cid:88) n ≤ N n − l/ (cid:104) − i (cid:16) n − k + n l (cid:17) b n e n + i (cid:16) n − k + n l (cid:17) a n f n (cid:105) (cid:98) ⊗ id= i √ (cid:98) ⊗ id (cid:98) ⊗ γ (cid:16)(cid:16) −| d | − k − l/ + | d | l/ (cid:17) J g (cid:17) + − √ (cid:98) ⊗ γ ∗ (cid:16)(cid:16) | d | − k − l/ + | d | l/ (cid:17) J g (cid:17) (cid:98) ⊗ id . Since g ∈ U fin , the above is a finite sum and it is bounded. The operator norm of this difference isbounded above by the L m - norm of J g because m ≥ k + l ≥ − k − l/ , l/ g ∈ U L m , L ( g,z ) preserves dom( (cid:19) ∂ ), and g ( (cid:19) ∂ ) − (cid:19) ∂ is bounded, because both of (cid:0) | d | − k − l/ + | d | l/ (cid:1) J g and (cid:0) −| d | − k − l/ + | d | l/ (cid:1) J g belong to Lie( U L m ) .Since all the maps U L m (cid:51) g (cid:55)→ (cid:0) | d | − k − l/ + | d | l/ (cid:1) J g ∈ Lie( U L k ), U L m (cid:51) g (cid:55)→ (cid:0) −| d | − k − l/ + | d | l/ (cid:1) J g ∈ Lie( U L k ),Lie( U L m ) (cid:51) v (cid:55)→ id (cid:98) ⊗ γ ( v ) (cid:98) ⊗ id ∈ L ( H )are continuous, the map g (cid:55)→ g (id (cid:98) ⊗ (cid:19) ∂ ) − id (cid:98) ⊗ (cid:19) ∂ is continuous. Remark . We have chosen strange Hilbert space and operator in order to consider the L k -topology for arbitrary k > / A ( U L k ) act there. If k were 1 / and l were R (cid:19) ∂ isactually equivariant and it is the setting dealt in [T4].This lemma gives an element of KK τU L m ( A HKT ( U L k ) , S ). By the pullback via ι (cid:98) ⊗ id, and thanksto Definition-Proposition 5.23 (2), it is possible to obtain a substitute for the index element atthe KK -theory level . However, we explicitly describe the unbounded Kasparov module which isnaturally defined by the pullback of the above index element via ι (cid:98) ⊗ id for the following reasons.First, the Kasparov module we are constructing is an ingredient of the index element of the wholemanifold M L k and it is defined by the tensor product of two KK -elements. Unbounded Kasparovmodules have an advantage when dealing with exterior tensor products. Second, in our opinion, the C ∗ -algebra A ( X ) constructed by [Yu] should be studied much more. It has an advantage comparingwith that constructed in [HKT], in that the former looks more geometrical. On the other hand,the former has a big disadvantage, in that the definition is abstract. We believe the followingconstructions can be useful to study the A ( X ) overcoming this disadvantage.75 efinition-Proposition 5.29. The triple ( S ε (cid:98) ⊗H , π, id (cid:98) ⊗ (cid:19) ∂ ) defines an unbounded U L m -equivariantKasparov ( A ( U L k ) , S ε )-module. The corresponding KK -element is denoted by [ (cid:101) (cid:19) ∂ ] and called the index element .The non-trivial thing is only the following: there is a dense subalgebra of A ( U L k ) consisting of“ C -elements” , where we say an element a of A ( U L k ) is a C -element if it preserves dom(id (cid:98) ⊗ (cid:19) ∂ )and [id (cid:98) ⊗ (cid:19) ∂, π ( a )] is bounded. It is highly non-trivial because A HKT ( U L k ) fin ∩ ι (cid:98) ⊗ id( A ( U L k )) is empty.Notice that the set of all C -elements is a subalgebra.We divide the proof into the following steps. Lemma 5.30. X (cid:98) ⊗ id + id (cid:98) ⊗ C has dense range.Proof. We first notice that C has dense range. We have essentially proved it as [T4, Proposition4.24]. We need to modify the definition of the Clifford operator and to replace the phrase “ C N ±√− C N has dense range”. It is correct because C N is self-adjoint, it is injective, and ran( C N ) ⊥ = ker( C N ) = 0 (note that we are working on “ L -spaces” noton “ C -spaces”).We prove that arbitrary element of S (cid:98) ⊗H can be approximated by elements in the range of X (cid:98) ⊗ id + id (cid:98) ⊗ C . It is sufficient to prove that g (cid:98) ⊗ φ can be approximated for φ ∈ H and g ∈ S satisfying the following: g is compactly supported and (cid:107) g (cid:107) = 1. Let (cid:15) >
0. Since C has denserange, there exists ψ ∈ H satisfying (cid:107) Cψ − φ (cid:107) < (cid:15)/
2. Thus, if we can find A ∈ S (cid:98) ⊗H such that (cid:107) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C ) A − g (cid:98) ⊗ Cψ (cid:107) < (cid:15)/
2, we have (cid:107) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C ) A − g (cid:98) ⊗ φ (cid:107) < (cid:15) . Therefore, we mayassume that φ = Cψ , from the beginning. In addition, we suppose that (cid:107) ψ (cid:107) ≤ (cid:107) φ (cid:107) ≤ A by the approximate spectral decomposition of theoperator X . Pick up a positive real number ∆ < (cid:15)/
16 such that | s − t | < ∆ implies | g ( t ) − g ( s ) | <(cid:15)/
16. This is possible because g is compactly supported. We define a bump function ρ around 0by ρ ( t ) := t ∆ t ∈ [ − ∆ , − t ∆ t ∈ [0 , ∆]0 otherwiseand we define a bump function at x by ρ x ( t ) := ρ ( t − x ). Let x n := n ∆ for n ∈ Z . We notice thefollowing properties:(1 ρ ) (cid:107) Xρ x − xρ x (cid:107) ≤ ∆;(2 ρ ) { ρ x n } n ∈ Z gives a continuous partition of unity; and(3 ρ ) For all t ∈ R , { n ∈ Z ≥ | ρ x n ( t ) (cid:54) = 0 } is at most 2.Let us consider A := g (0) ρ (cid:98) ⊗ ψ + (cid:88) n ∈ Z > (cid:18) ρ x n (cid:98) ⊗ g ( x n ) x n + g ( − x n ) Cx n + C φ + ρ − x n (cid:98) ⊗ − g ( − x n ) x n + g ( − x n ) Cx n + C φ (cid:19) Note that A is a finite sum because g is compactly supported. We verify the inequality (cid:107) ( X (cid:98) ⊗ id +76d (cid:98) ⊗ C ) A − g (cid:98) ⊗ φ (cid:107) < (cid:15) . Since the grading homomorphism on S exchanges ρ x n and ρ − x n , ( X (cid:98) ⊗ id + id (cid:98) ⊗ C ) A − g (cid:98) ⊗ φ = Xg (0) ρ (cid:98) ⊗ ψ + g (0) ρ (cid:98) ⊗ Cψ + (cid:88) n> (cid:18) Xρ x n (cid:98) ⊗ g ( x n ) x n + g ( − x n ) Cx n + C φ + ρ − x n (cid:98) ⊗ g ( x n ) x n C + g ( − x n ) C x n + C φ (cid:19) + (cid:88) n> (cid:18) Xρ − x n (cid:98) ⊗ − g ( − x n ) x n + g ( x n ) Cx n + C φ + ρ x n (cid:98) ⊗ − g ( − x n ) x n C + g ( x n ) C x n + C φ (cid:19) − (cid:88) n ∈ Z ρ x n g (cid:98) ⊗ φ = Xg (0) ρ (cid:98) ⊗ ψ + g (0) ρ (cid:98) ⊗ Cψ − ρ g (cid:98) ⊗ φ + (cid:88) n> (cid:18) Xρ x n (cid:98) ⊗ g ( − x n ) Cx n + C φ + ρ x n (cid:98) ⊗ − g ( − x n ) x n Cx n + C φ + ρ − x n (cid:98) ⊗ g ( x n ) x n Cx n + C φ + Xρ − x n (cid:98) ⊗ g ( x n ) Cx n + C φ (cid:19) + (cid:88) n> (cid:18) Xρ x n (cid:98) ⊗ g ( x n ) x n x n + C φ + ρ x n (cid:98) ⊗ g ( x n ) C x n + C φ − ρ x n g (cid:98) ⊗ φ (cid:19) + (cid:88) n> (cid:18) Xρ − x n (cid:98) ⊗ − g ( − x n ) x n x n + C φ + ρ − x n (cid:98) ⊗ g ( − x n ) C x n + C φ − ρ − x n g (cid:98) ⊗ φ (cid:19) = Xg (0) ρ (cid:98) ⊗ ψ + g (0) ρ (cid:98) ⊗ φ − ρ g (cid:98) ⊗ φ + (cid:88) n> (cid:18) ( Xρ x n − x n ρ x n ) (cid:98) ⊗ g ( − x n ) Cx n + C φ + ( x n ρ − x n + Xρ − x n ) (cid:98) ⊗ g ( x n ) Cx n + C φ (cid:19) + (cid:88) n> (cid:18) ( Xρ x n − x n ρ x n ) (cid:98) ⊗ g ( x n ) x n x n + C φ + g ( x n ) ρ x n (cid:98) ⊗ φ − ρ x n g (cid:98) ⊗ φ (cid:19) + (cid:88) n> (cid:18) ( Xρ − x n + x n ρ x n ) (cid:98) ⊗ − g ( − x n ) x n x n + C φ + g ( − x n ) ρ − x n (cid:98) ⊗ φ − ρ − x n g (cid:98) ⊗ φ (cid:19) . The norm of the first line is less than (cid:15)/ (cid:107) g (0) Xρ (cid:98) ⊗ ψ (cid:107) < | g (0) | · ∆ · (cid:107) ψ (cid:107) ≤ (cid:15)/
16 and (cid:107) g (0) ρ − ρ g (cid:107) ≤ (cid:15)/ (cid:15)/ (cid:107) Xρ ± x n ∓ x n ρ ± x n (cid:107) < ∆ < (cid:15)/ | g ( x n ) | ≤ (cid:107) Cx n + C φ (cid:107) = (cid:107) C x n + C ψ (cid:107) ≤ n ∈ Z (cid:54) =0 . Thanks to the property (3 ρ ),we notice the norm of the sum is less than (cid:15)/ (cid:15)/ (cid:107) g ( x n ) ρ x n (cid:98) ⊗ φ − ρ x n g (cid:98) ⊗ φ (cid:107) = (cid:107) g ( x n ) ρ x n − ρ x n g (cid:107) · (cid:107) φ (cid:107) < (cid:15)/ , and (cid:13)(cid:13)(cid:13)(cid:13) x n x n + C φ (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) x n Cx n + C ψ (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) C/x n C/x n ) ψ (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:107) ψ (cid:107) ≤ , where C/x n C/x n ) is obtained by the operator calculus F ( C/x n ) for F ( t ) = t t . The norm of thesum is less than (cid:15)/ C l = id (cid:98) ⊗ (cid:110)(cid:80) ∞ n =1 n − l [2 x n (cid:98) ⊗ c ( e n ) + 2 y n (cid:98) ⊗ c ( f n )] (cid:111) . Thanksto the above lemma, we can define an operator C l { X (cid:98) ⊗ id + id (cid:98) ⊗ C } − on the range of X (cid:98) ⊗ id + id (cid:98) ⊗ C which is dense in S (cid:98) ⊗H . It extends to a bounded operator on S (cid:98) ⊗H by the following lemma. Lemma 5.31. C l { X (cid:98) ⊗ id + id (cid:98) ⊗ C } − is bounded on ran (cid:0) X (cid:98) ⊗ id + id (cid:98) ⊗ C (cid:1) . roof. It suffices to prove that x n (cid:98) ⊗ c ( e n ) { X (cid:98) ⊗ id + id (cid:98) ⊗ C } − is bounded and its norm is at most 1.For any g (cid:98) ⊗ φ ∈ S (cid:98) ⊗H such that Xg ∈ S and Cφ ∈ H , we have (cid:107){ id (cid:98) ⊗ x n (cid:98) ⊗ c ( e n ) } ( X (cid:98) ⊗ id + id (cid:98) ⊗ C ) − ( X (cid:98) ⊗ id + id (cid:98) ⊗ C )( g (cid:98) ⊗ φ ) (cid:107) = (cid:107){ id (cid:98) ⊗ x n (cid:98) ⊗ c ( e n ) } ( g (cid:98) ⊗ φ ) (cid:107) = (cid:107) (cid:0) { id (cid:98) ⊗ x n (cid:98) ⊗ c ( e n ) } ( g (cid:98) ⊗ φ ) (cid:12)(cid:12) { id (cid:98) ⊗ x n (cid:98) ⊗ c ( e n ) } ( g (cid:98) ⊗ φ ) (cid:1) S (cid:107) S = (cid:107) g ∗ g (cid:107) (cid:0) x n (cid:98) ⊗ c ( e n )( φ ) (cid:12)(cid:12) x n (cid:98) ⊗ c ( e n )( φ ) (cid:1) H = (cid:107) g ∗ g (cid:107) (cid:0) x n (cid:98) ⊗ id( φ ) (cid:12)(cid:12) φ (cid:1) H . On the other hand, (cid:107) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C )( g (cid:98) ⊗ φ ) (cid:107) = (cid:107) (cid:0) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C )( g (cid:98) ⊗ φ ) (cid:12)(cid:12) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C )( g (cid:98) ⊗ φ ) (cid:1) S (cid:107) = (cid:107) ( Xg ) ∗ Xg ( φ | φ ) + g ∗ g ( Cφ | Cφ ) (cid:107)≥ (cid:107) g ∗ g (cid:107) ( Cφ | Cφ ) , where we have used the definition of the norm of S to prove the last inequality: The norm is definedby the maximum value, and hence the norm of the sum of two non-negative elements is not lessthan the norm of each summand. Noticing that( Cφ | Cφ ) = (cid:88) m ( (cid:0) x m (cid:98) ⊗ id( φ ) (cid:12)(cid:12) φ (cid:1) + (cid:0) y m (cid:98) ⊗ id( φ ) (cid:12)(cid:12) φ (cid:1) ) ≥ (cid:0) x n (cid:98) ⊗ id( φ ) (cid:12)(cid:12) φ (cid:1) H , we have the inequality (cid:107){ id (cid:98) ⊗ x n (cid:98) ⊗ c ∗ ( e n ) }{ X (cid:98) ⊗ id + id (cid:98) ⊗ C } − ( X (cid:98) ⊗ id + id (cid:98) ⊗ C )( g (cid:98) ⊗ φ ) (cid:107) ≤ (cid:107) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C )( g (cid:98) ⊗ φ ) (cid:107) . Since the subspace spanned by (cid:8) g (cid:98) ⊗ φ ∈ S (cid:98) ⊗H (cid:12)(cid:12) such that Xg ∈ S and Cφ ∈ H (cid:9) is dense, we findthat C l { X (cid:98) ⊗ id + id (cid:98) ⊗ C } − is bounded on ran (cid:0) X (cid:98) ⊗ id + id (cid:98) ⊗ C (cid:1) .We have defined π in an abstract and algebraic way. We need the following analytic andquantitative property for even elements of S . The even part of a Z -graded algebra A is denotedby A . The following proof reminds us of the chain rule. Lemma 5.32. If g ∈ S is the C -limit of a sequence of ( S fin ) , π ( β ∞ ( g )) preserves dom(id (cid:98) ⊗ (cid:19) ∂ ) and the commutator with id (cid:98) ⊗ (cid:19) ∂ is bounded.Proof. As a preliminary, we study the commutator [ π ( a ) , id (cid:98) ⊗ (cid:19) ∂ ] more, for a ∈ A HKT ( U L k ) fin . Anelement f of ( S fin ) can be written as f ( t ) = p ( f ev ( t )) = p (cid:16) t (cid:17) for some polynomials p . This isbecause f is given by a linear combination of f α od · f β ev for β ∈ N and α ∈ N , and f = f ev − f . Letus compute the commutator [id (cid:98) ⊗ (cid:19) ∂, π ( β ∞ ( f ))]. First, we compute it for the monomial p ( X ) = X m .By a standard algebraic technique on commutators, (cid:2) id (cid:98) ⊗ (cid:1) ∂, π ( β ∞ ( p ( f ev ))) (cid:3) = m − (cid:88) α =0 (cid:18)
11 + X (cid:98) ⊗ id + id (cid:98) ⊗ C (cid:19) α +1 (cid:32) − (cid:88) n id (cid:98) ⊗ n − l/ (cid:0) x n (cid:98) ⊗ c ( e n ) + y n (cid:98) ⊗ c ( f n ) (cid:1)(cid:33) (cid:18)
11 + X (cid:98) ⊗ id + id (cid:98) ⊗ C (cid:19) m − α = (cid:0) C l ( X (cid:98) ⊗ id + id (cid:98) ⊗ C ) − (cid:1) · ( − m )( X (cid:98) ⊗ id + id (cid:98) ⊗ C ) (cid:18)
11 + X (cid:98) ⊗ id + id (cid:98) ⊗ C (cid:19) m +1 . C l commutes with (cid:16) X (cid:98) ⊗ id+id (cid:98) ⊗ C (cid:17) α +1 . Since [ p ◦ f ev ] (cid:48) ( t ) = − mt (1 + t ) − m − , wenotice that (cid:2) id (cid:98) ⊗ (cid:19) ∂, π ( β ∞ ( f )) (cid:3) = C l { X (cid:98) ⊗ id + id (cid:98) ⊗ C } − f (cid:48) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C )= C l { X (cid:98) ⊗ id + id (cid:98) ⊗ C } − π ( β ∞ ( f (cid:48) )) . Therefore, for general polynomial p and f = p ◦ f ev , (cid:2) id (cid:98) ⊗ (cid:19) ∂, π ( β ∞ ( p ◦ f ev )) (cid:3) = C l { X (cid:98) ⊗ id + id (cid:98) ⊗ C } − π ( β ∞ ( f (cid:48) ))for f ∈ ( S fin ) .Let us prove the statement. By the assumption, there exists a sequence { g m } ⊆ ( S fin ) suchthat g m → g in the C -topology. We need to prove that π ( β ∞ ( g )) is a C -element. We have provedthat π ( β ∞ ( g m )) φ ∈ dom(id (cid:98) ⊗ (cid:19) ∂ ) for φ ∈ dom(id (cid:98) ⊗ (cid:19) ∂ ) in (1) of ( A ) of the proof of Lemma 5.27. Bythe above preliminary,id (cid:98) ⊗ (cid:19) ∂ ◦ π ( β ∞ ( g m )) φ = [id (cid:98) ⊗ (cid:19) ∂, π ( β ∞ ( g m ))] φ + π ( β ∞ ( g m ))id (cid:98) ⊗ (cid:19) ∂φ = C l { X (cid:98) ⊗ id + id (cid:98) ⊗ C } − π ( β ∞ ( g (cid:48) m )) φ + π ( β ∞ ( g m ))id (cid:98) ⊗ (cid:19) ∂φ → C l { X (cid:98) ⊗ id + id (cid:98) ⊗ C } − π ( β ∞ ( g (cid:48) )) φ + π ( β ∞ ( g ))id (cid:98) ⊗ (cid:19) ∂φ as m → ∞ . It means that π ( β ∞ ( g )) φ ∈ dom(id (cid:98) ⊗ (cid:19) ∂ ) and id (cid:98) ⊗ (cid:19) ∂π ( β ∞ ( g )) φ = lim m id (cid:98) ⊗ (cid:19) ∂π ( β ∞ ( g m )) φ .Moreover, [id (cid:98) ⊗ (cid:19) ∂, π ( β ∞ ( g ))] = C l { X (cid:98) ⊗ id+id (cid:98) ⊗ C } − π ( β ∞ ( g (cid:48) )) is a bounded operator. Thus π ( β ∞ ( g ))is a C -element.The above lemma gives an analytic condition to judge whether an element of A HKT ( U L k ) is of C or not. Next, we need to know the C -closure of β ∞ (( S fin ) ). Lemma 5.33.
An even function f ∈ C ( R ) such that f ∈ S and ( X + 1) f (cid:48) ∈ S , is the C -limit ofa sequence of ( S fin ) . In particular, any element of ι (( S ε ) ) ∩ C ( R ) is the C -limit of a sequenceof ( S fin ) .Proof. Since f = f ev − f , S fin = f ev C [ f ev ] ⊕ f od C [ f ev ]as a vector space. Since ( X + 1) f (cid:48) ∈ S , there is a sequence of polynomials { p n } such that { f od p n ( f ev ) } uniformly converges to ( X + 1) f (cid:48) . Thus, for any (cid:15) , there exists n such that n ≥ n implies that | ( t + 1) f (cid:48) ( t ) − f od ( t ) p n ( f ev ( t )) | < (cid:15)/π on R , and hence (cid:12)(cid:12)(cid:12)(cid:12) f (cid:48) ( t ) − t + 1 f od ( t ) p n ( f ev ( t )) (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15)π ( t + 1) . It means that (cid:12)(cid:12)(cid:12)(cid:12) f ( t ) − (cid:90) t −∞ s + 1 f od ( s ) p n ( f ev ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t −∞ (cid:18) f (cid:48) ( s ) − s + 1 f od ( s ) p n ( f ev ( s )) (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) t −∞ (cid:12)(cid:12)(cid:12)(cid:12) f (cid:48) ( s ) − s + 1 f od ( s ) p n ( f ev ( s )) (cid:12)(cid:12)(cid:12)(cid:12) ds< (cid:90) t −∞ (cid:15)π ( s + 1) ds = (cid:15)π (cid:16) arctan( t ) + π (cid:17) < (cid:15). f α ev ] (cid:48) = − αf od [ f ev ( t )] α , we have (cid:90) t −∞ s + 1 f od ( s )( f ev ( s )) β ds = − β + 1) [ f ev ( t )] β +1 . Hence, the function t (cid:55)→ (cid:82) t −∞ s +1 f od ( s ) p n ( f ev ( s )) ds belongs to S fin . This sequence converges to f in the C -topology. Lemma 5.34.
Any element of ( S ε ) ∩ C ( R ) is the C -limit of a sequence of f od ( S fin ) .Proof. Let f ∈ ( S ε ) ∩ C ( R ). Then, t − f ( t ) is defined on R and of C (the value of this function at 0is defined by f (cid:48) (0)). Hence, f − · f ( t ) := t t f ( t ) is of C . Note that it is an even element. Moreover,( X + 1)[ f − · f ] (cid:48) ∈ S ε ⊆ S . Thanks to the previous lemma, we have a sequence { f n } ⊆ ( S fin ) converging to f − · f in the C -topology. Then, the sequence { f od f n } converges to f od · f − · f = f in the C -topology. Proof of Definition-Proposition 5.29 : The necessary change from the proof of Lemma 5.27 isjust the following:
The set of C -elements of A ( U L k ) is dense. Combining Lemma 5.30–5.34, wenotice that all the elements of ∪ N β ∞ N ([ S ε ∩ C ( R )] (cid:98) ⊗ alg Cl τ, S ( U L k ,N )) are of C . The other conditionsare obvious. (cid:3) Let us define the substitute for the index element on the whole manifold M L k .Recall the isomorphism A ( M L k ) ∼ = A ( U L k ) (cid:98) ⊗ Cl τ ( (cid:102) M ). We prepare the index element for the (cid:102) M -direction. Let L| (cid:102) M be the restriction of L to (cid:102) M , which is a τ -twisted T × Π T -equivariant line bundle.It admits a ( T × Π T ) τ -invariant connection, by the averaging procedure using a cut-off function withrespect to the T × Π T -action on (cid:102) M . We have assumed that ( S M , γ M ) is a T -equivariant Spinorbundle over M . We define ( S (cid:102) M , γ (cid:102) M ) by the lift of this bundle to (cid:102) M , and similarly for ( S ∗ (cid:102) M , γ ∗ (cid:102) M ).Then, we define a ( T × Π T ) τ -equivariant Dirac operator D (cid:102) M by D (cid:102) M := (cid:88) n c (cid:102) M ( v n ) ◦ ∇ L| (cid:102) M (cid:98) ⊗ S ∗ (cid:102) M (cid:98) ⊗ S (cid:102) M v n , where c (cid:102) M ( v ) := √ (cid:16) id (cid:98) ⊗ γ (cid:102) M ( v ) − √− γ ∗ (cid:102) M ( v ) (cid:98) ⊗ id (cid:17) and { v n } is an orthonormal base of the tangentspace. Moreover, H := L ( (cid:102) M , L| (cid:102) M (cid:98) ⊗ S ∗ (cid:102) M (cid:98) ⊗ S (cid:102) M ) admits a ∗ -representation µ of Cl τ ( (cid:102) M ) given by theClifford multiplication c ∗ (cid:102) M ( v ) := √− √ (cid:16) id (cid:98) ⊗ γ (cid:102) M ( v ) + √− γ ∗ (cid:102) M ( v ) (cid:98) ⊗ id (cid:17) for v ∈ T (cid:102) M . We denote the KK -element corresponding to ( H, µ, D (cid:102) M ) by [ (cid:103) D (cid:102) M ] ∈ KK τT × Π T ( Cl τ ( (cid:102) M ) , C ) (the reformulated indexelement for even-dimensional Spin c -manifold).We define D := (cid:19) ∂ (cid:98) ⊗ id + id (cid:98) ⊗ D (cid:102) M on H (cid:98) ⊗ H . Definition 5.35.
The index element [ (cid:101) D ] ∈ KK τLT L m ( A ( M L k ) , S ε ) is the corresponding KK -element to the unbounded τ -twisted LT L m -equivariant Kasparov ( A ( M L k ) , S ε )-module (cid:0) S ε (cid:98) ⊗H (cid:98) ⊗ H, π (cid:98) ⊗ µ, id (cid:98) ⊗D (cid:1) . l and m . Although the conditions that l > m ≥ l + k are the keys of our quantitative arguments, we hope that our object isindependent of them. As we expect, at the KK τLT -theory level, the index element is independent ofthem.We have fixed l and m so far. From now on, we compare two Kasparov modules defined usingtwo different parameters ( l, m ) and ( l (cid:48) , m (cid:48) ). In order to distinguish them, (cid:63) l denotes an object (cid:63) which is defined with ( l, m ), for example, H l , (cid:19) ∂ l , D l , π l , vac l , dL (cid:48) l ( z n ) and so on. The objects definedindependently l is denoted without the subscript l , for example S U and z n depend only on k . Proposition 5.36.
For l, l (cid:48) > , the equivariant unbounded Kasparov module constructed with ( l, m ) is homotopic to that constructed with ( l (cid:48) , m (cid:48) ) , as LT L m (cid:48)(cid:48) -equivariant Kasparov modules for m (cid:48)(cid:48) ≥ max { m, m (cid:48) } , that is to say, i ∗ m (cid:48)(cid:48) ,m [ (cid:101) D l ] = i ∗ m (cid:48)(cid:48) ,m (cid:48) [ (cid:101) D l (cid:48) ] in KK τLT L m (cid:48)(cid:48) ( A ( M L k ) , S ε ) .Remark . By this result, the corresponding element of KK τLT ( A ( M L k ) , S ε ) to [ (cid:101) D l ] is indepen-dent of l . The resulting element is denoted by [ (cid:101) D ]. Proof.
It suffices to deal with the U L k -part. For −→ α = ( α , α , · · · ), −→ β = ( β , β , · · · ), −→ ξ =( ξ , ξ , · · · , ξ M ), and −→ η = ( η , η , · · · , η N ) such that α i , β i ∈ Z ≥ , α i = β i = 0 except for finitelymany i ’s, ξ i , η i ∈ N , N, M ∈ Z ≥ , 0 < ξ < ξ < · · · < ξ M , and 0 < η < η < · · · < η N , we put (cid:101) φ −→ α , −→ β , −→ ξ , −→ η ( s ):= (cid:16) dR (cid:48) (1 − s ) l + sl (cid:48) ( z ) α dL (cid:48) (1 − s ) l + sl (cid:48) ( z ) β · · · (cid:17) vac (1 − s ) l + sl (cid:48) (cid:98) ⊗ [ z ξ ∧ · · · ∧ z ξ M ] (cid:98) ⊗ [ z η ∧ · · · ∧ z η N ] ∈ S ε (cid:98) ⊗H (1 − s ) l + sl (cid:48) . We construct a homotopy, that is to say, an U L m (cid:48)(cid:48) -equivariant Kasparov ( A ( U L k ) , S ε (cid:98) ⊗ C ( I ))-module( E, σ, (cid:26)(cid:26) D ). The U L m (cid:48)(cid:48) -equivariant Hilbert S ε (cid:98) ⊗ C ( I )-module E is given as follows: • Let E fin be the pre-Hilbert S ε (cid:98) ⊗ C ( I )-module given by the C ( I )-linear span of the (cid:96) s ∈ [0 , S ε (cid:98) ⊗H (1 − s ) l + sl (cid:48) -valued section s (cid:55)→ f (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( s )for f ∈ S ε and multi-indices −→ α , −→ β , −→ ξ , −→ η satisfying the above condition. • The pre-Hilbert module structure is given by the following formulas: For ψ, ψ , ψ ∈ E fin and f (cid:98) ⊗ F ∈ S ε (cid:98) ⊗ C ( I ), [ ψ · f (cid:98) ⊗ F ]( s ) := [ ψ ( s ) · f ] · F ( s ) ∈ S ε (cid:98) ⊗H (1 − s ) l + sl (cid:48) ( ψ | ψ ) E ( s ) := ( ψ ( s ) | ψ ( s )) S ε (cid:98) ⊗H (1 − s ) l + sl (cid:48) . • The Hilbert module E is the completion of E fin with respect to the above inner product. • The U τL m (cid:48)(cid:48) -action is given by the pointwise action { ( g, z ) · ( ψ ) } ( s ) := L ( g,z ) ( ψ ( s )).We need to prove that the inner product of two elements ψ , ψ ∈ E fin is indeed an element of S ε (cid:98) ⊗ C ( I ). For this aim, it suffices to prove that the function s (cid:55)→ (cid:16) f (cid:98) ⊗ (cid:101) φ −→ α , −→ β , −→ ξ , −→ η ( s ) (cid:12)(cid:12)(cid:12) f (cid:48) (cid:98) ⊗ (cid:101) φ −→ α (cid:48) , −→ β (cid:48) , −→ ξ (cid:48) , −→ η (cid:48) ( s ) (cid:17)
81s continuous. If −→ α = −→ α (cid:48) , −→ β = −→ β (cid:48) , −→ ξ = −→ ξ (cid:48) , and −→ η = −→ η (cid:48) , that function is given by s (cid:55)→ f ∗ f (cid:48) −→ α ! −→ β ! ∞ (cid:89) m =1 (2 m (1 − s ) l + sl (cid:48) ) α m + β m , and if not, the function is identically zero. Both functions are continuous (note that α m = 0 and β m = 0 except for finitely many m ’s, and hence the above infinite product is in fact a finite productof continuous functions).We define σ : A ( U L k ) → L S ε (cid:98) ⊗ C ( I ) ( E ) by σ ( a )( ψ )( s ) := π (1 − s ) l + sl (cid:48) ( a )( ψ ( s )), and we define (cid:26)(cid:26) D by (cid:26)(cid:26) D ( ψ )( s ) := (cid:19) ∂ (1 − s ) l + sl (cid:48) [ ψ ( s )]. Then, the triple ( E, σ, (cid:26)(cid:26) D ) is obviously a U L m (cid:48)(cid:48) -equivariant unboundedKasparov ( A ( U L k ) , S ε (cid:98) ⊗ C ( I ))-module.Let us prove that the evaluation of ( E, σ, (cid:26)(cid:26) D ) at s = 0 gives ( S ε (cid:98) ⊗H l , π l , D l ) and similarly for s = 1. We prove it for arbitrary s ∈ I . What we need to prove is that ev s ( E ) is isomorphic to S ε (cid:98) ⊗H (1 − s ) l + sl (cid:48) . For this aim, it suffices to prove that the evaluation at s is surjective. It suffices tofind an isometric embedding as a Banach space j s : S ε (cid:98) ⊗H (cid:44) → E satisfying j s ( v )( s ) = v . We defineit as follows: For f ∈ S ε and ψ −→ α , −→ β , −→ ξ , −→ η ( s ) ∈ H (1 − s ) l + sl (cid:48) , we define j s ( f (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( s )) by t (cid:55)→ (cid:115)(cid:89) m (2 m ) [( t − s ) l +( s − t ) l (cid:48) ]( α m + β m ) f (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( t ) . Since the function appearing as the coefficient is continuous, the above function belongs to E . Inthe remainder of this proof, we prove that j s is isometric.If −→ α = −→ α (cid:48) , −→ β = −→ β (cid:48) , −→ ξ = −→ ξ (cid:48) , and −→ η = −→ η (cid:48) , the inner product of j s ( f (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( s )) and j s ( f (cid:48) (cid:98) ⊗ ψ −→ α (cid:48) , −→ β (cid:48) , −→ ξ (cid:48) , −→ η (cid:48) ( s )) is given by the function on t (cid:16) j s ( f (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( s )) (cid:12)(cid:12)(cid:12) j s ( f (cid:48) (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( s )) (cid:17) S ε (cid:98) ⊗ C ( I ) ( t )= f ∗ f (cid:48) −→ α ! −→ β ! (cid:89) m (2 m ) [( t − s ) l +( s − t ) l (cid:48) ]( α m + β m ) (cid:89) m (2 m ) [(1 − t ) l + tl (cid:48) ]( α m + β m ) = f ∗ f (cid:48) −→ α ! −→ β ! (cid:89) m (2 m ) [(1 − s ) l + sl (cid:48) ]( α m + β m ) = (cid:16) f (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( s ) (cid:12)(cid:12)(cid:12) f (cid:48) (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( s ) (cid:17) S ε . Taking the C ∗ -norm of both sides, we find that j s is isometric. Consequently, j s extends to S ε (cid:98) ⊗H (1 − s ) l + sl (cid:48) , and it satisfies ev s ( j s ( v )) = v for all v ∈ S ε (cid:98) ⊗H (1 − s ) l + sl (cid:48) .If not, both sides are zero, and we finish the proof.In the following, we again fix l and m , and we do not study this kind of problems. The C ∗ -norm of the right hand side is the maximum norm of the function u (cid:55)→ f ( u ) f (cid:48) ( u ) (cid:16) ψ −→ α , −→ β , −→ ξ , −→ η ( s ) (cid:12)(cid:12)(cid:12) ψ −→ α , −→ β , −→ ξ , −→ η ( s ) (cid:17) , and that of the left hand side is the maximum norm of the function( u, t ) (cid:55)→ (cid:16) j s ( f (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( s )) (cid:12)(cid:12)(cid:12) j s ( f (cid:48) (cid:98) ⊗ ψ −→ α , −→ β , −→ ξ , −→ η ( s )) (cid:17) S ε (cid:98) ⊗ C ( I ) ( t, u ), where u is the variable for S ε and t is that for C ( I ). Since it is constant with respect to t , j s is isometric. .3 The Poincar´e duality homomorphism for proper LT -spaces The task of this subsection is to apply the Poincar´e duality homomorphism to the index elementconstructed in the previous subsection.We consider the restriction of the local Bott element [ (cid:94) Θ M L k , ] to M L m and we denote it by thesame symbol: [ (cid:94) Θ M L k , ] ∈ R KK LT L m ( M L m ; S ε (cid:98) ⊗ C ( M L m ) , A ( M L k ) (cid:98) ⊗ C ( M L m )) . Let [ L ] ∈ R KK τLT L m ( M L m ; C ( M L m ) , C ( M L m )) be the R KK -element defined by (cid:16) {L x } x ∈M L m , { } x ∈M L m , { } x ∈M L m (cid:17) . The goal of this subsection is to prove that PD([ (cid:101) D ]) = σ S ε ([ L ]). For this aim, it is convenient todivide the problem into the U L m -part and the (cid:102) M -part. Thus, we rewrite the local Bott element bythe tensor product of those of U L m and (cid:102) M . We begin with a technical lemma. Lemma 5.38.
We define the following ∗ -homomorphisms: ∆ : S → S (cid:98) ⊗S by ∆( f ) := f ( X (cid:98) ⊗ id +id (cid:98) ⊗ X ) and ev : S → C by ev ( f ) = f (0) . The same formulas define homomorphisms on S ε . (1) (id (cid:98) ⊗ ev ) ◦ ∆ = id . The same formula holds for S ε . (2) Let A and B be Z -graded C ∗ -algebras equipped with odd unbounded multipliers with compactresolvent D A and D B . Suppose that ∆( f ) = (cid:80) f i (cid:98) ⊗ f i . Then, we have f ( D A (cid:98) ⊗ id + id (cid:98) ⊗ D B ) = (cid:80) i f i ( D A ) (cid:98) ⊗ f i ( D B ) on A (cid:98) ⊗ B ,Proof. (1) It is mentioned in [HG, P. 14] as an exercise. For S ε , one can prove it by considering thecommutative diagram S ε ∆ −−−−→ S ε (cid:98) ⊗S ει (cid:121) (cid:121) ι (cid:98) ⊗ ι S ∆ −−−−→ S (cid:98) ⊗S . (2) The homomorphism S (cid:51) f (cid:55)→ f ( D A ) ∈ A is denoted by β A , and similarly for B . With thesenotations, the statement can be written as β A (cid:98) ⊗ β B ◦ ∆ = β A (cid:98) ⊗ B , where D A (cid:98) ⊗ B := D A (cid:98) ⊗ id + id (cid:98) ⊗ D B .Since both sides are ∗ -homomorphisms, it suffices to prove it for generators. We choose f e ( t ) := e − t and f o ( t ) = te − t as generators. Thanks to∆( f e ) = f e (cid:98) ⊗ f e and ∆( f o ) = f o (cid:98) ⊗ f e + f e (cid:98) ⊗ f o ,and thanks to computations on functional calculus in [HKT, Appendix A.4], we have β A (cid:98) ⊗ β B ◦ ∆( f e ) = f e ( D A ) (cid:98) ⊗ f e ( D B ) = β A (cid:98) ⊗ B ( f e ); β A (cid:98) ⊗ β B ◦ ∆( f o ) = f o ( D A ) (cid:98) ⊗ f e ( D B ) + f e ( D A ) (cid:98) ⊗ f o ( D B ) = β A (cid:98) ⊗ B ( f o ) . The local Bott homomorphism for a Hilbert manifold X at x is denoted by β Xx when it isnecessary to emphasize the manifold to consider it.83 emma 5.39. Let [ (cid:94) Θ M L k , (cid:48) ] be the M L m (cid:111) LT L m -equivariant Kasparov ( S ε (cid:98) ⊗ C ( M L m ) , A ( M L k ) (cid:98) ⊗ C ( M L m )) -module defined by the exterior tensor product of the following Kasparov modules: {A ( U L k ) } g ∈ U L m , (cid:26) β U L k g (cid:27) g ∈ U L m , { } g ∈ U L m and (cid:16) { Cl τ ( V x ) } x ∈ (cid:102) M , { } x ∈ (cid:102) M , (cid:8) ε − Θ x (cid:9) x ∈ (cid:102) M (cid:17) , where V x is the ε -neighborhood of x in (cid:102) M . Then, [ (cid:94) Θ M L k , (cid:48) ] is homotopic to [ (cid:94) Θ M L k , ] .Proof. We define [ (cid:94) Θ M L k , (cid:48)(cid:48) ] ∈ R KK LT L m ( M LT L m ; S ε (cid:98) ⊗S ε (cid:98) ⊗ C ( M L m ) , A ( U L m ) (cid:98) ⊗A ( (cid:102) M ) (cid:98) ⊗ C ( M L m ))by the exterior tensor product of the following Kasparov modules: {A ( U L k ) } g ∈ U L k , (cid:26) β U L k g (cid:27) g ∈ U L k , { } g ∈ U L k and (cid:16) {S ε (cid:98) ⊗ Cl τ ( V x ) } x ∈ (cid:102) M , { S ε (cid:98) ⊗ id } x ∈ (cid:102) M , (cid:8) S ε (cid:98) ⊗ ε − Θ x (cid:9) x ∈ (cid:102) M (cid:17) , where 1 S ε (cid:98) ⊗ id( f )( g (cid:98) ⊗ φ ) := f g (cid:98) ⊗ φ for f, g ∈ S ε and φ ∈ Cl τ ( (cid:102) M ).We define ev (cid:98) ⊗ id Cl τ ( (cid:102) M ) : A ( (cid:102) M ) (cid:51) f (cid:98) ⊗ h (cid:55)→ f (0) h ∈ Cl τ ( (cid:102) M ). It gives a KK -element [ev (cid:98) ⊗ id] ∈ KK T × Π T ( A ( (cid:102) M ) , Cl τ ( (cid:102) M )). It suffices to prove the following formulas in the R KK -group:(1) [ (cid:94) Θ M L k , (cid:48) ] = [∆] (cid:98) ⊗ [ (cid:94) Θ M L k , (cid:48)(cid:48) ] (cid:98) ⊗ [ev (cid:98) ⊗ id Cl τ ( (cid:102) M ) ].(2) [ (cid:94) Θ M L k , ] = [∆] (cid:98) ⊗ [ (cid:94) Θ M L k , (cid:48)(cid:48) ] (cid:98) ⊗ [ev (cid:98) ⊗ id Cl τ ( (cid:102) M ) ].(1) By Lemma 3.18, [ (cid:93) Θ (cid:102) M, ] (cid:98) ⊗ [ev (cid:98) ⊗ id] = [ev ] (cid:98) ⊗ C [Θ (cid:102) M, ] . Thus, [ (cid:94) Θ M L k , (cid:48)(cid:48) ] (cid:98) ⊗ [ev (cid:98) ⊗ id Cl τ ( (cid:102) M ) ] is givenby[ (cid:94) Θ M L k , (cid:48)(cid:48) ] (cid:98) ⊗ [ev (cid:98) ⊗ id Cl τ ( (cid:102) M ) ] = [ (cid:94) Θ U L k , ] (cid:98) ⊗ C (cid:16) [ev ] (cid:98) ⊗ C [Θ (cid:102) M, ] (cid:17) = {A ( U L k ) (cid:98) ⊗ Cl τ ( V x ) } ( g,x ) ∈ U L k × (cid:102) M , (cid:26) β U L k g (cid:98) ⊗ ev (cid:98) ⊗ id Cl τ ( (cid:102) M ) (cid:27) ( g,x ) ∈ U L k × (cid:102) M , { id (cid:98) ⊗ ε − Θ x } ( g,x ) ∈ U L k × (cid:102) M . Let us compute the triple Kasparov product. First, we notice that β U L k g (cid:98) ⊗ ev (cid:98) ⊗ id Cl τ ( (cid:102) M ) ◦ ∆( f )is given by β U L k g ( f ) (cid:98) ⊗ id Cl τ ( (cid:102) M ) . In fact, for ∆( f ) = (cid:80) i f i (cid:98) ⊗ f i , β U L k g (cid:98) ⊗ ev (cid:98) ⊗ id Cl τ ( (cid:102) M ) ◦ ∆( f ) = (cid:88) i β U L k g ( f i ) (cid:98) ⊗ ev ( f i )id Cl τ ( (cid:102) M ) = β U L k g (cid:32)(cid:88) i f i ev ( f i ) (cid:33) (cid:98) ⊗ id Cl τ ( (cid:102) M ) = β U L k g ( f ) (cid:98) ⊗ id Cl τ ( (cid:102) M ) , (cid:98) ⊗ ev ) ◦ ∆ = id S ε at the second equality. Therefore, the tripleKasparov product is given by [ (cid:94) Θ M L k , (cid:48) ].(2) By the proof of Proposition 3.19, (cid:16) {S ε (cid:98) ⊗ Cl τ ( V x ) } x ∈ (cid:102) M , { S ε (cid:98) ⊗ id } x ∈ (cid:102) M , (cid:8) S ε (cid:98) ⊗ ε − Θ x (cid:9) x ∈ (cid:102) M (cid:17) ishomotopic to (cid:16) {A ( V x ) } x ∈ (cid:102) M , { β (cid:102) Mx } x ∈ (cid:102) M , { } x ∈ (cid:102) M (cid:17) . Thus, the Kasparov product (cid:16) {S ε (cid:98) ⊗ Cl τ ( V x ) } x ∈ (cid:102) M , { S ε (cid:98) ⊗ id } x ∈ (cid:102) M , (cid:8) S ε (cid:98) ⊗ ε − Θ x (cid:9) x ∈ (cid:102) M (cid:17) (cid:98) ⊗ [ev (cid:98) ⊗ id]is represented by (cid:16) { Cl τ ( V x ) } x ∈ (cid:102) M , (cid:110) (ev (cid:98) ⊗ id) ◦ β (cid:102) Mx (cid:111) x ∈ (cid:102) M , { } x ∈ (cid:102) M (cid:17) . It is easy to see that (ev (cid:98) ⊗ id) ◦ β (cid:102) Mx : S ε → Cl τ ( V x ) is given by f (cid:55)→ [ y (cid:55)→ f ( C x ( y ))]. We denote itby β (cid:102) M, x .Thus, the triple Kasparov product [∆] (cid:98) ⊗ [ (cid:94) Θ M L k , (cid:48)(cid:48) ] (cid:98) ⊗ [ev (cid:98) ⊗ id] is given by (cid:110) A ( U L k ) (cid:98) ⊗ Cl τ ( V x ) (cid:111) ( g,x ) ∈ U L m × (cid:102) M , (cid:26)(cid:18) β U L k g (cid:98) ⊗ β (cid:102) M, x (cid:19) ◦ ∆ (cid:27) ( g,x ) ∈ U L m × (cid:102) M , { } ( g,x ) ∈ U L m × (cid:102) M . We would like to “apply” Lemma 5.38 (2) to β U L k g (cid:98) ⊗ β (cid:102) M, x ◦ ∆ : S ε → S ε (cid:98) ⊗S ε → A ( U L k ) (cid:98) ⊗ Cl τ ( V x ).For this aim, we extend this homomorphism to S by embedding Cl τ ( V x ) into another C ∗ -algebra.Let W x be the 2 ε -ball of x equipped with the new Riemannian metric ds given by (the old oneis denoted by ds ) ds ( y ) := ρ ( r ( x, y )) ds ( y ) , where ρ is a smooth function : [0 , ε ) → R > such that ρ (cid:48) ( s ) ≥ s ∈ [0 , ε ), ρ ( s ) = 1 on s ∈ [0 , ε ]and lim s → ε (cid:82) s (cid:112) ρ ( s ) ds = ∞ . Then, the new metric is complete, and the Clifford operator on W x with respect to the new metric is with compact resolvent. Thus, we can define a ∗ -homomorphism S → Cl τ ( W x ). Since the ε -ball at x in W x is isometric to V x , we have the following commutativediagram: S ε ∆ (cid:47) (cid:47) ι (cid:15) (cid:15) S ε (cid:98) ⊗S ε β UL kg (cid:98) ⊗ β (cid:102) M, x (cid:47) (cid:47) ι (cid:98) ⊗ ι (cid:15) (cid:15) A ( U L k ) (cid:98) ⊗ Cl τ ( V x ) ( ι (cid:98) ⊗ id) (cid:98) ⊗ ι (cid:15) (cid:15) S ∆ (cid:47) (cid:47) S (cid:98) ⊗S β UL kg (cid:98) ⊗ β Wx, x (cid:47) (cid:47) A HKT ( U L k ) (cid:98) ⊗ Cl τ ( W x ) , where each zero-extension is denoted by ι . Thanks to the following commutative diagram verified85y Lemma 5.38 (2) and the fact that C U L k g (cid:98) ⊗ id + id (cid:98) ⊗ C W x x = C M L k ( g,x ) on V x , we finish the proof: S ∆ (cid:47) (cid:47) id (cid:15) (cid:15) S (cid:98) ⊗S β UL kg (cid:98) ⊗ β Wx, x (cid:47) (cid:47) A HKT ( U L k ) (cid:98) ⊗ Cl τ ( W x ) ∼ = (cid:15) (cid:15) S β UL k × Wx ( g,x ) (cid:47) (cid:47) A HKT ( U L k × W x ) S ει (cid:79) (cid:79) id (cid:15) (cid:15) β M L k ( g,x ) (cid:47) (cid:47) A ( U L k × V x ) ( ι (cid:98) ⊗ id) (cid:98) ⊗ ι (cid:79) (cid:79) id (cid:98) ⊗ ι (cid:15) (cid:15) S ε β M L k ( g,x ) (cid:47) (cid:47) A ( M L k ) . Remark . Since the Kasparov module (cid:16) { Cl τ ( V x ) } x ∈ (cid:102) M , { } x ∈ (cid:102) M , (cid:8) ε − Θ x (cid:9) x ∈ (cid:102) M (cid:17) represents[Θ X, ] (cid:98) ⊗ σ C ( X ) ([ S ∗ ]) (cid:98) ⊗ σ X,C ( X ) (fgt[ S ]), we write the above result as[ (cid:94) Θ M L k , ] = [ (cid:94) Θ U L k , ] (cid:98) ⊗ C (cid:16) [Θ (cid:102) M, ] (cid:98) ⊗ σ C ( (cid:102) M ) ([ S ∗ ]) (cid:98) ⊗ σ (cid:102) M,C ( (cid:102) M ) (fgt[ S ]) (cid:17) . Note that the bounded transformation D (cid:55)→ D √ D commutes with the group action D (cid:55)→ g ( D ) = L ( g,z ) ◦ D ◦ L ( − g,z − ) . By Corollary 4.40, σ U L m , C ( U L m ) ([ (cid:101) (cid:19) ∂ ]) is represented by (cid:16) {S ε (cid:98) ⊗H} g ∈ U L m , { π } g ∈ U L m , { g ( (cid:19) ∂ ) } g ∈ U L m (cid:17) . With the preparation so far, we can prove the main theorem of this section.
Theorem 5.41.
PD([ (cid:101) D ]) = σ S ε ([ L ]) .Proof. Thanks to [ (cid:101) D ] = [ (cid:101) (cid:19) ∂ ] (cid:98) ⊗ C [ (cid:103) D (cid:102) M ] and Lemma 5.39, we can divide the problem into the U L m -partand the (cid:102) M -part. It suffices to check that[ (cid:94) Θ U L k , ] (cid:98) ⊗ A ( U L k ) [ (cid:101) (cid:19) ∂ ] = σ S ε (cid:16) [ L| U L k ] (cid:17) , and (cid:16) [Θ (cid:102) M, ] (cid:98) ⊗ σ C ( (cid:102) M ) ([ S ∗ ]) (cid:98) ⊗ σ (cid:102) M,C ( (cid:102) M ) (fgt[ S ]) (cid:17) (cid:98) ⊗ Cl τ ( (cid:102) M ) (cid:16) [ (cid:103) D (cid:102) M ] (cid:17) = [ L| (cid:102) M ] . For the second one, see Proposition 3.15 and Example 3.20.Let us prove the first one. Since we would like to discuss it in the unbounded picture, we rewritethe statement using A HKT ( U L k ). We introduce and recall several notations: • The KK -element corresponding to the Kasparov ( A HKT ( U L k ) , S )-module constructed in Lemma5.27 is denoted by [ (cid:101) (cid:19) ∂ HKT ] ∈ KK U L m ( A HKT ( U L k ) , S ) in this proof.86 The canonical embedding A ( U L k ) → A HKT ( U L k ) is denoted by ι (cid:98) ⊗ id. • The canonical embedding S ε → S is denoted by j , in order to distinguish it from ι .Then, it is obvious that [ (cid:101) (cid:19) ∂ ] (cid:98) ⊗ [ j ] = [ ι (cid:98) ⊗ id] (cid:98) ⊗ [ (cid:101) (cid:19) ∂ HKT ]. Since [ j ] is invertible in KK -group, we have[ (cid:101) (cid:19) ∂ ] = [ ι (cid:98) ⊗ id] (cid:98) ⊗ [ (cid:101) (cid:19) ∂ HKT ] (cid:98) ⊗ [ j ] − . Associated to it, we consider[ (cid:101) Θ U L k , , HKT ] := [ j ] − (cid:98) ⊗ [ (cid:94) Θ U L k , ] (cid:98) ⊗ [ ι (cid:98) ⊗ id] ∈ R KK U L m ( U L m ; S (cid:98) ⊗ C ( U L m ) , A HKT ( U L k ) (cid:98) ⊗ C ( U L m )) . Then, we have [ (cid:94) Θ U L k , ] (cid:98) ⊗ [ (cid:101) (cid:19) ∂ ] = [ j ] (cid:98) ⊗ [ (cid:101) Θ U L k , , HKT ] (cid:98) ⊗ [ (cid:101) (cid:19) ∂ HKT ] (cid:98) ⊗ [ j ] − . Thus, it suffices to prove that [ (cid:101) Θ U L k , , HKT ] (cid:98) ⊗ [ (cid:101) (cid:19) ∂ HKT ] = σ S ([ L| U L m ]).Note that [ (cid:101) Θ U L k , , HKT ] is represented by (cid:18) {A HKT ( U L k ) } g ∈ U L m , { β g } g ∈ U L m , { } g ∈ U L m (cid:19) . In fact, [ j ] (cid:98) ⊗ (cid:18) {A HKT ( U L k ) } g ∈ U L m , { β g } g ∈ U L m , { } g ∈ U L m (cid:19) is clearly [ (cid:94) Θ U L k , ] (cid:98) ⊗ [ ι (cid:98) ⊗ id]. Thus, theKasparov product [ (cid:101) Θ U L k , , HKT ] (cid:98) ⊗ [ (cid:101) (cid:19) ∂ HKT ] is given by the field of Kasparov modules (cid:16) {S (cid:98) ⊗H} g ∈ U L m , { π ◦ β g } g ∈ U L m , { g (id (cid:98) ⊗ (cid:19) ∂ ) } g ∈ U L m (cid:17) . We prove this R KK -element is σ S (cid:16) [ L| U L m ] (cid:17) . This is equivalent to the following by Lemma2.18: [ b ± ] (cid:98) ⊗ (cid:16) {S ε (cid:98) ⊗H} g ∈ U L m , { π ◦ β g } g ∈ U L m , { g (id (cid:98) ⊗ (cid:19) ∂ ) } g ∈ U L m (cid:17) (cid:98) ⊗ [ d ± ] = [ L| U L m ] , [ b ± ] (cid:98) ⊗ (cid:16) {S ε (cid:98) ⊗H} g ∈ U L m , { π ◦ β g } g ∈ U L m , { g (id (cid:98) ⊗ (cid:19) ∂ ) } g ∈ U L m (cid:17) (cid:98) ⊗ [ d ∓ ] = 0(double signs are in the same order).We prove only [ b + ] (cid:98) ⊗ ( · · · ) (cid:98) ⊗ [ d + ] = [ L| U L m ]. By a direct computation,( · · · ) (cid:98) ⊗ [ d + ] = (cid:16) { L ( R ) gr (cid:98) ⊗H} g ∈ U L m , { f (cid:55)→ f ( X (cid:98) ⊗ id + id (cid:98) ⊗ C g ) } g ∈ U L m , { (cid:15)d (cid:98) ⊗ id + id (cid:98) ⊗ g ( (cid:19) ∂ ) } g ∈ U L m (cid:17) , where Xφ ( t ) := tφ ( t ) for φ ∈ L ( R ) gr . Next, S (cid:98) ⊗ S (cid:8) L ( R ) gr (cid:98) ⊗H (cid:9) is isomorphic to L ( R ) gr (cid:98) ⊗H , by f (cid:98) ⊗ φ (cid:98) ⊗ ψ (cid:55)→ f ( X (cid:98) ⊗ id + id (cid:98) ⊗ C g ) φ (cid:98) ⊗ ψ . Under this isomorphism, X (cid:98) ⊗ S id L ( R ) gr (cid:98) ⊗H is given by X (cid:98) ⊗ id +id (cid:98) ⊗ C g . Noticing it, we find that the triple Kasparov product is given by (cid:18)(cid:8) L ( R ) gr (cid:98) ⊗H (cid:9) g ∈ U L m , { } g ∈ U L m , (cid:8) ( (cid:15)d + X ) (cid:98) ⊗ id + id (cid:98) ⊗ g ( (cid:19) ∂ ) (cid:9) g ∈ U L m (cid:19) . In fact, it is obvious that { ( (cid:15)d + X ) (cid:98) ⊗ id + id (cid:98) ⊗ g ( (cid:19) ∂ ) } is an ( (cid:15)d (cid:98) ⊗ id + id (cid:98) ⊗ g ( (cid:19) ∂ ))-connection for S .Moreover,[ X (cid:98) ⊗ id+id (cid:98) ⊗ C g , ( (cid:15)d + X ) (cid:98) ⊗ id+id (cid:98) ⊗ g ( (cid:19) ∂ )] = (2 X + (cid:15) ) (cid:98) ⊗ id+ g (cid:32) id (cid:98) ⊗ (cid:88) n n − l/ { c ( e n ) c ∗ ( e n ) + c ( f n ) c ∗ ( f n ) } (cid:33) . (cid:19) ∂ commutes with C , whichis because the vector x n J e n + y n J f n (the potential of the n -th summand of (cid:19) ∂ ) is orthogonal to x n e n + y n f n (the n -th summand of C ) at each point. This formal argument is justified as follows.The positivity condition to be a Kasparov product is, strictly speaking, the following: The quadraticform ψ (cid:98) ⊗ φ (cid:55)→ (cid:0) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C g )( ψ (cid:98) ⊗ φ ) (cid:12)(cid:12) id (cid:98) ⊗ g ( (cid:1) ∂ )( ψ (cid:98) ⊗ φ ) (cid:1) + (cid:0) id (cid:98) ⊗ g ( (cid:1) ∂ )( ψ (cid:98) ⊗ φ ) (cid:12)(cid:12) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C g )( ψ (cid:98) ⊗ φ ) (cid:1) is positive modulo bounded. Since C g = g ◦ C ◦ g − , g ( (cid:19) ∂ ) = g ◦ (cid:19) ∂ ◦ g − and the U L m -action isisometric, it is given by lim N →∞ (cid:8)(cid:0) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C N )( ψ (cid:98) ⊗ g ( φ )) (cid:12)(cid:12) id (cid:98) ⊗ (cid:1) ∂ ( ψ (cid:98) ⊗ g · φ ) (cid:1) + (cid:0) id (cid:98) ⊗ (cid:1) ∂ ( ψ (cid:98) ⊗ g · φ ) (cid:12)(cid:12) ( X (cid:98) ⊗ id + id (cid:98) ⊗ C N )( ψ (cid:98) ⊗ g · φ ) (cid:1)(cid:9) , thanks to the definition of C . In this expression, we can do the above formal argument.The Kasparov module (cid:18)(cid:8) L ( R ) gr (cid:98) ⊗H (cid:9) g ∈ U L m , { } g ∈ U L m , (cid:8) ( (cid:15)d + X ) (cid:98) ⊗ id + id (cid:98) ⊗ g ( (cid:19) ∂ ) (cid:9) g ∈ U L m (cid:19) is ho-motopic to (cid:18)(cid:8) ker( (cid:15)d + X ) (cid:98) ⊗ ker( g ( (cid:19) ∂ )) (cid:9) g ∈ U L m , { } g ∈ U L m , { } g ∈ U L m (cid:19) by the family version of the argument to prove that KK ( C , C ) ∼ = Z . Since ker( (cid:15)d + X ) (cid:98) ⊗ ker( g ( (cid:19) ∂ )) ∼ = C and the U τL m -action is at level 1, it is [ L| U L m ]. LT -spaces The aim of this section is to construct an infinite-dimensional version of the topological assemblymap for proper LT -spaces, and to compute it. The main result of this section is the following: The value of the topological assembly map at the value of the Poincar´e duality of the index element,coincides with the analytic index constructed in [T3] . This is an analogous result of Proposition3.15. In addition, as a concluding remark, we will explain what we should do after the presentpaper. LT -spaces We have explained the description of crossed products and the descent homomorphism for R KK τG -theory in terms of fields of Hilbert modules in Section 3.3. Imitating it, we define substitutes forthe descent homomorphisms for proper LT -spaces. For this aim, we need to introduce a substitutefor “ L ( U L k , L ⊗ q )” for each q ∈ Z . Definition 6.1. (1) For q ∈ Z , we define a new U τL m -action R at level − q on L ( U L k , L ) by R ( g,z ) φ ( x ) := z − q φ ( x + g ) τ ( g, x ) q for g ∈ U fin and φ ∈ L ( U L k , L ) fin . This representation extends to U τL m as a continuous homomor-phism. This − qτ -twisted U L m -representation space is denoted by L ( U L k , L ⊗ q ).(2) The − qτ -twisted T × Π T -action on L ( T × Π T , L ⊗ q ) induced by the right regular represen-tation of ( T × Π T ) τ , is denoted by R . 883) The Hilbert space L ( LT L k , L ⊗ q ) is defined by the tensor product L ( LT L k , L ⊗ q ) := L ( U L k , L ⊗ q ) (cid:98) ⊗ L ( T × Π T , L ⊗ q ) , and it is equipped with a − qτ -twisted representation R of LT τL m = U τL m (cid:2) U (1) ( T × Π T ) τ definedby R ( g ,g ,z ) φ (cid:98) ⊗ φ := R ( g ,z ) φ (cid:98) ⊗ R ( g , φ for ( g , g , z ) ∈ LT τL m , φ ∈ L ( U L k , L ⊗ q ) and φ ∈ L ( T × Π T , L ⊗ q ). Definition-Proposition 6.2. (1) Let A = ( { A x } x ∈M L m , Γ A ) be an M L m (cid:111) LT L m -equivariantu.s.c. field of C ∗ -algebras. Then, the set of invariant sections C (cid:18) (cid:102) M × T × Π T (cid:110) K (cid:16) L ( LT L k , L ⊗ q ) (cid:17) (cid:98) ⊗ A x (cid:111) x ∈ (cid:102) M (cid:19) for the T × Π T -action defined by the restriction of the LT L m -action Ad R (cid:98) ⊗ α A , is an ( M/T ) (cid:111) { e } -equivariant C ∗ -algebra . We regard the constructed C ∗ -algebra as the “ qτ -twisted crossedproduct of A by LT L m ” and we denote it by A (cid:111) qτ LT L m .(2) Let B = ( { B x } x ∈M L m , Γ B ) be an M L m (cid:111) LT L m -equivariant u.s.c. field of C ∗ -algebras,and let E = ( { E x } x ∈M L m , Γ E ) be a pτ -twisted M L m (cid:111) LT L m -equivariant u.s.c. field of Hilbert B -modules. Then, the set of invariant sections C (cid:18) (cid:102) M × T × Π T (cid:110) K (cid:16) L ( LT L k , L ⊗ q ) , L ( LT L k , L ⊗ ( q − p ) ) (cid:17) (cid:98) ⊗ E x (cid:111) x ∈ (cid:102) M (cid:19) for the T × Π T -action defined by the restriction of the LT L m -action Ad R (cid:98) ⊗ α E (this action is un-twisted. See the exposition before Proposition 3.44), is a Hilbert B (cid:111) qτ LT L m -module. We regardthe constructed Hilbert module as the “ qτ -twisted crossed product of E by LT L m ”, and wedenote it by E (cid:111) qτ LT L m .(3) Let ( E , π, F ) be a pτ -twisted M L m (cid:111) LT L m -equivariant Kasparov ( A , B )-module. We definethe ∗ -homomorphism id (cid:98) ⊗ π : A (cid:111) ( q − p ) τ LT L m → L B (cid:111) qτ LT L m ( E (cid:111) qτ LT L m )by id (cid:98) ⊗ π x ( k (cid:98) ⊗ a x )( k (cid:98) ⊗ e x ) := k ◦ k (cid:98) ⊗ π x ( a x )( e x ) and the operatorid (cid:98) ⊗ F ∈ L B (cid:111) qτ LT L m ( E (cid:111) qτ LT L m )by id (cid:98) ⊗ F x ( k (cid:98) ⊗ e x ) := k (cid:98) ⊗ F x ( e x ), for k ∈ K (cid:16) L ( LT L k , L ⊗ q ) , L ( LT L k , L ⊗ ( q − p ) ) (cid:17) , k ∈ K (cid:16) L ( LT L k , L ⊗ q ) (cid:17) , a x ∈ A x and e x ∈ E x . Then, the triple (cid:16) E (cid:111) qτ LT L m , { id (cid:98) ⊗ π x } x ∈M L m , { id (cid:98) ⊗ F x } x ∈M L m (cid:17) is an ( M/T ) (cid:111) { e } -equivariant Kasparov ( A (cid:111) ( q − p ) τ LT L m , B (cid:111) qτ LT L m ) -module. It is denoted by j qτLT L m ( E , π, F ) and the correspondence ( E , π, F ) (cid:55)→ j qτLT L m ( E , π, F ) is called the partial descenthomomorphism . 89 roof. (1) This is obvious from the fact that C (cid:18) (cid:102) M × T × Π T (cid:110) K (cid:16) L ( LT L k , L ⊗ q ) (cid:17) (cid:98) ⊗ A x (cid:111) x ∈ (cid:102) M (cid:19) ∼ = C (cid:16) (cid:102) M × T × Π T (cid:8) K (cid:0) L ( T × Π T , L ⊗ q ) (cid:1) (cid:98) ⊗ A x (cid:9) x ∈ (cid:102) M (cid:17) (cid:98) ⊗ K (cid:16) L ( LT L k , L ⊗ q ) (cid:17) ∼ = C ( (cid:102) M ) (cid:111) qτ ( T × Π T ) (cid:98) ⊗ K (cid:16) L ( LT L k , L ⊗ q ) (cid:17) . (2) One can prove it in the same way.(3) For a ∈ Γ A , k ∈ K (cid:16) L ( LT L k , L ⊗ ( q − p ) ) (cid:17) , e ∈ Γ E and k (cid:48) ∈ K (cid:16) L ( LT L k , L ⊗ q ) , L ( LT L k , L ⊗ ( q − p ) ) (cid:17) ,[id (cid:98) ⊗ π x ( k (cid:98) ⊗ a x ) , id (cid:98) ⊗ F x ]( k (cid:48) (cid:98) ⊗ e ) = k ◦ k (cid:48) (cid:98) ⊗ [ π x ( a x ) , F x ]( e x )= ( k ◦ (cid:63) ) (cid:98) ⊗ [ π x ( a x ) , F x ]( k (cid:48) (cid:98) ⊗ e x ) , where k ◦ (cid:63) means the operator k (cid:48) (cid:55)→ k ◦ k (cid:48) . The second component x (cid:55)→ [ π x ( a x ) , F x ] is an elementof Γ K ( E ) because ( E , { π x } , { F x } ) is a Kasparov ( A , B )-module. The first one k ◦ (cid:63) : K (cid:16) L ( LT L k , L ⊗ q ) , L ( LT L k , L ⊗ ( q − p ) ) (cid:17) → K (cid:16) L ( LT L k , L ⊗ q ) , L ( LT L k , L ⊗ ( q − p ) ) (cid:17) is K (cid:16) L ( LT L k , L ⊗ q ) (cid:17) -compact as proved in the next paragraph. Consequently, the section x (cid:55)→ [id (cid:98) ⊗ π x ( k (cid:98) ⊗ a ) , id (cid:98) ⊗ F x ] is a B (cid:111) LT L m -compact operator.What we need to prove is that k ◦ (cid:63) is a K (cid:16) L ( LT L k , L ⊗ q ) (cid:17) -compact operator on K (cid:16) L ( LT L k , L ⊗ q ) , L ( LT L k , L ⊗ ( q − p ) ) (cid:17) . For this aim, we may assume that k is a single Schattenform φ (cid:98) ⊗ ψ ∗ as an operator on the Hilbert space. Then, the operator k ◦ (cid:63) can be written as thecomposition of the following: For a unit vector λ ∈ L ( LT L k , L ⊗ q ), K (cid:16) L ( LT L k , L ⊗ q ) , L ( LT L k , L ⊗ ( q − p ) ) (cid:17) ( ψ (cid:98) ⊗ λ ∗ ) ∗ (cid:15) (cid:15) K (cid:16) L ( LT L k , L ⊗ q ) (cid:17) φ (cid:98) ⊗ λ ∗ (cid:15) (cid:15) K (cid:16) L ( LT L k , L ⊗ q ) , L ( LT L k , L ⊗ ( q − p ) ) (cid:17) . This is a single Schatten form on K (cid:16) L ( LT L k , L ⊗ q ) , L ( LT L k , L ⊗ ( q − p ) ) (cid:17) as the Hilbert K (cid:16) L ( LT L k , L ⊗ q ) (cid:17) -module. In particular, it is compact.For the same reason, the section x (cid:55)→ id (cid:98) ⊗ π x ( k (cid:98) ⊗ a x )(1 − id (cid:98) ⊗ F x ) = ( k ◦ (cid:63) ) (cid:98) ⊗ π x ( a x )(1 − F x )defines a B (cid:111) qτ LT L m -compact operator. 90 emarks . (1) Since we have assumed that the LT L m -action on M L m is co-compact, the crossedproduct is not C ( · · · ) but C ( · · · ). If the action were just “co-locally compact”, the crossed productwould become C ( · · · ) which is the set of continuous sections whose norms vanish at infinity in thequotient space M L m /LT L m . Note that the norm function is defined on the quotient space becausethe norm is invariant under the LT L m -action.(2) L ( LT L k , L ⊗ q ) and L ( LT L k , L ⊗ ( q − p ) ) are trivially graded, and hence signs coming from thegradings do not appear.We must prove that j pτLT L m makes sense at the level of KK -theory. Proposition 6.4.
Let A and B be M L m (cid:111) LT L m -equivariant u.s.c. fields of C ∗ -algebras. Then,the correspondence j qτLT L m is homotopy invariant and hence it defines a homomorphism R KK pτLT L m ( M L m ; A , B ) → R KK (cid:16) M/T ; A (cid:111) ( q − p ) τ LT L m , B (cid:111) qτ LT L m (cid:17) . Proof.
Let (cid:16) (cid:101) E , { (cid:101) π x } x ∈M L m , { (cid:101) F x } x ∈M L m (cid:17) be a homotopy between ( E , π , F ) and ( E , π , F ), whichis a pτ -twisted M L m (cid:111) LT L m -equivariant Kasparov ( A , B (cid:98) ⊗ C ( I ))-module. The Kasparov (cid:16) A (cid:111) ( q − p ) τ LT L m , B (cid:111) qτ LT L m (cid:98) ⊗ C ( I ) (cid:17) -module (cid:16) (cid:101) E (cid:111) qτ LT L m , { id (cid:98) ⊗ (cid:101) π x } , { id (cid:98) ⊗ (cid:101) F x } (cid:17) gives a homotopy between j qτLT L m ( E , π , F ) and j qτLT L m ( E , π , F ).Let us introduce an infinite-dimensional version of [ c X ], imitating Lemma 3.25.Take a cut-off function c (cid:102) M : (cid:102) M → R ≥ with respect to the T × Π T -action. It defines a function c (cid:102) M,x : T × Π T → R ≥ by c (cid:102) M,x ( g ) := c (cid:102) M ( g · x ). Note that T × Π T is unimodular and the modularfunction does not appear here.The Hilbert space L ( U L k ) has a unit vector “vac”. We regard it as the “square root of acut-off function on U L k ”. Recall that a cut-off function on a locally compact group with respect toleft translation is just a non-negative-valued L -unit function. The rank one projection onto theone-dimensional subspace C vac is denoted by P vac . Definition 6.5.
Under the identification C ( M L m ) (cid:111) LT L m ∼ = C (cid:16) (cid:102) M × T × Π T (cid:110) K (cid:16) L ( U L k ) (cid:17) (cid:98) ⊗ K ( L ( T × Π T ) (cid:111)(cid:17) , the Mishchenko line bundle (cid:104) c M L m (cid:105) ∈ KK (cid:16) C , C ( M L m ) (cid:111) LT L m (cid:17) is defined by the equivariantfamily of rank one projections P : x (cid:55)→ P x := P vac (cid:98) ⊗ P √ c (cid:102) M,x . Proposition 6.6. (cid:104) c M L m (cid:105) is represented by the following Kasparov module: (cid:18) C (cid:18) (cid:102) M × T × Π T L (cid:16) LT L k (cid:17) ∗ (cid:19) , , (cid:19) , here the Hilbert module structure is given as follows: For equivariant sections f, f , f : (cid:102) M → L ( LT L k ) ∗ and b : (cid:102) M → K (cid:16) L ( LT L k ) (cid:17) , • ( f · b )( x ) := f ( x ) ◦ b ( x ) ; and • ( f | f ) C ( M L m ) (cid:111) LT L m ( x ) := f ( x ) ∗ ◦ f ( x ) , where f ( x ) ∗ is an element of (cid:20) L (cid:16) LT L k (cid:17) ∗ (cid:21) ∗ ∼ = L ( LT L k ) by the Riesz representation theorem.Proof. We construct an isomorphismΦ : P · (cid:104) C (cid:16) M L m × LT L m K (cid:16) L ( LT L k ) (cid:17)(cid:17)(cid:105) → C (cid:18) (cid:102) M × T × Π T L (cid:16) LT L k (cid:17) ∗ (cid:19) by the formula Φ( P · f )( x ) := [vac (cid:98) ⊗ (cid:112) c (cid:102) M,x ] ∗ ◦ f ( x ) , where [vac (cid:98) ⊗ (cid:112) c (cid:102) M,x ] ∗ is the Riesz dual of vac (cid:98) ⊗ (cid:112) c (cid:102) M,x . We need to verify the following: (1) Φ iswell-defined; (2) Φ is isometric right module homomorphism; and (3) Φ is surjective.(1) We need to check that Φ( P · a ) is T × Π T -equivariant, that is to say, Φ( P · a )( g − x ) ◦ R − g =Φ( P · a )( x ) . By definition, a satisfies the equivariant condition R g ◦ a ( g − x ) ◦ R − g = a ( x ) . Since Φ( P · a )( x ) = [vac (cid:98) ⊗ (cid:112) c (cid:102) M,x ] ∗ ◦ a ( x ), it is sufficient to verify that [vac (cid:98) ⊗ (cid:112) c (cid:102) M,x ] ∗ is T × Π T -invariant. Since “vac” is T × Π T -invariant, it suffices to deal with the (cid:112) c (cid:102) M,x -part. We can provethe invariance as follows: R g (cid:16)(cid:112) c (cid:102) M,g − · x (cid:17) ( h ) = (cid:112) c (cid:102) M,g − · x ( hg )= (cid:113) c (cid:102) M ( hgg − · x )= (cid:113) c (cid:102) M ( hx )= (cid:112) c (cid:102) M,x ( h ) . (2) Φ is clearly a module homomorphism. We check that it is isometric. For a , a ∈ C (cid:16) M L m × LT L m K (cid:16) L ( LT L k ) (cid:17)(cid:17) , since P = P ∗ = P ,( P · a | P · a ) C ( M L m ) (cid:111) LT L m ( x ) = [ P · a ( x )] ∗ ◦ P · a ( x )= a ( x ) ∗ ◦ P ◦ a ( x )= a ( x ) ∗ ◦ [vac (cid:98) ⊗ (cid:112) c (cid:102) M,x ] ◦ [vac (cid:98) ⊗ (cid:112) c (cid:102) M,x ] ∗ ◦ a ( x )= (cid:16) [vac (cid:98) ⊗ (cid:112) c (cid:102) M,x ] ∗ ◦ a ( x ) (cid:17) ∗ ◦ (cid:16) [vac (cid:98) ⊗ (cid:112) c (cid:102) M,x ] ∗ ◦ a ( x ) (cid:17) = (Φ( P · a ) | Φ( P · a )) ( x ) . (3) For φ ∈ C (cid:16) (cid:102) M × T × Π T L (cid:0) LT L m (cid:1) ∗ (cid:17) , let us consider the compact operator-valued function a defined by x (cid:55)→ [vac (cid:98) ⊗ (cid:112) c (cid:102) M,x ] ◦ φ ( x ) . It gives an equivariant section thanks to the same argument for (1), and clearly Φ( P · a ) = φ .92n the remainder of this subsection, we prove several properties of the crossed products and thedescent homomorphism for proper LT -spaces, in order to show how appropriate our constructionsare. Since these results will not be used in the following, the reader can skip them. For the samereason, we will often skip the detailed proof.We begin with an infinite-dimensional version of the following fundamental property: Thecrossed product by a semi-direct product group is obtained by the iterated crossed product. Weare interested in this property for the following reason: LT L m has the canonical decomposition U LT L m × T × Π T , and this decomposition is partially inherited to the central extension LT τL m = U τLT L m (cid:2) ( T × Π T ) τ ; It has an alternative description ( U τL m × Π T ) (cid:111) T, where the T -action is definedby Ad t (( u, z ) , n ) := (( u, zκ τ − n ( t )) , n )for ( u, z ) ∈ U τL m , n ∈ Π T and t ∈ T . This factorization is valid also for LG for a compact Liegroup G . Although LG does not have a subgroup corresponding to U L m , it does have a subgroupcorresponding to U τL m × Π T . Thus, the following observation can be useful even for the case of LG .We can define the concept of (twisted) crossed products by U L m and U L m × Π T as follows. Definition 6.7. (1) For q ∈ Z , we define L ( U L k × Π T , L ⊗ q ) := L ( U L k , L ⊗ q ) (cid:98) ⊗ L (Π T , L ⊗ q ), andwe define a T -action on it by the following: For φ ∈ L ( U L k , L ⊗ q ), ψ ∈ C c (Π T , L ⊗ q ), t ∈ T and n ∈ Π T , t · [ φ (cid:98) ⊗ ψ ]( n ) := [ κ τn ( t )] q φ (cid:98) ⊗ ψ ( n ) . It admits the “right regular representation” R : U τL m × Π T → Aut (cid:16) L ( U L k × Π T , L ⊗ q ) (cid:17) , which iscompatible with the above T -action.(2) For an M L m (cid:111) LT L m -equivariant u.s.c. field of C ∗ -algebras A , we define the qτ -twistedcrossed product of A by U L m × Π T by A (cid:111) qτ ( U L m × Π T ) := C (cid:18) (cid:102) M × Π T (cid:110) K (cid:16) L ( U L k × Π T , L ⊗ q ) (cid:17) (cid:98) ⊗ A x (cid:111) x ∈ (cid:102) M (cid:19) . (3) In the same setting, we define a substitute for the qτ -twisted crossed product of A by U L m A (cid:111) qτ U L m := C (cid:18) (cid:102) M , (cid:110) K (cid:16) L ( U L k , L ⊗ q ) (cid:17) (cid:98) ⊗ A x (cid:111) x ∈ (cid:102) M (cid:19) similarly. Remark . Since (cid:102) M is non-compact, we need to use C ( · · · ) instead of C ( · · · ) in (3).It is obvious that A (cid:111) qτ ( U L m × Π T ) is an M (cid:111) T -equivariant C ∗ -algebra and A (cid:111) qτ U L m is an (cid:102) M (cid:111) ( T × Π T )-equivariant C ∗ -algebra. We will prove that these algebras are related in a naturalway which is analogous to the following fundamental result. Lemma 6.9 (See [Wil, Proposition 3.11]) . Let G = N (cid:111) H be the semi-direct product of two locallycompact amenable groups N and H , and let A be a G - C ∗ -algebra. Suppose that N admits a U (1) -central extension N τ : → U (1) i −→ N τ p −→ N → , and suppose that the H -action on N lifts to N τ . Then, the semi-direct product N (cid:111) H admits a U (1) -central extension N τ (cid:111) H by the naturalhomomorphisms: → U (1) i (cid:111) −−→ N τ (cid:111) H p (cid:111) id −−−→ N (cid:111) H → . Then, we have a ∗ -isomorphism A (cid:111) kτ G ∼ = ( A (cid:111) kτ N ) (cid:111) H. roposition 6.10. For an M L m (cid:111) LT L m -equivariant u.s.c. field of C ∗ -algebras A and q ∈ Z , A (cid:111) qτ ( U L m × Π T ) ∼ = A (cid:111) qτ U L m (cid:111) qτ Π T as M (cid:111) T -equivariant C ∗ -algebras, and A (cid:111) qτ LT L m ∼ = A (cid:111) qτ ( U L m × Π T ) (cid:111) T ∼ = A (cid:111) qτ U L m (cid:111) qτ ( T × Π T ) as ( M/T ) (cid:111) { e } - C ∗ -algebras.Proof. We prove only A (cid:111) qτ LT L m ∼ = A (cid:111) qτ U L m (cid:111) qτ ( T × Π T ), and the others are left to the reader.Recall that A (cid:111) qτ U L m is defined by C (cid:18) (cid:102) M , (cid:110) K (cid:16) L ( U L k , L ⊗ q ) (cid:17) (cid:98) ⊗ A x (cid:111) x ∈ (cid:102) M (cid:19) . Since the T × Π T -action on K (cid:16) L ( U L k , L ⊗ q ) (cid:17) is trivial, A (cid:111) qτ U L m is isomorphic to C (cid:16) (cid:102) M , { A x } x ∈ (cid:102) M (cid:17) (cid:98) ⊗ K (cid:16) L ( U L k , L ⊗ q ) (cid:17) as (cid:102) M (cid:111) ( T × Π T )- C ∗ -algebras. Thus, the qτ -twisted crossed product of this C ∗ -algebra by T × Π T is isomorphic to C (cid:16) (cid:102) M , { A x } x ∈ (cid:102) M (cid:17) (cid:111) qτ ( T × Π T ) (cid:98) ⊗ K (cid:16) L ( U L k , L ⊗ q ) (cid:17) ∼ = C (cid:16) (cid:102) M × T × Π T (cid:8) K ( L ( T × Π T , L ⊗ q )) (cid:98) ⊗ A x (cid:9) x ∈ (cid:102) M (cid:17) (cid:98) ⊗ K (cid:16) L ( U L k , L ⊗ q ) (cid:17) ∼ = C (cid:18) (cid:102) M × T × Π T (cid:110) K ( L ( LT L k , L ⊗ q )) (cid:98) ⊗ A x (cid:111) x ∈ (cid:102) M (cid:19) = A (cid:111) qτ LT L m . Remarks . (1) A (cid:111) qτ ( U L m × Π T ) carries information of the central extension of T × Π T as the T -action on (cid:110) K (cid:16) L ( U L k × Π T , L ⊗ q ) (cid:17) (cid:98) ⊗ A x (cid:111) x ∈ (cid:102) M . However, A (cid:111) qτ U L m does not know this kindof information. This is the reason why two kinds of crossed products (cid:111) and (cid:111) qτ appear in thestatement.(2) The isomorphism A (cid:111) qτ ( U L m × Π T ) (cid:111) T ∼ = A (cid:111) qτ U L m (cid:111) qτ ( T × Π T ) can be proved by theisomorphism A (cid:111) qτ ( U L m × Π T ) ∼ = A (cid:111) qτ U L m (cid:111) qτ Π T and Lemma 6.9.In a finite-dimensional setting, a crossed product is a noncommutative analogue of the quotientspace. A fixed-point algebra is a more straightforward substitute. It is especially useful for a properfree action. Since the U L m × Π T -action on M L m is proper and free, it is worth trying to constructfixed-point algebras for the U L m × Π T -action and those for the U L m -action. Definition 6.12. (1) For an M L m (cid:111) LT L m -equivariant u.s.c. field of C ∗ -algebras A = (cid:16) { A x } x ∈M L m , Γ A (cid:17) ,we define the fixed-point algebra A U L m by C ( (cid:102) M , { A x } x ∈ (cid:102) M ).(2) In the same setting, we define A U L m × Π T by C ( (cid:102) M × T × Π T { A x } x ∈ (cid:102) M ).94 emma 6.13. (1) A U L m × Π T is an M (cid:111) T -equivariant C ∗ -algebra. (2) A U L m is an (cid:102) M (cid:111) ( T × Π T ) -equivariant C ∗ -algebra. (3) The generalized fixed-point algebra of A U L m with respect to the Π T -action (cid:16) A U L m (cid:17) Π T , isisomorphic to A U L m × Π T as M (cid:111) T -equivariant C ∗ -algebras. (4) A U L m is R KK T × Π T ( (cid:102) M ; − , − ) -equivalent to A (cid:111) U L m , and similarly for A U L m × Π T and A (cid:111) (cid:0) U L m × Π T (cid:1) .Proof. We leave the proofs of the first three statements to the reader.For (4), we must specify the R KK -elements giving the equivalences. We will do that only for A U L m . We define [ I A ,U L m ] ∈ R KK T × Π T ( (cid:102) M ; A U L m , A (cid:111) U L m ) and[ J A ,U L m ] ∈ R KK T × Π T ( (cid:102) M ; A (cid:111) U L m , A U L m )by [ I A ,U L m ] := (cid:16) C ( (cid:102) M , { A x (cid:98) ⊗ L ( U L k ) ∗ } x ∈ (cid:102) M ) , { id (cid:98) ⊗ } x ∈ (cid:102) M , (cid:17) and[ J A ,U L m ] := (cid:16) C ( (cid:102) M , { A x (cid:98) ⊗ L ( U L k ) } x ∈ (cid:102) M ) , { id (cid:98) ⊗ Op } x ∈ (cid:102) M , (cid:17) , where id (cid:98) ⊗ a )( a (cid:48) (cid:98) ⊗ φ ) := aa (cid:48) (cid:98) ⊗ φ and id (cid:98) ⊗ Op( a (cid:98) ⊗ k )( a (cid:48) (cid:98) ⊗ ψ ) := aa (cid:48) (cid:98) ⊗ kψ for a, a (cid:48) ∈ A x , φ ∈ L ( U L k ) ∗ , ψ ∈ L ( U L k ) and k ∈ K ( L ( U L k )). It is clear that [ I A ,U L m ] and [ J A ,U L m ] are mutually inverse,thanks to the isomorphisms V (cid:98) ⊗ C V ∗ ∼ = K ( V ) and V ∗ (cid:98) ⊗ K ( V ) V ∼ = C for a Hilbert space V . Example . As everyone expects, we have isomorphisms C (cid:0) M L m (cid:1) U L m ∼ = C ( (cid:102) M ) and C (cid:0) M L m (cid:1) U L m × Π T ∼ = C ( M ).With the fixed-point algebra construction, we can define another kind of descent homomorphismfollowing [Kas2, Theorem 3.4]. This construction looks quite natural, and it is perhaps much moreacceptable than j LT L m . Definition 6.15. (1) Let B = (cid:16) { B x } x ∈M L m , Γ B (cid:17) be an M L m (cid:111) LT L m -equivariant u.s.c. field of C ∗ -algebras. For an M L m (cid:111) LT L m -equivariant Kasparov ( A , B )-module ( E , π, F ), we define the fixed-point module E U L m × Π T by C ( (cid:102) M × Π T { E x } x ∈ (cid:102) M ) . The restriction of { π x : A x → L B x ( E x ) } x ∈M L m to the fixed-point algebra A U L m × Π T is denoted by π U L m × Π T : A U L m × Π T → L B UL m × Π T (cid:16) E U L m × Π T (cid:17) ,and the restriction of { F x } x ∈M L m to E U L m × Π T is denoted by F U L m × Π T . We denote (cid:16) E U L m × Π T , π U L m × Π T , F U L m × Π T (cid:17) by λ U L m × Π T ( E , π, F ).(2) Similarly, we define λ U L m ( E , π, F ). Note that V (cid:98) ⊗ C V ∗ is not a Hilbert space but a Hilbert K ( V )-module. The correspondence φ (cid:98) ⊗ ψ (cid:55)→ [ λ (cid:55)→ φ (cid:104) ψ, λ (cid:105) ]gives an isometric isomorphism. On the other hand, V ∗ (cid:98) ⊗ K ( V ) V is isomorphic to C by the correspondence ψ (cid:98) ⊗ φ (cid:55)→(cid:104) ψ, φ (cid:105) . This is isomorphic because the tensor product is taken over K ( V ).
95e define two homomorphisms j U L m × Π T and j U L m in the same way of j LT L m . Five “decenthomomorphisms” are related to each other as follows. Proposition 6.16. (1) λ U L m × Π T ( E , π, F ) is an M (cid:111) T -equivariant Kasparov (cid:16) A U L m × Π T , B U L m × Π T (cid:17) -module. Similarly, λ U L m ( E , π, F ) is an (cid:102) M (cid:111) ( T × Π T ) -equivariant Kasparov (cid:16) A U L m , B U L m (cid:17) -module. (2) Both constructions λ U L m × Π T and λ U L m are homotopy invariant. Thus, they define homo-morphisms λ U L m × Π T : R KK LT L m ( M L m ; A , B ) → R KK T ( M ; A U L m × Π T , B U L m × Π T ) and λ U L m : R KK LT L m ( M L m ; A , B ) → R KK T × Π T ( (cid:102) M ; A U L m , B U L m ) . (3) λ U L m × Π T = λ Π T ◦ λ U L m . (4) Under the R KK -equivalence of crossed products and fixed-point algebras, j ’s correspond to λ ’s, that is to say, the following two diagrams commute: R KK LT L m ( M L m ; A , B ) j UL m × Π T (cid:47) (cid:47) λ UL m × Π T (cid:44) (cid:44) R KK T ( M ; A (cid:111) ( U L m × Π T ) , B (cid:111) ( U L m × Π T )) ∼ = (cid:15) (cid:15) R KK T ( M ; A U L m × Π T , B U L m × Π T ) , R KK LT L m ( M L m ; A , B ) j UL m (cid:47) (cid:47) λ UL m (cid:44) (cid:44) R KK T × Π T ( M ; A (cid:111) U L m , B (cid:111) U L m ) ∼ = (cid:15) (cid:15) R KK T × Π T ( M ; A U L m , B U L m ) . Proof. (1) and (3) are clear. For (2), consider the parallel construction of Proposition 6.4 (2).(4) We prove only the latter one. Let ( E , π, F ) ∈ R KK LT L m ( M L m ; A , B ). Then, j U L m ( E , π, F ) = (cid:16) C ( (cid:102) M , { E x (cid:98) ⊗ K ( L ( U L k )) } x ∈ (cid:102) M ) , { π x (cid:98) ⊗ id } x ∈ (cid:102) M , { F x (cid:98) ⊗ id } x ∈ (cid:102) M (cid:17) and λ U L m ( E , π, F ) = (cid:16) C ( (cid:102) M , { E x } x ∈ (cid:102) M ) , { π x } x ∈ (cid:102) M , { F x } x ∈ (cid:102) M (cid:17) . We may assume the non-degeneracy A x E x = E x for every x . Thus, the Kasparov module (cid:16) C ( (cid:102) M , { E x } x ∈ (cid:102) M ) , { π x } x ∈ (cid:102) M , { F x } x ∈ (cid:102) M (cid:17) represents the Kasparov product [ I A ,U L m ] (cid:98) ⊗ j U L m ( E , π, F ) (cid:98) ⊗ [ J B ,U L m ], because V ∗ (cid:98) ⊗ K ( V ) K ( V ) (cid:98) ⊗ K ( V ) V ∼ = C for a Hilbert space V .A parallel property of (3) holds for j ’s. Proposition 6.17.
At the level of Kasparov modules, j pτLT L m = j T ◦ j pτU L m × Π T = j T ◦ j pτ Π T ◦ j pτU L m . emarks . (1) This is clear from the same argument of Proposition 6.10.(2) The equality j U L m × Π T = j Π T ◦ j U L m can be proved as a corollary of Proposition 6.16.As the final result of this subsection, we prove that the descent map preserves the Kasparovproduct for proper LT -spaces. Proposition 6.19.
Let A , A , B be M L m (cid:111) LT L m -equivariant locally separable u.s.c. fields of C ∗ -algebras. Then, the following diagram commutes R KK q τLT L m ( M L m ; A , A ) × R KK q τLT L m ( M L m ; A , B ) (cid:98) ⊗ −−−−→ R KK ( q + q ) τLT L m ( M L m ; A , B ) j ( p − q τLTL m × j pτLTL m (cid:121) (cid:121) j pτLTL m R KK ( M/T ; (cid:63) , (cid:63) ) × R KK ( M/T ; (cid:63) , (cid:63) ) (cid:98) ⊗ −−−−→ R KK ( M/T ; (cid:63) , (cid:63) ) , where the (cid:63) ’s stand for the following: (cid:63) := A (cid:111) ( p − q − q ) τ LT L m ,(cid:63) := A (cid:111) ( p − q ) τ LT L m ,(cid:63) := B (cid:111) pτ LT L m . The same is true for j pτU L m , j pτU L m × Π T , λ U L m and λ U L m × Π T .Proof. We deal with only the most important one j pτLT L m . Let ( E , π , F ) be a q τ -twisted M L m (cid:111) LT L m -equivariant Kasparov ( A , A )-module, and let ( E , π , F ) be a q τ -twisted M L m (cid:111) LT L m -equivariant Kasparov ( A , B )-module. Let ( E , π, F ) be a representative of the Kasparov product ofthese two Kasparov modules, where E = E (cid:98) ⊗ E . We need to prove that j pτLT L m ( E , π, F ) = j ( p − q ) τLT L m ( E , π , F ) (cid:98) ⊗ j pτLT L m ( E , π , F ) . Let us prove that the bimodules are isomorphic. Since E (cid:111) ( p − q ) τ LT L m = C (cid:18) (cid:102) M × T × Π T (cid:110) E ,x (cid:98) ⊗ K ( L ( LT L k , L ⊗ ( p − q ) ) , L ( LT L k , L ⊗ ( p − q − q ) )) (cid:111) x ∈ (cid:102) M (cid:19) ; E (cid:111) pτ LT L m = C (cid:18) (cid:102) M × T × Π T (cid:110) E ,x (cid:98) ⊗ K ( L ( LT L k , L ⊗ p ) , L ( LT L k , L ⊗ ( p − q ) )) (cid:111) x ∈ (cid:102) M (cid:19) ; A (cid:111) ( p − q ) τ LT L m = C (cid:18) (cid:102) M × T × Π T (cid:110) A ,x (cid:98) ⊗ K ( L ( LT L k , L ⊗ ( p − q ) )) (cid:111) x ∈ (cid:102) M (cid:19) ; E (cid:111) pτ LT L m = C (cid:18) (cid:102) M × T × Π T (cid:110) E x (cid:98) ⊗ K ( L ( LT L k , L ⊗ p ) , L ( LT L k , L ⊗ ( p − q − q ) )) (cid:111) x ∈ (cid:102) M (cid:19) , we have a natural isomorphism E (cid:111) pτ LT L m ∼ = E (cid:111) ( p − q ) τ LT L m (cid:98) ⊗ A (cid:111) ( p − q τ LT L m E (cid:111) pτ LT L m as ( A (cid:111) ( p − q − q ) τ LT L m , B (cid:111) pτ LT L m )-bimodules, thanks to the isomorphism K ( V , V ) (cid:98) ⊗ K ( V ) K ( V , V ) ∼ = K ( V , V ) for Hilbert spaces V , V , V The conditions on F can be easily proved, and we leave the details to the reader.97 .2 The “descent of the Dirac element” The final tool to define the topological assembly map is the “descent of the reformulated Diracelement”. In this subsection, we define an infinite-dimensional version of it, by imitating Proposition3.45.In this formula, a Hilbert
C (cid:111) τ G -module C (cid:111) − τ G appears. We begin with the LT L m -versionof it. Definition 6.20. (1) We define a Hilbert
C (cid:111) τ LT L m -module C (cid:111) − τ LT L m by the following: • We define C ∗ -algebras C (cid:111) − τ U L m and C (cid:111) − τ LT L m in the same way of Definition 4.18 byreplacing K (cid:16) L ( R ∞ ) ∗ (cid:17) and C (cid:111) τ ( T × Π T ) with K (cid:16) L ( R ∞ ) (cid:17) and C (cid:111) − τ ( T × Π T ). Whenwe regard an element φ of C (cid:111) − τ U L m as an element of K (cid:16) L ( R ∞ ) (cid:17) , we denote it by Op( φ ). • For a function b on ( T × Π T ) τ , we define b ∨ ( g ) := b ( g − ). This operation exchanges C (cid:111) τ ( T × Π T )and C (cid:111) − τ ( T × Π T ). • Similarly, we define φ ∨ := Op − ( t Op( φ )) for φ ∈ C (cid:111) − τ U L m or C (cid:111) τ U L m . This correspon-dence exchanges C (cid:111) τ U L m and C (cid:111) − τ U L m . • We define a Hilbert
C (cid:111) τ LT L m -module structure on C (cid:111) − τ LT L m by the following: For φ (cid:98) ⊗ ψ, φ (cid:98) ⊗ ψ , φ (cid:98) ⊗ ψ ∈ C (cid:111) − τ U L m (cid:98) ⊗ [ C(cid:111) − τ ( T × Π T )] = C (cid:111) − τ LT L m and a (cid:98) ⊗ b ∈ C (cid:111) τ U L m (cid:98) ⊗ [ C(cid:111) τ ( T × Π T )] = C (cid:111) τ LT L m , φ (cid:98) ⊗ ψ · a (cid:98) ⊗ b := a ∨ ∗ φ (cid:98) ⊗ b ∨ ∗ ψ ; (cid:0) φ (cid:98) ⊗ ψ (cid:12)(cid:12) φ (cid:98) ⊗ ψ (cid:1) := ( φ ∗ φ ∗ ) ∨ (cid:98) ⊗ ( ψ ∗ ψ ∗ ) ∨ . (2) We define a dense subspace ( C (cid:111) − τ LT L m ) fin just like Definition 4.18 (3). It has a pre-Hilbert( C (cid:111) τ LT L m ) fin -module structure.The above Hilbert module is equipped with an LT τL m -action denoted by “rt”. See also Definition4.21. Definition 6.21. (1) We define an LT τL m -action “rt” on the Hilbert ( C (cid:111) τ LT L m )-module C (cid:111) − τ LT L m by the tensor product of the following actions: • For φ ∈ C (cid:111) − τ U L m and u ∈ U τL m , we define Op(rt u φ ) := Op( φ ) ◦ ρ u − , and • For ψ ∈ C c ( T × Π T , − τ ) ⊆ C (cid:111) − τ ( T × Π T ) and γ ∈ ( T × Π T ) τ , we define rt γ ψ ( x ) := ψ ( xγ ).(2) The infinitesimal version of “rt” is denoted by d rt X for X ∈ Lie( LT fin ). It is an operatordefined on ( C (cid:111) − τ LT L m ) fin . Its extension is also denoted by the same symbol.With this Hilbert module, we define a substitute for “the descent of the Dirac element”. Wedenote the exterior tensor product of the Spinor bundles S U and S (cid:102) M by S M . Take a cut-off function c : (cid:102) M → R ≥ with respect to the T × Π T -action. We refer to [T3, Section 6.2.1] for the “fiber”of the following Kasparov module. See Remark 3.31 (2) for the strict definitions of the followingsymbolic formulas. 98 efinition 6.22. (1) We define a Hilbert C (cid:111) τ LT L m -module L ( M L k , S M ) (cid:111) τ LT L m = L (cid:18) (cid:102) M × T × Π T (cid:26) S U (cid:98) ⊗ S (cid:102) M,x (cid:98) ⊗ L (cid:16) LT L k , L (cid:17) (cid:98) ⊗ C (cid:111) − τ LT L m (cid:27) x ∈ (cid:102) M (cid:19) by the completion of C (cid:18) (cid:102) M × T × Π T (cid:26) S U (cid:98) ⊗ S (cid:102) M,x (cid:98) ⊗ L (cid:16) LT L k , L (cid:17) (cid:98) ⊗ C (cid:111) − τ LT L m (cid:27) x ∈ (cid:102) M (cid:19) by the follow-ing operations: For φ, φ , φ : (cid:102) M → S U (cid:98) ⊗ S (cid:102) M (cid:98) ⊗ L (cid:16) LT L k , L (cid:17) and ψ, ψ , ψ : (cid:102) M → C (cid:111) − τ LT L m suchthat k = φ (cid:98) ⊗ ψ , k = φ (cid:98) ⊗ ψ ∈ L ( M L k , S M ) (cid:111) τ LT L m , and b ∈ C (cid:111) τ LT L m , • [ φ (cid:98) ⊗ ψ · b ]( x ) := φ ( x ) (cid:98) ⊗ [ ψ ( x ) · b ]; and • (cid:0) φ (cid:98) ⊗ ψ (cid:12)(cid:12) φ (cid:98) ⊗ ψ (cid:1) C(cid:111) τ LT L m := (cid:82) (cid:102) M ( φ ( x ) | φ ( x )) ( ψ ( x ) | ψ ( x )) C(cid:111) τ LT L m c ( x ) dx .(2) We define a left C (cid:0) M L m (cid:1) (cid:111) τ LT L m -module structure π (cid:111) τ lt : C (cid:0) M L m (cid:1) (cid:111) τ LT L m → L C(cid:111) τ LT L m (cid:16) L ( M L k , S M ) (cid:111) τ LT L m (cid:17) as follows: For a ∈ C (cid:0) M L m (cid:1) (cid:111) τ LT L m , φ : (cid:102) M → S U (cid:98) ⊗ S (cid:102) M (cid:98) ⊗ L (cid:16) LT L k , L (cid:17) , ψ : (cid:102) M → C (cid:111) − τ LT L m such that k = φ (cid:98) ⊗ ψ ∈ L ( M L k , S M ) (cid:111) τ LT L m ,[ π (cid:111) τ lt( a )( φ (cid:98) ⊗ ψ )]( x ) := [ π x ( a ( x ))( φ ( x ))] (cid:98) ⊗ ψ ( x ) . (3) By using the identification L ( M L k , S M ) (cid:111) τ LT L m ∼ = L (cid:16) (cid:102) M × T × Π T (cid:110) S (cid:102) M,x (cid:98) ⊗ L ( T × Π T , L ) (cid:98) ⊗ [ C (cid:111) − τ ( T × Π T )] (cid:111) x ∈ (cid:102) M (cid:17) (cid:100)(cid:79) L (cid:16) U L k , L (cid:17) (cid:98) ⊗ C (cid:111) − τ U L m (cid:98) ⊗ S U , we define an operator (cid:101) D by (cid:101) D := (cid:32)(cid:88) i c ( e i ) ∇ S (cid:102) M e i (cid:98) ⊗ id C(cid:111) − τ ( T × Π T ) (cid:33) (cid:100)(cid:79) id+ id (cid:100)(cid:79) (cid:32)(cid:88) n (cid:110)(cid:16) n − l dR (cid:48) ( z n ) (cid:98) ⊗ id + id (cid:98) ⊗√ nd rt z n (cid:17) (cid:98) ⊗ γ ( z n ) + (cid:16) n − l dR (cid:48) ( z n ) (cid:98) ⊗ id + id (cid:98) ⊗√ nd rt z n (cid:98) ⊗ γ ( z n ) (cid:17)(cid:111)(cid:33) . We denote (cid:80) i c ( e i ) ∇ S (cid:102) M e i (cid:98) ⊗ id C(cid:111) − τ ( T × Π T ) by (cid:94) D base .(4) The KK -element corresponding to the unbounded Kasparov ( C ( M L m ) (cid:111) τ LT L m , C (cid:111) τ LT L m )-module (cid:16) L ( M L k , S M ) (cid:111) τ LT L m , π (cid:111) τ lt , (cid:101) D (cid:17) is denoted by j τLT L m (cid:16) fgt[ S ] (cid:98) ⊗ (cid:104) (cid:94) d M L m (cid:105)(cid:17) ∈ KK ( C ( M L m ) (cid:111) τ LT L m , C (cid:111) τ LT L m )(we will prove that the above triple is actually an unbounded Kasparov module later). We regardit as the “ descent of the reformulated Dirac element ”.99 emarks . (1) The “fiber” of this KK -element is almost the same with [ (cid:101) (cid:19) ∂ R ] discussed in [T3,Theorem 6.8].(2) The tensor product between the (cid:102) M -part and the U L m -part is denoted by (cid:99)(cid:78) , and others aredenoted by (cid:98) ⊗ .(3) We have defined neither j τLT L m nor fgt[ S ]. Although we have defined a KK -element “[ (cid:94) d M L m ]”in [T4], we have not proved any relationships between “[ (cid:94) d M L m ]” and “ j τLT L m (cid:16) fgt[ S ] (cid:98) ⊗ (cid:104) (cid:94) d M L m (cid:105)(cid:17) ”.We will give a comment on this issue in Section 6.4. Theorem 6.24.
The triple (cid:16) L ( M L k , S M ) (cid:111) τ LT L m , π (cid:111) τ lt , (cid:101) D (cid:17) is an unbounded Kasparov (cid:16) C ( M L m ) (cid:111) τ LT L m , C (cid:111) τ LT L m (cid:17) -module.Proof. We need to prove the following properties: (1) The operator (cid:101) D is well-defined and essentiallyself-adjoint; (2) For “smooth” a ∈ C ( M L m ) (cid:111) τ LT L m , the commutator (cid:104) π (cid:111) τ lt( a ) , (cid:101) D (cid:105) is bounded;(3) For any a ∈ C ( M L m ) (cid:111) τ LT L m , π (cid:111) τ lt( a ) (cid:16) (cid:101) D (cid:17) − is C (cid:111) τ LT L m -compact. We refer to [T3,Theorem 6.8] for several estimates of this proof.(1) Let us consider the dense subspace C ∞ (cid:16) (cid:102) M × T × Π T (cid:8) S (cid:102) M (cid:98) ⊗ L ( T × Π T , L ) (cid:98) ⊗ [ C (cid:111) − τ ( T × Π T )] (cid:9)(cid:17) (cid:100)(cid:79) alg L (cid:16) U L k , L (cid:17) fin (cid:98) ⊗ alg [ C (cid:111) − τ U L m ] fin (cid:98) ⊗ alg S U, fin of L ( M L k , S M ) (cid:111) τ LT L m . Note that the ( T × Π T )-action on L (cid:16) U L k , L (cid:17) fin (cid:98) ⊗ [ C (cid:111) − τ U L m ] fin (cid:98) ⊗ S U, fin is trivial, and hence the above is actually a subspace of L ( M L k , S M ) (cid:111) τ LT L m . On this subspace, (cid:101) D acts as (cid:94) D base (cid:100)(cid:79) id + id (cid:100)(cid:79) (cid:88) n (cid:16) n − l dR (cid:48) ( z n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( z n ) + n − l dR (cid:48) ( z n ) (cid:98) ⊗ id (cid:98) ⊗ γ ( z n ) (cid:17) + id (cid:100)(cid:79) (cid:88) n (cid:8) id (cid:98) ⊗√ nd rt z n (cid:98) ⊗ γ ( z n ) + id (cid:98) ⊗√ nd rt z n (cid:98) ⊗ γ ( z n ) (cid:9) =: (cid:94) D base (cid:100)(cid:79) id + id (cid:100)(cid:79) (cid:102) D + id (cid:100)(cid:79) (cid:102) D . (cid:94) D base is well-defined and essentially self-adjoint by a finite-dimensional argument. (cid:102) D is well-definedand essentially self-adjoint just like Definition-Theorem 5.29. For (cid:102) D , see [T3, Lemma 6.9].(2) The “smooth algebra” is the dense subspace C ∞ (cid:16) (cid:102) M × T × Π T F ( C ∞ c ( T × Π T , L )) (cid:17) (cid:100)(cid:79) alg F (cid:18) L (cid:16) U L k , L (cid:17) fin (cid:19) , where F ( V (cid:48) ) for a Hilbert space V and its dense subspace V (cid:48) ⊆ V is the set of finite rank operatorson V preserving V (cid:48) . Let us verify that the commutator of a (cid:100)(cid:79) k ∈ C ∞ (cid:16) (cid:102) M × T × Π T F ( C ∞ c ( T × Π T , L )) (cid:17) (cid:100)(cid:79) alg F (cid:18) L (cid:16) U L k , L (cid:17) fin (cid:19) (cid:101) D is a bounded operator. Since d rt commutes with k , and a commutes with the Clifford actionson S (cid:102) M and S U , we obtain [ a (cid:100)(cid:79) k, (cid:101) D ] = [ a, (cid:94) D base ] (cid:100)(cid:79) k + a (cid:100)(cid:79) [ k, (cid:102) D ] . The first term is a bounded operator, thanks to the ordinary argument of the descent ho-momorphism for unbounded Kasparov modules. For the second one, we put k = (cid:80) φ i (cid:98) ⊗ ψ ∗ i for φ i , ψ i ∈ L (cid:16) U L k , L (cid:17) fin . Since [ k, (cid:102) D ] = (cid:80) i [ φ i (cid:98) ⊗ ψ ∗ i , (cid:102) D ], it is sufficient to prove that each com-mutator is bounded. Thus, we may assume that k is a single Schatten form k = φ (cid:98) ⊗ ψ ∗ . For ξ (cid:98) ⊗ s (cid:98) ⊗ b ∈ L (cid:16) U L k , L (cid:17) fin (cid:98) ⊗ S U, fin (cid:98) ⊗ [ C (cid:111) − τ U L m ] fin ,[ k, (cid:102) D ]( ξ (cid:98) ⊗ s (cid:98) ⊗ b )= k (cid:88) n (cid:110) n − l dR (cid:48) ( z n )( ξ ) (cid:98) ⊗ γ ( z n ) ( s ) + n − l dR (cid:48) ( z n )( ξ ) (cid:98) ⊗ γ ( z n ) ( s ) (cid:111) (cid:98) ⊗ b − (cid:88) n (cid:110) n − l dR (cid:48) ( z n ) ◦ k ( ξ ) (cid:98) ⊗ γ ( z n ) ( s ) + n − l dR (cid:48) ( z n ) ◦ k ( ξ ) (cid:98) ⊗ γ ( z n ) ( s ) (cid:111) (cid:98) ⊗ b = (cid:88) n (cid:110) φ (cid:16) ψ (cid:12)(cid:12)(cid:12) n − l dR (cid:48) ( z n )( ξ ) (cid:17) (cid:98) ⊗ γ ( z n ) ( s ) + φ (cid:16) ψ (cid:12)(cid:12)(cid:12) n − l dR (cid:48) ( z n )( ξ ) (cid:17) (cid:98) ⊗ γ ( z n ) ( s ) (cid:111) (cid:98) ⊗ b − (cid:88) n (cid:110) n − l dR (cid:48) ( z n )[ φ ( ψ | ξ )] (cid:98) ⊗ γ ( z n ) ( s ) + n − l dR (cid:48) ( z n )[ φ ( ψ | ξ )] (cid:98) ⊗ γ ( z n ) ( s ) (cid:111) (cid:98) ⊗ b = (cid:88) n (cid:110) φ (cid:16) { n − l dR (cid:48) ( z n ) } ∗ ( ψ ) (cid:12)(cid:12)(cid:12) ξ (cid:17) (cid:98) ⊗ γ ( z n ) ( s ) + φ (cid:16) { n − l dR (cid:48) ( z n ) } ∗ ( ψ ) (cid:12)(cid:12)(cid:12) ξ (cid:17) (cid:98) ⊗ γ ( z n ) ( s ) (cid:111) (cid:98) ⊗ b − (cid:88) n (cid:110) n − l dR (cid:48) ( z n )( φ ) ( ψ | ξ ) (cid:98) ⊗ γ ( z n ) ( s ) + n − l dR (cid:48) ( z n )( φ ) ( ψ | ξ ) (cid:98) ⊗ γ ( z n ) ( s ) (cid:111) (cid:98) ⊗ b = (cid:88) n (cid:16)(cid:110)(cid:104) φ (cid:98) ⊗ [ { n − l dR (cid:48) ( z n ) } ∗ ( ψ )] ∗ (cid:105) (cid:98) ⊗ γ ( z n ) + (cid:104) φ (cid:98) ⊗ [ { n − l dR (cid:48) ( z n ) } ∗ ( ψ )] ∗ (cid:105) (cid:98) ⊗ γ ( z n ) (cid:111) (cid:98) ⊗ id (cid:17) ξ (cid:98) ⊗ s (cid:98) ⊗ b − (cid:88) n (cid:16)(cid:110)(cid:104) n − l dR (cid:48) ( z n )( φ ) (cid:98) ⊗ ψ ∗ (cid:105) (cid:98) ⊗ γ ( z n ) + (cid:104) n − l dR (cid:48) ( z n )( φ ) (cid:98) ⊗ ψ ∗ (cid:105) (cid:98) ⊗ γ ( z n ) (cid:111) (cid:98) ⊗ id (cid:17) ξ (cid:98) ⊗ s (cid:98) ⊗ b, where φ (cid:98) ⊗ [ { n − l dR (cid:48) ( z n ) } ∗ ( ψ )] ∗ stands for the Schatten form λ (cid:55)→ φ (cid:0) { n − l dR (cid:48) ( z n ) } ∗ ( ψ ) (cid:12)(cid:12) λ (cid:1) , andsimilarly for other terms of the last two lines. Thus, the commutator can be divided into four parts: (cid:88) n φ (cid:98) ⊗ [ { n − l dR (cid:48) ( z n ) } ∗ ( ψ )] ∗ (cid:98) ⊗ γ ( z n ) + (cid:88) n φ (cid:98) ⊗ [ { n − l dR (cid:48) ( z n ) } ∗ ( ψ )] ∗ (cid:98) ⊗ γ ( z n ) − (cid:88) n n − l dR (cid:48) ( z n )( φ ) (cid:98) ⊗ ψ ∗ (cid:98) ⊗ γ ( z n ) − (cid:88) n n − l dR (cid:48) ( z n )( φ ) (cid:98) ⊗ ψ ∗ (cid:98) ⊗ γ ( z n ) . Since dR (cid:48) ( z n ) ∗ = − dR (cid:48) ( z n ) is an “annihilator”, and since φ, ψ ∈ L ( U L k , L ) fin , the second andthird terms are finite sums of finite rank operators, which are obviously bounded.For the first and fourth terms, we prove that the infinite sums converge in operator norm. Wedeal with only the first one. For this aim, we take an orthonormal base of L ( U L k , L ). First, we put (cid:101) φ −→ α , −→ β := (cid:16) dR (cid:48) ( z ) α dL (cid:48) ( z ) β dR (cid:48) ( z ) α dL (cid:48) ( z ) β · · · (cid:17) vac , φ −→ α , −→ β := (cid:107) (cid:101) φ −→ α , −→ β (cid:107) − (cid:101) φ −→ α , −→ β . Since (cid:107) (cid:101) φ −→ α , −→ β (cid:107) = (cid:81) n (2 n l ) α n + β n α n ! β n !, we obtain dR (cid:48) ( z n ) φ −→ α , −→ β = (cid:113) n l ( α n + 1) φ −→ α + e n , −→ β , where −→ α + e n := ( · · · , α n − , α n + 1 , α n +1 , · · · ).We have assumed that ψ is a finite linear combination of φ −→ α , −→ β ’s. Thus, we may assume that ψ = φ −→ α , −→ β from the beginning, since a finite sum of bounded operators is again bounded. We have (cid:107) φ (cid:98) ⊗ [ { n − l dR (cid:48) ( z n ) } ∗ ( φ −→ α , −→ β )] ∗ (cid:98) ⊗ γ ( z n ) (cid:107) ≤ n − l/ (cid:107) φ (cid:107)√ α n + 1 . Since α n = 0 except for finitely many n ’s, the infinite sum (cid:80) n φ (cid:98) ⊗ [ { n − l dR (cid:48) ( z n ) } ∗ ( ψ )] ∗ (cid:98) ⊗ γ ( z n )converges in norm.(3) See also [T3, Lemma 6.12] for details of the following argument. We prove this propertyby the following two steps: For a ∈ C ( (cid:102) M × T × Π T K ( L ( T × Π T , L ))) and k ∈ K (cid:18) L (cid:16) U L k , L (cid:17)(cid:19) ,( a ) a (cid:99)(cid:78) k (cid:98) ⊗ id(1 + [ (cid:94) D base (cid:99)(cid:78) id + id (cid:99)(cid:78) (cid:102) D ] ) − is compact; ( b ) the difference a (cid:99)(cid:78) k (1 + [ (cid:94) D base (cid:99)(cid:78) id +id (cid:99)(cid:78) (cid:102) D ] ) − − a (cid:99)(cid:78) k (1 + [ (cid:94) D base (cid:99)(cid:78) id + id (cid:99)(cid:78) (cid:102) D + id (cid:99)(cid:78) (cid:102) D ] ) − is also compact.( a ) First, we recall how C (cid:111) τ LT L m -compact operators look like. Note that the tensor productof a C (cid:111) τ U L m -compact operator and a C (cid:111) τ ( T × Π T )-compact operator is a C (cid:111) τ LT L m -compactoperator. Since C (cid:111) τ U L m = K ( L ( R ∞ ) ∗ ), a Hilbert C (cid:111) τ U L m -module is isomorphic to V (cid:98) ⊗ L ( R ∞ )for some Hilbert space V . Thus, C (cid:111) τ U L m -compact operator is k (cid:98) ⊗ id L ( R ∞ ) for k ∈ K ( V ). Forexample, C (cid:111) − τ U L m = K ( L ( R ∞ )) can be regarded as L ( R ∞ ) (cid:98) ⊗ L ( R ∞ ) ∗ ∼ = L ( R ∞ ) ∗ (cid:98) ⊗ L ( R ∞ ) . With this observation, we consider the spectral decomposition of (cid:102) D . By the identification[ C (cid:111) − τ U L m ] fin (cid:98) ⊗ alg S U, fin ∼ = L ( R ∞ ) fin (cid:98) ⊗ alg L ( R ∞ ) ∗ fin (cid:98) ⊗ alg S U, fin , we rewrite (cid:102) D asid L ( U L k , L ) (cid:98) ⊗ id L ( R ∞ ) (cid:98) ⊗ D alg := id L ( U L k , L ) (cid:98) ⊗ id L ( R ∞ ) (cid:98) ⊗ (cid:88) n (cid:8) √ ndρ ∗ z n (cid:98) ⊗ γ ( z n ) + √ ndρ ∗ z n (cid:98) ⊗ γ ( z n ) (cid:9) . The spectral decomposition of D alg can be computed just like the proof of (2) of ( A ) of Lemma 5.27.The property we need is that D has discrete spectrum bounded below with finite multiplicity,that is to say, D = (cid:80) n λ n P n , where λ n ≥
0, { λ n | λ n < K } is finite for any K > P n isthe orthogonal projection onto a finite-dimensional subspace of L ( R ∞ ) ∗ fin (cid:98) ⊗ alg S U, fin , and P n ’s aremutually orthogonal.Thanks to this spectral decomposition, and thanks to the fact that (cid:94) D base (cid:99)(cid:78) id anti-commuteswith id (cid:99)(cid:78) (cid:102) D ,1 + [ (cid:94) D base (cid:100)(cid:79) id + id (cid:100)(cid:79) (cid:102) D ] = 1 + (cid:94) D base2 (cid:100)(cid:79) id + id (cid:100)(cid:79) id L ( U L k , L ) (cid:98) ⊗ id L ( R ∞ ) (cid:98) ⊗ (cid:88) n λ n P n = (cid:88) n (cid:26) λ n + (cid:94) D base2 (cid:27) (cid:100)(cid:79) id L ( U L k , L ) (cid:98) ⊗ id L ( R ∞ ) ∗ (cid:98) ⊗ P n . Thus, the operator a (cid:99)(cid:78) k (1 + (cid:94) D base2 (cid:99)(cid:78) id + id (cid:99)(cid:78) (cid:102) D ) − is rewritten as (cid:88) n a (cid:26) λ n + (cid:94) D base2 (cid:27) − (cid:100)(cid:79) k (cid:98) ⊗ id L ( R ∞ ) ∗ (cid:98) ⊗ P n . (cid:94) D base is the operator for the descent of the index element of a Dirac operator “ D base ”,and hence a (cid:26) λ n + (cid:94) D base2 (cid:27) − is C(cid:111) τ ( T × Π T )-compact. Moreover, since k is a compact operatorand since P n is a finite rank operator, k (cid:98) ⊗ id L ( R ∞ ) (cid:98) ⊗ P n is C (cid:111) τ U L m -compact. Thus, each summandis C (cid:111) τ LT L m -compact. Finally, thanks to the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a (cid:26) λ n + (cid:94) D base2 (cid:27) − (cid:100)(cid:79) k (cid:98) ⊗ id L ( R ∞ ) (cid:98) ⊗ P n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:107) a (cid:107) ·
11 + λ n · (cid:107) k (cid:107) , and the fact that P n ’s are mutually orthogonal, the above infinite sum converges in norm because λ n → ∞ as m → ∞ . Therefore, it is a C (cid:111) τ LT L m -compact operator.( b ) One can prove the statement by the argument of the proof of the latter half of [T3, Lemma6.12]. We leave the details to the reader. LT -spaces In this subsection, we define an proper LT -spaces version of the topological assembly map, and wewill prove that the value of it at [ L ] is given by the analytic index constructed in [T3]. Combiningthis result and Theorem 5.41, we will prove the main result of the present paper. Definition 6.25.
We define a homomorphism substituting for the topological assembly map ν τLT L m : R KK τLT L m ( M L m ; C ( M L m ) , C ( M L m )) → KK ( C , C (cid:111) τ LT L m )by ν τLT L m ( y ) := [ c M L m ] (cid:98) ⊗ fgt (cid:18) j τLT L m ( y ) (cid:19) (cid:98) ⊗ j τLT L m (cid:16) fgt[ S ] (cid:98) ⊗ (cid:104) (cid:94) d M L m (cid:105)(cid:17) . According to the index theorem explained in Section 3.2, the value of the topological assemblymap at the R KK -element corresponding to a vector bundle E , coincides with the index of the Diracoperator twisted by E . In order to prove a parallel result for our case, we recall the definition ofthe analytic index from [T3, Definition 6.18] Definition 6.26. (1) We define a pre-Hilbert
C (cid:111) τ LT L m -module structure on C ∞ c ( (cid:102) M , L (cid:102) M (cid:98) ⊗ S (cid:102) M ) (cid:100)(cid:79) alg ( C (cid:111) − τ U L m ) fin (cid:98) ⊗ S U, fin by the following operations: For f, f , f ∈ C ∞ c ( (cid:102) M , L (cid:102) M (cid:98) ⊗ S (cid:102) M ), φ, φ , φ ∈ ( C (cid:111) − τ U ) fin , s, s , s ∈ S U, fin , b (cid:99)(cid:78) b ∈ [ C (cid:111) τ ( T × Π T )] (cid:99)(cid:78) C (cid:111) τ U L m and γ ∈ ( T × Π T ) τ , • ( f (cid:99)(cid:78) φ (cid:98) ⊗ s ) · ( b (cid:99)(cid:78) b ) := (cid:82) ( T × Π T ) τ γ ( f ) b ( γ − ) dγ (cid:99)(cid:78) ( b ∨ ∗ φ ) (cid:98) ⊗ s ; and • (cid:16) f (cid:99)(cid:78) φ (cid:98) ⊗ s (cid:12)(cid:12)(cid:12) f (cid:99)(cid:78) φ (cid:98) ⊗ s (cid:17) C(cid:111) τ LT L m ( γ ) := (cid:82) (cid:102) M (cid:0) f ( x ) (cid:12)(cid:12) γ.f ( γ − .x ) (cid:1) dx [ φ ∗ ∗ φ ] ∨ ( s | s ) S U . E L .(2) On this Hilbert module, we define an unbounded operator D L by the following: D L := (cid:88) n c ( e i ) ◦ ∇ L (cid:102) M (cid:98) ⊗ S (cid:102) M e i (cid:100)(cid:79) id ( C(cid:111) − τ LT L m ) fin (cid:98) ⊗ S U + id L ( (cid:102) M, L (cid:102) M (cid:98) ⊗ S (cid:102) M ) (cid:100)(cid:79) (cid:88) n (cid:0) √ nd rt z n (cid:98) ⊗ γ ( z n ) + √ nd rt z n (cid:98) ⊗ γ ( z n ) (cid:1) . The self-adjoint extension of D L is also denoted by the same symbol. The analytic index ind( D L )is defined by [( E L , D L )] ∈ KK ( C , C (cid:111) τ LT L m ). Remark . The index is given by the exterior tensor product of the index of D base and the KK -element given in [T3, Definition 6.18]. Theorem 6.28. ν τLT L m ([ L ]) = ind( D L ) . Proof.
We prove the result by the following steps. (1) We will compute the Kasparov product (cid:104) c M L m (cid:105) (cid:98) ⊗ fgt (cid:18) j τLT L m ([ L ]) (cid:19) . Then, we will prove ν τLT L m ([ L ]) = ind( D L ). For this aim, we willprove the following: (2) The modules of both sides are isomorphic; (3) D L satisfies the connectioncondition; (4) It satisfies the positivity condition. In fact, (4) is immediate from the fact that theoperator of the KK -element given in (1) is 0.(1) Let us compute the Kasparov product (cid:104) c M L m (cid:105) (cid:98) ⊗ fgt (cid:18) j τLT L m ([ L ]) (cid:19) . As proved in Proposition6.6, (cid:104) c M L m (cid:105) is represented by (cid:16) C (cid:16) (cid:102) M × T × Π T L ( LT L k ) ∗ (cid:17) , , (cid:17) . As defined in Definition-Proposition 6.2, j τLT L m ([ L ]) is represented by (cid:18) C (cid:18) (cid:102) M × T × Π T (cid:26) K (cid:18) L (cid:16) LT L k , L (cid:17) , L (cid:16) LT L k (cid:17)(cid:19) (cid:98) ⊗L (cid:102) M (cid:27)(cid:19) , π (cid:111) τ lt , (cid:19) . Therefore, the Kasparov product we are computing is given by (cid:18) C (cid:18) (cid:102) M × T × Π T (cid:26) L (cid:16) LT L k , L (cid:17) ∗ (cid:98) ⊗L (cid:102) M (cid:27)(cid:19) , , (cid:19) . (2) We would like to prove that the Kasparov product of the above and j τLT L m (cid:16) fgt[ S ] (cid:98) ⊗ (cid:104) (cid:94) d M L m (cid:105)(cid:17) coincides with ind( D L ). First, we prove that the modules are isomorphic. C (cid:18) (cid:102) M × T × Π T [ L (cid:16) LT L k , L (cid:17) ∗ (cid:98) ⊗L (cid:102) M ] (cid:19) (cid:98) ⊗ C (cid:32)(cid:102) M × T × Π T K (cid:32) L (cid:18) LT L k , L (cid:19)(cid:33)(cid:33) L (cid:18) (cid:102) M × T × Π T (cid:26) L (cid:16) LT L k , L (cid:17) (cid:98) ⊗ C (cid:111) − τ LT L m (cid:98) ⊗ [ S U (cid:98) ⊗ S (cid:102) M ] (cid:27)(cid:19) ∼ = L (cid:102) M × T × Π T (cid:18) L (cid:16) LT L k , L (cid:17) ∗ (cid:98) ⊗L (cid:102) M,x (cid:19) (cid:98) ⊗ K (cid:32) L (cid:18) LT L k , L (cid:19)(cid:33) (cid:18) L (cid:16) LT L k , L (cid:17) (cid:98) ⊗ C (cid:111) − τ LT L m (cid:98) ⊗ S M ,x (cid:19) x ∈ (cid:102) M ∼ = L (cid:16) (cid:102) M × T × Π T (cid:110) C (cid:111) − τ LT L m (cid:98) ⊗L (cid:102) M (cid:98) ⊗ S M | (cid:102) M (cid:111)(cid:17) , V ∗ (cid:98) ⊗ K ( V ) V ∼ = C for a Hilbert space V . The isomorphism is given by[ φ (cid:98) ⊗ l ] (cid:98) ⊗ C (cid:32)(cid:102) M × T × Π T K (cid:32) L (cid:18) LT L k , L (cid:19)(cid:33)(cid:33) [ ψ (cid:98) ⊗ b (cid:98) ⊗ s ] (cid:55)→ (cid:104) φ, ψ (cid:105) b (cid:98) ⊗ l (cid:98) ⊗ s for φ : (cid:102) M → L (cid:16) LT L k , L (cid:17) ∗ , l : (cid:102) M → L (cid:102) M , ψ : (cid:102) M → L (cid:16) LT L k , L (cid:17) , b : (cid:102) M → C (cid:111) − τ LT L m and s : (cid:102) M → S M | (cid:102) M such that φ (cid:98) ⊗ l and ψ (cid:98) ⊗ b (cid:98) ⊗ s are equivariant sections.For the next step, we describe this isomorphism in detail. We can factorize L (cid:16) LT L k , L (cid:17) , L (cid:16) LT L k , L (cid:17) ∗ and C (cid:111) − τ LT L m into the T × Π T -part and the U L m -part. Since T × Π T acts on thelatter factor trivially, φ can be approximated by finite sums of functions of the form x (cid:55)→ φ ( x ) (cid:98) ⊗ φ for φ : (cid:102) M → L ( T × Π T , L ) ∗ and φ ∈ L (cid:16) U L k , L (cid:17) ∗ . Similarly, ψ , b and s can be approximatedby finite sums of functions of the following forms, respectively: x (cid:55)→ ψ ( x ) (cid:98) ⊗ ψ , x (cid:55)→ b ( x ) (cid:98) ⊗ b and x (cid:55)→ s ( x ) (cid:98) ⊗ s for ψ : (cid:102) M → L ( T × Π T , L ), ψ ∈ L (cid:16) U L k , L (cid:17) , b : (cid:102) M → C (cid:111) − τ ( T × Π T ), b ∈ C (cid:111) − τ U L m , s : (cid:102) M → S (cid:102) M and s ∈ S U . Using these factorizations, we identify the vectorspaces as follows: C (cid:18) (cid:102) M × T × Π T (cid:26) L (cid:16) LT L k , L (cid:17) ∗ (cid:98) ⊗L (cid:102) M (cid:27)(cid:19) ∼ = C (cid:16) (cid:102) M × T × Π T (cid:8) L ( T × Π T , L ) ∗ (cid:98) ⊗L (cid:102) M (cid:9)(cid:17) (cid:100)(cid:79) L (cid:16) U L k , L (cid:17) ∗ ; L (cid:18) (cid:102) M × T × Π T (cid:26) L (cid:16) LT L k , L (cid:17) (cid:98) ⊗ C (cid:111) − τ LT L m (cid:98) ⊗ S M | (cid:102) M (cid:27)(cid:19) ∼ = L (cid:16) (cid:102) M × T × Π T (cid:8) L ( T × Π T , L ) (cid:98) ⊗ C (cid:111) − τ ( T × Π T ) (cid:98) ⊗ S (cid:102) M (cid:9)(cid:17) (cid:100)(cid:79) L (cid:16) U L k , L (cid:17) (cid:98) ⊗ C (cid:111) − τ U L m (cid:98) ⊗ S U ; L (cid:16) (cid:102) M × T × Π T (cid:110) C (cid:111) − τ LT L m (cid:98) ⊗L (cid:102) M (cid:98) ⊗ S M | (cid:102) M (cid:111)(cid:17) ∼ = L (cid:16) (cid:102) M × T × Π T (cid:8) ( C (cid:111) − τ ( T × Π T )) (cid:98) ⊗L (cid:102) M (cid:98) ⊗ S (cid:102) M (cid:9)(cid:17) (cid:100)(cid:79) C (cid:111) − τ U L m (cid:98) ⊗ S U . Under these identifications, the isomorphism [ φ (cid:98) ⊗ l ] (cid:98) ⊗ [ ψ (cid:98) ⊗ b (cid:98) ⊗ s ] (cid:55)→ (cid:104) φ, ψ (cid:105) b (cid:98) ⊗ l (cid:98) ⊗ s can be described as (cid:18) φ (cid:98) ⊗ l (cid:100)(cid:79) φ (cid:19) (cid:98) ⊗ (cid:18) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:100)(cid:79) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:19) (cid:55)→ (cid:0) (cid:104) φ , ψ (cid:105) b (cid:98) ⊗ l (cid:98) ⊗ s (cid:1) (cid:100)(cid:79) (cid:0) (cid:104) φ , φ (cid:105) b (cid:98) ⊗ s (cid:1) . Note that (cid:104) φ , φ (cid:105) ( b (cid:98) ⊗ s ) is independent of x ∈ (cid:102) M .(3) Let us check the connection condition. Let us consider C ∞ c (cid:16) (cid:102) M × T × Π T (cid:2) L ( T × Π T , L ) ∗ (cid:98) ⊗L (cid:102) M (cid:3)(cid:17) (cid:100)(cid:79) alg L (cid:16) U L k , L (cid:17) ∗ fin . It is a dense subspace of C (cid:18) (cid:102) M × T × Π T (cid:26) L (cid:16) LT L k , L (cid:17) ∗ (cid:98) ⊗L (cid:102) M (cid:27)(cid:19) . We prove that the commutator (cid:20)(cid:18) D L (cid:101) D (cid:19) , (cid:18) T e T ∗ e (cid:19)(cid:21) = (cid:32) D L ◦ T e − T e ◦ (cid:101) D (cid:101) D ◦ T ∗ e − T ∗ e ◦ D L (cid:33) e of the above dense subspace. We may assume that e is a finite sum ofelements of the form φ (cid:98) ⊗ l (cid:99)(cid:78) φ for φ : (cid:102) M → L ( T × Π T , L ) ∗ , l : (cid:102) M → L (cid:102) M and φ ∈ L (cid:16) U L k , L (cid:17) ∗ fin satisfying the equivariance condition g · [ φ (cid:98) ⊗ l ( g − x )] = φ (cid:98) ⊗ l ( x ) for g ∈ T × Π T . We may assume e = φ (cid:98) ⊗ l (cid:99)(cid:78) φ from the beginning, because a finite sum of bounded operators is again bounded. D L ◦ T φ (cid:98) ⊗ l (cid:99)(cid:78) φ (cid:18) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:100)(cid:79) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:19) − T φ (cid:98) ⊗ l (cid:99)(cid:78) φ ◦ (cid:101) D (cid:18) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:100)(cid:79) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:19) = D L (cid:104) φ , ψ (cid:105) ( b (cid:98) ⊗ l (cid:98) ⊗ s ) (cid:100)(cid:79) (cid:104) φ , ψ (cid:105) ( b (cid:98) ⊗ s ) − T φ (cid:98) ⊗ l (cid:99)(cid:78) φ (cid:88) j c ( e j ) ◦ ∇ S (cid:102) M e j (cid:18) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:100)(cid:79) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:19) − T φ (cid:98) ⊗ l (cid:99)(cid:78) φ (cid:32)(cid:0) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:1) (cid:100)(cid:79) (cid:88) n (cid:110)(cid:16) n − l dR (cid:48) ( z n ) ψ (cid:98) ⊗ b + ψ (cid:98) ⊗√ nd rt z n b (cid:17) (cid:98) ⊗ γ ( z n ) ( s ) (cid:111)(cid:33) − T φ (cid:98) ⊗ l (cid:99)(cid:78) φ (cid:32)(cid:0) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:1) (cid:100)(cid:79) (cid:88) n (cid:110)(cid:16) n − l dR (cid:48) ( z n ) ψ (cid:98) ⊗ b + ψ (cid:98) ⊗√ nd rt z n b (cid:17) (cid:98) ⊗ γ ( z n ) ( s ) (cid:111)(cid:33) = (cid:88) j c ( e j ) ◦ ∇ L (cid:102) M (cid:98) ⊗ S (cid:102) M e j (cid:8) (cid:104) φ , ψ (cid:105) ( b (cid:98) ⊗ l (cid:98) ⊗ s ) (cid:9) (cid:100)(cid:79) (cid:104) φ , ψ (cid:105) ( b (cid:98) ⊗ s )+ (cid:104) φ , ψ (cid:105) ( b (cid:98) ⊗ l (cid:98) ⊗ s ) (cid:100)(cid:79) (cid:104) φ , ψ (cid:105) (cid:88) n (cid:0) √ nd rt z n b (cid:98) ⊗ γ ( z n ) ( s ) + √ nd rt z n b (cid:98) ⊗ γ ( z n ) ( s ) (cid:1) − T φ (cid:98) ⊗ l (cid:99)(cid:78) φ (cid:88) j c ( e j ) ◦ ∇ S (cid:102) M e j (cid:18) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:100)(cid:79) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:19) − T φ (cid:98) ⊗ l (cid:99)(cid:78) φ (cid:32)(cid:0) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:1) (cid:100)(cid:79) (cid:88) n (cid:110)(cid:16) n − l dR (cid:48) ( z n ) ψ (cid:98) ⊗ b + ψ (cid:98) ⊗√ nd rt z n b (cid:17) (cid:98) ⊗ γ ( z n ) ( s ) (cid:111)(cid:33) − T φ (cid:98) ⊗ l (cid:99)(cid:78) φ (cid:32)(cid:0) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:1) (cid:100)(cid:79) (cid:88) n (cid:110)(cid:16) n − l dR (cid:48) ( z n ) ψ (cid:98) ⊗ b + ψ (cid:98) ⊗√ nd rt z n b (cid:17) (cid:98) ⊗ γ ( z n ) ( s ) (cid:111)(cid:33) =: ∆ + ∆ + ∆ + ∆ + ∆ . We prove that the unbounded terms are cancelled out. Let us begin with the computation of ∆ .The trivial connection on the trivial bundles (cid:102) M × L (cid:16) U L k , L (cid:17) , (cid:102) M × C (cid:111) − τ U L m or (cid:102) M × S U , is106enoted by ∇ . Thanks to the Leibniz rule,∆ = (cid:88) j c ( e j ) (cid:8)(cid:10) ∇ e j φ , ψ (cid:11) b (cid:98) ⊗ l (cid:98) ⊗ s + (cid:10) φ , ∇ e j ψ (cid:11) b (cid:98) ⊗ l (cid:98) ⊗ s (cid:9) (cid:100)(cid:79) (cid:104) φ , ψ (cid:105) b (cid:98) ⊗ s + (cid:88) j (cid:104) φ , ψ (cid:105) c ( e j ) (cid:16) ∇ e j b (cid:98) ⊗ l (cid:98) ⊗ s + b (cid:98) ⊗∇ L e j l (cid:98) ⊗ s + b (cid:98) ⊗ l (cid:98) ⊗∇ S (cid:102) M e j s (cid:17) (cid:100)(cid:79) (cid:104) φ , ψ (cid:105) b (cid:98) ⊗ s = (cid:88) j c ( e j ) (cid:110)(cid:10) ∇ e j φ , ψ (cid:11) b (cid:98) ⊗ l (cid:98) ⊗ s + (cid:104) φ , ψ (cid:105) b (cid:98) ⊗∇ L e j l (cid:98) ⊗ s (cid:111) (cid:100)(cid:79) (cid:104) φ , ψ (cid:105) b (cid:98) ⊗ s + (cid:88) j (cid:104) φ , ψ (cid:105) c ( e j ) (cid:16) ∇ e j b (cid:98) ⊗ l (cid:98) ⊗ s + b (cid:98) ⊗ l (cid:98) ⊗∇ S (cid:102) M e j s (cid:17) (cid:100)(cid:79) (cid:104) φ , ψ (cid:105) b (cid:98) ⊗ s + (cid:88) j (cid:10) φ , ∇ e j ψ (cid:11) b (cid:98) ⊗ l (cid:98) ⊗ c ( e j ) s (cid:100)(cid:79) (cid:104) φ , ψ (cid:105) b (cid:98) ⊗ s . Note that the assignment ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:99)(cid:78) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:55)→ “the first line of the above” does not containderivations. Therefore, it is a bounded operator (notice that a finite sum of bounded operators isbounded). The remainder terms are canceled by ∆ thanks to the Leibniz rule.Finally, ∆ + ∆ + ∆ is − T φ (cid:98) ⊗ l (cid:99)(cid:78) φ (cid:32) [ ψ (cid:98) ⊗ b (cid:98) ⊗ s ] (cid:100)(cid:79) (cid:88) n (cid:8) n − l dR (cid:48) ( z n ) ψ (cid:98) ⊗ b (cid:98) ⊗ γ ( z n ) ( s ) + n − l dR (cid:48) ( z n ) ψ (cid:98) ⊗ b (cid:98) ⊗ γ ( z n ) ( s ) (cid:9)(cid:33) = − (cid:104) φ , ψ (cid:105) b (cid:98) ⊗ l (cid:98) ⊗ s (cid:100)(cid:79) (cid:88) n (cid:0)(cid:10) φ , n − l dR (cid:48) ( z n ) ψ (cid:11) b (cid:98) ⊗ γ ( z n ) ( s ) + (cid:10) φ , n − l dR (cid:48) ( z n ) ψ (cid:11) b (cid:98) ⊗ γ ( z n ) ( s ) (cid:1) = − (cid:104) φ , ψ (cid:105) b (cid:98) ⊗ l (cid:98) ⊗ s (cid:100)(cid:79) (cid:88) n (cid:0)(cid:10) n − l [ dR (cid:48) ( z n )] ∗ φ , ψ (cid:11) b (cid:98) ⊗ γ ( z n ) ( s ) + (cid:10) n − l [ dR (cid:48) ( z n )] ∗ φ , ψ (cid:11) b (cid:98) ⊗ γ ( z n ) ( s ) (cid:1) . For the same reason of Theorem 6.24, the correspondence ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:100)(cid:79) ψ (cid:98) ⊗ b (cid:98) ⊗ s (cid:55)→ ∆ + ∆ + ∆ is bounded.One can prove that (cid:101) D ◦ T ∗ e − T ∗ e ◦ D L is bounded in the same way of the above. Therefore, D L satisfies the connection condition.(4) Since [0 (cid:98) ⊗ id , D L ] = 0 ≥ D L satisfies the positivity condition.Combining it and Theorem 5.41, we obtain the following main result of the present paper. Corollary 6.29. ν τLT (PD[ (cid:101) D ]) = σ S ε (cid:16) ind( D L ) (cid:17) . As a concluding remark of the present paper, we explain what we should do in order to com-plete the index theory for proper LT -spaces. Then, we will give a comment to improve ourtheory as mentioned in Remarks 5.18 (3). Roughly speaking, this “improvement” is to replace KK LT L m ( A ( M L k ) , S ε ) with KK LT L k ( A ( M L k ) , S ε ). We will explain why we needed to use theformer one in the present paper, and why we want to replace it.107ecall the big diagram in the proof of Proposition 3.9. The infinite-dimensional version of itshould be of the following form. Dotted arrows have not been defined. Undefined symbols will beexplained later as Conjecture 6.30. We omit the subscripts of LT and M for simplicity. KK τLT ( A ( M ) , S ε ) (cid:103) j τLT (cid:15) (cid:15) PD (cid:44) (cid:44) KK τLT ( A ( M ) , A ( M )) − (cid:98) ⊗ [ (cid:103) d M(cid:48) ] (cid:111) (cid:111) (cid:103) j τLT (cid:15) (cid:15) R KK τLT ( M ; S ε (cid:98) ⊗ C ( M ) , S ε (cid:98) ⊗ C ( M )) (cid:102) fgt (cid:111) (cid:111) j τLT (cid:15) (cid:15) KK ( S ε (cid:98) ⊗ (cid:63) , S ε (cid:111) τ LT ) σ S ε [ c M ] (cid:98) ⊗− (cid:15) (cid:15) KK ( S ε (cid:98) ⊗ (cid:63) , S ε (cid:98) ⊗ (cid:63) ) − (cid:98) ⊗ j τLT (fgt[ S ] (cid:98) ⊗ [ (cid:103) d M ]) (cid:111) (cid:111) σ S ε [ c M ] (cid:98) ⊗− (cid:15) (cid:15) R KK ( M /LT ; S ε (cid:98) ⊗ (cid:63) , S ε (cid:98) ⊗ (cid:63) ) fgt (cid:111) (cid:111) KK ( S ε , S ε (cid:111) τ LT ) KK ( S ε , S ε (cid:98) ⊗ (cid:63) ) , − (cid:98) ⊗ (cid:103) j τLT (fgt[ S ] (cid:98) ⊗ [ (cid:103) d M ]) (cid:111) (cid:111) where (cid:63) stands for C ( M ) (cid:111) LT and (cid:63) stands for C ( M ) (cid:111) τ LT . [ (cid:103) d M(cid:48) ] ∈ KK LT ( A ( M ) , S ε ) is thereformulated Dirac element defined in [T4] (we change the symbol from this paper). Conjecture 6.30. (1)
We can define a substitute for the descent homomorphism (cid:93) j pτLT,q : KK qτLT ( A ( M ) , S ε ) → KK ( S ε (cid:98) ⊗ C ( M ) (cid:111) ( p − q ) τ LT , S ε (cid:111) pτ LT ) and satisfies j τLT (fgt[ S ] (cid:98) ⊗ [ (cid:103) d M ]) = (cid:93) j τLT, ([ (cid:103) d M(cid:48) ]) . (2) We can define a substitute for the descent homomorphism (cid:93) j pτLT,q : KK qτLT ( A ( M ) , A ( M )) → KK ( S ε (cid:98) ⊗ C ( M ) (cid:111) ( p − q ) τ LT , S ε (cid:98) ⊗ C ( M ) (cid:111) pτ LT ) satisfying the following: For x ∈ KK q τLT ( A ( M ) , A ( M )) and y ∈ KK q τLT ( A ( M ) , S ε ) , j pτLT,q + q ( x (cid:98) ⊗ y ) = j ( p − q ) τLT,q ( x ) (cid:98) ⊗ j pτLT,q ( y ) . As a corollary, the top left corner of the above big diagram commutes. (3)
We can define a homomorphism (cid:102) fgt : R KK qτLT ( X ; S ε (cid:98) ⊗ C ( M ) , S ε (cid:98) ⊗ C ( M )) → KK qτLT ( A ( M ) , A ( M )) and it satisfies the following commutative diagram R KK qτLT ( M ; S ε (cid:98) ⊗ C ( M ) , S ε (cid:98) ⊗ C ( M )) (cid:102) fgt −−−−→ KK qτLT ( A ( M ) , A ( M )) (cid:103) j pτLT (cid:121) (cid:121) j pτLT R KK ( M /LT ; S ε (cid:98) ⊗ (cid:63) , S ε (cid:98) ⊗ (cid:63) ) fgt −−−−→ KK ( S ε (cid:98) ⊗ (cid:63) , S ε (cid:98) ⊗ (cid:63) ) , where (cid:63) := C ( M ) (cid:111) ( p − q ) τ LT and (cid:63) := C ( M ) (cid:111) pτ LT . As a corollary, the top right corner of theabove big diagram commutes. (4) The homomorphisms PD and y (cid:55)→ (cid:102) fgt( y ) (cid:98) ⊗ [ (cid:103) d M(cid:48) ] are mutually inverse. µ τLT ( x ) := σ S ε [ c M ] (cid:98) ⊗ (cid:103) j τLT ( x ) for x ∈ KK LT ( A ( M ) , S ε ), and the index theorem type equal-ity µ τLT ( x ) = ν τLT (PD( x )) will hold. By the results of the present paper, we have µ τLT ([ (cid:101) D ]) = σ S ε (cid:16) ind( D L ) (cid:17) . If the index theorem is completed, we will have various applications as we haveexplained in Section 1.Next, let us discuss the “improvement” we have in mind. We will explain the reason why weneed it. Unfortunately, we do not have concrete ideas on this problem. From now on, we write thesubscripts L k ’s again, because it is essential.Seeing Definition 5.35, everyone would wish the following. Conjecture 6.31.
There is a “collect” index element [ (cid:101) D ] ∈ KK LT L k ( A ( M L k ) , S ε ) such that i ∗ m,k [ (cid:101) D ] ∈ KK LT L m ( A ( M L k ) , S ε ) is the index element defined in this paper, where i m,k : LT L m → LT L k is the canonical embeddingand i ∗ m,k is the restriction of the LT L k -action to LT L m . Probably, our direct and naive method does not work for this problem. In order to explain thereason of this, we explain the reason why we needed to use KK LT L m ( A ( M L k ) , S ε ).Let us attempt to do the same thing of Section 5.2 for l = 0 and m = k . Then, vac ∈ L ( U L k )looks like the Gaussian “ (cid:81) N π − / e − ( x N + y N ) / ”. Thus, if we try to define π : A HKT ( U L k ) →L S ( S (cid:98) ⊗H ) in the same way of the present paper, for f e ( t ) := e − rt ∈ S for arbitrary r > π ( β ∞ ( f e ))vac looks like π −∞ / e − ( + r ) (cid:80) N ( x N + y N ) . By the eigenfunction expansion, vac NN := π − / e − ( + r )( x N + y N ) can be written as (cid:88) α,β c α,β dR (cid:48) ( z N ) α dL (cid:48) ( z N ) β vac NN (cid:107) dR (cid:48) ( z N ) α dL (cid:48) ( z N ) β vac NN (cid:107) , where dR (cid:48) ( z N )’s are defined for l = 0. Since c α,β is independent of N and | c α,β | < (cid:107) π ( β ∞ ( f e ))vac (cid:107) ≤ (cid:107) f e (cid:107)(cid:107) vac (cid:107) = 1), the infinite tensor product π ( β ∞ ( f e ))vac = (cid:79) N (cid:88) α N ,β N c α N ,β N dR (cid:48) ( z N ) α N dL (cid:48) ( z N ) β N vac NN (cid:107) dR (cid:48) ( z N ) α N dL (cid:48) ( z N ) β N vac NN (cid:107) = (cid:88) α ,β ,α ,β , ··· (cid:32)(cid:89) N c α N ,β N (cid:33) dR (cid:48) ( z ) α dL (cid:48) ( z ) β dR (cid:48) ( z ) α dL (cid:48) ( z ) β · · · vac (cid:107) dR (cid:48) ( z ) α dL (cid:48) ( z ) β dR (cid:48) ( z ) α dL (cid:48) ( z ) β · · · vac (cid:107) must be zero. For example, the coefficient of “vac” is “( c , ) ∞ ”, which is zero because | c , | < l is positive to define the ∗ -homomorphism π .This argument is rather algebraic. From the analytic or geometric point of view, the troubleof this observation is due to the fact that π ( β ∞ ( f e )) is “too localized to give an operator on the L -space”. Therefore, in [T4], a strange definition of π is adopted, so that π ( a ) looks like an“asymptotically constant function”. Conversely, in the present paper, we adopted a strange Hilbert109pace so that each element of H looks like an “asymptotically Dirac δ -function”. The cost we havepaid is that the natural U L m -action on our Hilbert space L ( U L k ) does not extend to U L k . Thismade us work in the strange setting: The Poincar´e duality homomorphism is a homomorphism fromthe KK -group of A ( M L k ) to R KK LT L m ( M L m ; − , − )-group.As long as we regard proper LT -spaces as ILH-manifolds, the above problem is not essential.The statement in the form of Section 4.4 looks natural. Moreover, this setting is convenient fromthe view point of LT -equivariant KK -theory, Definition 4.41.However, if one hopes to fix the Sobolev level and regard a proper LT -space as a fixed Hilbertmanifold, our construction is not satisfying, and the above conjecture must be investigated. Acknowledgments
An essential part of Section 5 was done during my stay in Pennsylvania State University. I amdeeply grateful to professor Nigel Higson, Yiannis Loizides and Shintaro Nishikawa for discussionsheld there. Especially, the primitive idea of R KK -theory for non-locally compact spaces is due toShintaro Nishikawa. I again thank him for permission to introduce it before our joint paper [NT].I am also grateful for the warm hospitality of Penn State during my stay.I am supported by JSPS KAKENHI Grant Number 18J00019. References [AMM] A. Alekseev, A. Malkin and E. Meinrenken, “
Lie group valued moment maps ”, J. Differen-tial Geom. 48 (1998), no. 3, 445-495.[ASe] M. Atiyah and G. Segal, “
The index of elliptic operators: II ”, Ann. of Math. (2) 87 1968531-545.[ASi1] M. Atiyah and I. Singer, “
The index of elliptic operators: I ”, Ann. of Math. (2) 87 1968484-530.[ASi2] M. Atiyah and I. Singer, “
The index of elliptic operators: III ”, Ann. of Math. (2) 87 1968546-604.[Ati] M. Atiyah, “
Bott periodicity and the index of elliptic operators ”, Quart. J. Math. Oxford Ser.(2) 19 (1968), 113-140.[BGV] N. Berline, E. Getzler and M. Vergne “
Heat kernels and Dirac operators ”, Corrected reprintof the 1992 original. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004.[Bil] L. Biliotti, “
Some results on infinite dimensional Riemannian geometry ” Acta Sci. Math.(Szeged) 72 (2006), no. 1-2, 387-405.[Blac] B. Blackadar, “ K -theory for operator algebras ”, Mathematical Sciences Research InstitutePublications, 5. Cambridge University Press, Cambridge, 1998.[Blan] E. Blanchard, “ D´eformations de C ∗ -alg`ebres de Hopf ”, Bull. Soc. Math. France 124 (1996),no. 1, 141-215. 110Dix] Dixmier, “ C ∗ -algebras ”, Translated from the French by Francis Jellett. North-Holland Math-ematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.[DT] D. Dumitrascu and J. Trout, “ On C ∗ -algebras and K -theory for infinite-dimensional Fredholmmanifolds ”, Topology Appl. 153 (2006), no. 14, 2528-2550.[EE] S. Echterhoff and H. Emerson, “ Structure and K -theory of crossed products by proper ac-tions ”, Expo. Math. 29 (2011), no. 3, 300-344.[FHT1] D. Freed, M. J. Hopkins, and C. Teleman, “ Loop groups and twisted K-theory I ”, J. Topol.4 (2011), no. 4, 737-798.[FHT2] D. Freed, M. J. Hopkins, and C. Teleman, “
Loop groups and twisted K-theory II ”, J. Amer.Math. Soc. 26 (2013), no. 3, 595-644.[FHT3] D. Freed, M. J. Hopkins, and C. Teleman, “
Loop groups and twisted K-theory III ”, Ann.of Math. (2) 174 (2011), no. 2, 947-1007.[Fre] D. Freed, “
The geometry of loop groups ”, J. Differential Geom. 28 (1988), no. 2, 223-276.[Fur] M. Furuta, “
Index theorem ”, (Japanese) Iwanami Shoten, 2008.[GWY] S. Gong, J. Wu and G. Yu, “
The Novikov conjecture, the group of volume preservingdiffeomorphisms and Hilbert-Hadamard spaces ”, preprint arXiv:1811.02086v3.[HG] N. Higson and E. Guentner, “
Group C ∗ -algebras and K -theory ”, Noncommutative geometry,137-251, Lecture Notes in Math., 1831, Fond. CIME/CIME Found. Subser., Springer, Berlin,2004.[HK] N. Higson and G. Kasparov, “ E -theory and KK -theory for groups which act properly andisometrically on Hilbert space ”, Invent. Math. 144 (2001), no. 1, 23-74.[HKT] N. Higson, G. Kasparov and J. Trout, “ A Bott periodicity theorem for infinite-dimensionalEuclidean space ”, Adv. Math. 135 (1998), no. 1, 1-40.[JT] K. Jensen and K. Thomsen, “
Elements of KK -theory ”, Mathematics: Theory and Applica-tions. Birkh¨auser Boston, Inc., Boston, MA, 1991.[Kas1] G. Kasparov, “ Operator K -theory and its applications: elliptic operators, group repre-sentations, higher signatures, C ∗ -extensions. ”, Proceedings of the International Congress ofMathematicians, Vol. 1, 2 (Warsaw, 1983), 987-1000, PWN, Warsaw, 1984.[Kas2] G. Kasparov, “ Equivariant KK -theory and the Novikov conjecture ”, Invent. Math. 91(1988), no. 1, 147-201.[Kas3] G. Kasparov, “ Elliptic and transversally elliptic index theory from the viewpoint of KK -theory ”, J. Noncommut. Geom. 10 (2016), no. 4, 1303-1378.[Kir] A. A. Kirillov, “ Lectures on the orbit methods ”, Graduate Studies in Mathematics, 64. Amer-ican Mathematical Society, Providence, RI, 2004.111KN1] S. Kobayashi and K. Nomizu, “
Foundations of differential geometry, volume I ”, Reprint ofthe 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley andSons, Inc., New York, 1996.[KN2] S. Kobayashi and K. Nomizu, “
Foundations of differential geometry, volume II ”, Reprint ofthe 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley andSons, Inc., New York, 1996.[Kos] B. Kostant, “
A cubic Dirac operator and the emergence of Euler number multiplets of repre-sentations for equal rank subgroups ”, Duke Math. J. 100 (1999), no. 3, 447-501.[Kuc] D. Kucerovsly, “
The KK -product of unbounded modules ”, K -Theory 11 (1997), no. 1, 17-34.[KW] E. Kirchberg and S. Wassermann, “ Operations on continuous bundles of C ∗ -algebras ”, Math.Ann. 303 (1995), no. 4, 677-697.[Lan] G. Landweber, “ Dirac operators on loop spaces ”, Thesis (Ph.D.)-Harvard University. 1999.56 pp.[LG] P. Y. Le Gall, Th´eorie de Kasparov ´equivariante et groupo¨ıdes. I, K-Theory 16 (1999), no. 4,361-390,[LM] M. Lesch and B. Mesland, “
Sums of regular selfadjoint operators in Hilbert- C ∗ -modules , J.Math. Anal. Appl. 472 (2019), no. 1, 947-980.[LMS] Y. Loizides, E. Meinrenken and Y. Song, “ Spinor modules for Hamiltonian loop groupspaces ”, arXiv:1706.07493.[LS] Y. Loizides and Y. Song, “
Quantization of Hamiltonian loop group spaces ”, Math. Ann. 374(2019), no. 1-2, 681-722.[Loi] Y. Loizides, “
Geometric K -homology and the Freed-Hopkins-Teleman theorem ”,arXiv:1804.05213.[McD] D. McDuff, “ The moment map for circle actions on symplectic manifolds ”, J. Geom. Phys.5 (1988), no. 2, 149-160.[Mei1] E. Meinrenken, “
Twisted K -homology and group-valued moment maps ”, Int. Math. Res.Not. IMRN 2012, no. 20, 4563-4618.[Mei2] E. Meinrenken, “ Clifford algebras and Lie theory ”, Ergebnisse der Mathematik und ihrerGrenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematicsand Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 58. Springer,Heidelberg, 2013.[Miy] S. Miyajima, “
Functional analysis ”, (in Japanese), Yokohama Tosho, 2005.[MS] J. Milnor and J. Stasheff, “
Characteristic classes ”, Annals of Mathematics Studies, No. 76.Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. vii+331pp.[Mur] G. Murphy, “ C ∗ -algebras and operator theory ”, Academic Press, Inc., Boston, MA, 1990.112MW] E. Meinrenken and C. Woodward, “ Hamiltonian loop group actions and Verlinde factoriza-tion ”, J. Differential Geom. 50 (1998), no. 3, 417-469.[Nil] M. Nilsen, “ C ∗ -bundles and C ( X ) -algebras ”, Indiana Univ. Math. J. 45 (1996), no. 2, 463-477.[NT] S. Nishikawa and D. Takata, “ Non-locally compact groupoid equivariant KK -theory ”, inpreparation.[Omo] H. Omori, “ Infinite-dimensiona Lie groups ”. Translated from the 1979 Japanese original andrevised by the author. Translations of Mathematical Monographs, 158. American MathematicalSociety, Providence, RI, 1997.[PS] A. Pressery and G. Segal, “
Loop groups ”, Oxford Science Publications. The Clarendon Press,Oxford University Press, New York, 1986.[Roe] J. Roe, “
Elliptic operators, topology and asymptotic methods ”, Second edition. Pitman Re-search Notes in Mathematics Series, 395. Longman, Harlow, 1998.[Sak] T. Sakai, “
Riemannian geometry ”,Translated from the 1992 Japanese original by the author.Translations of Mathematical Monographs, 149. American Mathematical Society, Providence,RI, 1996.[Son] Y. Song, “
Dirac operators on quasi-Hamiltonian G -spaces ”, J. Geom. Phys. 106 (2016), 70-86.[T1] D. Takata, “ An analytic LT -equivariant index and noncommutative geometry ”, J. Noncom-mut. Geom. 13 (2019), no. 2, 553-586.[T2] D. Takata, “ LT-equivariant index from the viewpoint of KK-theory. A global analysis on theinfinite-dimensional Heisenberg group ”, J. Geom. Phys. 150 (2020), 103591, 30 pp.[T3] D. Takata, “
A loop group equivariant analytic index theory for infinite-dimensional manifolds ”,PhD thesis, available on https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/232217/2/drigk04334.pdf [T4] D. Takata, “
An infinite-dimensional index theorem and the Higson-Kasparov-Trout algebra ”,arXiv:1811.06811.[Tro] J. Trout, “
A Thom isomorphism for infinite rank Euclidean bundle ”, Homology HomotopyAppl. 5 (2003), no. 1, 121-159.[TXLG] J. L. Tu, P. Xu, C. laurent-Gengoux “
Twisted K -theory of differential stacks ”, Ann. Sci.´Ecole Norm. Sup. (4) 37 (2004), no. 6, 841-910.[Was] A. Wasserman, “ Kac-Moody and Virasoro algebras ”, lecture notes, available asarXiv:1004.1287.[Wil] D. Williams, “
Crossed products of C ∗ -algebras ”, Mathematical Surveys and Monographs,134. American Mathematical Society, Providence, RI, 2007.[Wit] E. Witten, “ The index of the Dirac operator in loop space ”, Elliptic curves and modular formsin algebraic topology (Princeton, NJ, 1986), 161-181, Lecture Notes in Math., 1326, Springer,Berlin, 1988. 113Wor] S. Woronowicz, “
Unbounded elements affiliated with C ∗ -algebras and non-compact quantumgroups ”, Comm. Math. Phys. 136 (1991), no. 2, 399-432.[Yu] G. Yu, “ K -theory of C ∗ -algebras associated to infinite-dimensional manifolds (Talk) ”, NCGFestival 2019 held in Washington University in St. Louis, May 3rd (2019).Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba Meguro-kuTokyo, JapanE-mail address: [email protected]@ms.u-tokyo.ac.jp