An infinite-dimensional index theorem and the Higson-Kasparov-Trout algebra
AAn infinite-dimensional index theorem and theHigson-Kasparov-Trout algebra
Doman TakataThe University of TokyoNovember 19, 2018
Abstract
We have been studying the index theory for some special infinite-dimensional manifolds with a “propercocompact” actions of the loop group LT of the circle T , from the viewpoint of the noncommutativegeometry. In this paper, we will introduce the LT -equivariant KK -theory and we will construct three KK -elements: the index element, the Clifford symbol element and the Dirac element. These elements satisfya certain relation, which should be called the ( KK -theoretical) index theorem, or the KK -theoreticalPoincar´e duality for infinite-dimensional manifolds. We will also discuss the assembly maps. Contents LT -equivariant KK -theory and the problem 19 LT -equivariant KK -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 The definition of the Higson-Kasparov-Trout algebra S C ( U ) . . . . . . . . . . . . . . . . . . 254.3 A group action and a representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (cid:101) (cid:1) ∂ ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Dirac element [ (cid:102) d U ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Clifford symbol element [ (cid:103) σ Cl (cid:1) ∂ ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4 The KK -theoretical index theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 a r X i v : . [ m a t h . K T ] N ov Assembly maps 49
The Atiyah-Singer index theorem is one of the monumental work in differential topology [ASi68a, ASi68b].The original theorem was stated for closed manifolds, but it has been generalized to many cases, for example:closed manifolds with compact group action [ASe68], families of closed manifolds [ASi71], operator algebras[Con85, Con94], and complete Riemannian manifolds with isometric, proper and cocompact group actions[Kas83, Kas16]. Our dream is to formulate and prove an infinite-dimensional version of the Atiyah-Singerindex theorem. In particular, we would like to study an infinite-dimensional version of [Kas83, Kas16].As a first step of this dream, we have been studying the index problem for some special infinite-dimensionalmanifolds in [Tak1, Tak2, Tak3]. [Tak3] is the PhD thesis of the author, which contains [Tak2], most of [Tak1]and some detailed study on proper LT -spaces which is to be explained soon. The following is a rough versionof our problem of the present paper. Let T be the circle group, and let LT be its loop group C ∞ ( S , T ).Precise definitions will be explained later. Problem 1.1.
For an infinite-dimensional proper LT -space M equipped with a “ τ -twisted” LT -equivariantline bundle L → M , and an LT -equivariant Spinor bundle S → M , study an index problem of the “Diracoperator” D on L ⊗ S , from the viewpoint of noncommutative geometry.In order to explain the results of the present paper, we need to recall the Kasparov index theorem. Let X be a finite-dimensional complete Riemannian Spin c -manifold, and let Γ be a locally compact, secondcountable and Hausdorff group. Let Cl τ ( X ) := C ( X, Cliff ( T ∗ X )) be the Clifford algebra-valued functionalgebra. Suppose that Γ acts on X isometrically, properly and cocompactly. Then, the following twofundamental KK -elements are defined: the Dirac element [ d X ] ∈ KK Γ ( Cl τ ( X ) , C ) and the Mishchenkoline bundle [ c ] ∈ KK ( C , C ( X ) (cid:111) Γ). We also suppose that the Spinor bundle S of X is Γ-equivariant, anda Γ-equivariant Dirac operator D is given. Since X is not supposed to be compact, D can be non-Fredholm.However, in this situation, D is a “ C ∗ (Γ)-Fredholm operator” as follows. Fact 1.2 ([Kas83, Kas16]) . (1) An analytic index ind( D ) ∈ KK ( C , C ∗ (Γ)) can be defined.(2) By the given data, the following two KK -elements are defined: the index element [ D ] ∈ KK Γ ( C ( X ) , C )and the Clifford symbol element [ σ ClD ] ∈ KK Γ ( C ( X ) , Cl τ ( X )). These two elements are related as follows:[ D ] = [ σ ClD ] ⊗ Cl τ ( X ) [ d X ].(3) The analytic index ind( D ) is completely determined by the K -homology class [ D ] by the formula[ c ] ⊗ C ( X ) (cid:111) Γ j Γ ([ D ]). Consequently, ind( D ) is determined by the topological data [ σ ClD ].The main result of the present paper is an infinite-dimensional version of (2) above. In order to formulateit, we need a “function algebra of M ”. The Higson-Kasparov-Trout algebra (HKT algebra for short) playsthe role of such algebra [HKT, Tro, DT], which will be explained soon. The HKT algebra for M is denotedby S C ( M ) in the present paper. The main result, Definition-Theorem 5.14, 5.19, 5.23, and Theorem 5.24,is summarized as follows. Theorem 1.3.
Under the assumption of Problem 1.1, we can define three LT -equivariant Kasparov modulescorresponding to the index element, the Clifford symbol element and the Dirac element. The Kasparov modulecorresponding to the index element is a Kasparov product of the others. Formally, [ (cid:101) D ] = [ (cid:103) σ Cl D ] ⊗ S C ( M ) [ (cid:103) d M ] . Moreover, we also study an “assembly map”, or (3) of Fact 1.2. Unfortunately, it is not of satisfactoryform. The aim of Section 6 is, not only to describe the following result, but also to leave some ideas andobservations, which can be useful. 2 heorem 1.4 (Theorem 6.8) . The index element [ (cid:101) D ] is “assemblable”. The value of the “total assemblymap” coincides with the analytic index computed in [Tak1, Tak2, Tak3]. Let us explain several details. We must begin with the explanations of the difficulties of the global analysisof infinite-dimensional manifolds and several observation on these difficulties. After that, we will talk aboutprevious researches and the outline of our constructions of this paper.Recall the Kasparov index theorem, [Kas83, Kas16] or Fact 1.2. There, we need the following ingredients: C ∞ c ( X, S ), D and C ∗ (Γ) for (1), C ( X ), Cl τ ( X ), L ( X, S ), C ( X, S ), σ ClD and [ d X ] for (2), and C ( X ) (cid:111) Γ, j Γ and [ c ] for (3). See also Section 2.4 for details. All of them cannot be defined for infinite-dimensionalcases in classical ways. For example, our manifold is non-locally compact, and hence a compactly supportedcontinuous function on M is automatically trivial. Therefore, the classical C ( M ) is trivial. Accordingto a well-known theorem by Gelfand (see [Mur] for example), the category of locally compact Hausdorffspaces is contravariantly equivalent to that of commutative C ∗ -algebras. However, this is another reasonwhy commutative C ∗ -algebras are useless for the study of non-locally compact spaces . For the same reason,we can not define “ Cl τ ( M )” or “ C ( M , L ⊗ S )”. Moreover, an infinite-dimensional manifold does not have agood measure, unlike the Lebesgue measure on a Euclidean spaces. Therefore, we can not define “ C ∗ ( LT )”,“ L ( M , L ⊗ S )”, “ C ( M ) (cid:111) LT ” or “ j LT ”.However, there are many noncommutative C ∗ -algebras which are “equivalent” to the C -algebras in somesense. It can be possible to study an infinite-dimensional version of such noncommutative C ∗ -algebras. Infact, the HKT algebra is a C ∗ -algebra which plays a role of “ C ( R ) (cid:98) ⊗ C ( R ∞ , Cliff ( R ∞ ))”. Once we define a C ∗ -algebra which carries some topological information on M , we may forget M itself. In order to study theindex theory of M , it is enough to study the LT -equivariant KK -theory of this algebra.By this observation, our abstract and naive problem becomes slightly concrete: Find a C ∗ -algebra whichcarries some information on M , and consider some KK -elements of this algebra .Then, how can we define the Hilbert space which plays a role of “ L -space”? In other words, how will wedefine the K -homology class? The answer seems to lie on the representation theory. Recall the Peter-Weyltheorem [KO, Kna]: For a compact group G , there is an isomorphism L ( G ) ∼ = ⊕ λ ∈ (cid:98) G V λ ⊗ V ∗ λ as representationspaces of G × G , where (cid:98) G is the set of isomorphism classes of irreducible unitary representations of G , and V λ is the corresponding representation space. In order to define the left hand side, we need a Haar measureon G , but, for the right hand side, we do not need it. Since our manifold is, up to some finite-dimensionalstuffs, an infinite-dimensional Lie group, such observation can be useful.In order to carry out such ideas, we need to recall several previous researches: The HKT algebras and therepresentation theory of loop groups. Then, we review some related researches on Hamiltonian loop groupspaces, in order to explain the geometrical background.Let us begin with the HKT algebra, which is one of the main objects in the present paper. Recall the Bottperiodicity K ( Cl τ ( R n )) = K − n ( R n ) ∼ = Z , and the Bott periodicity map K ( Cl τ ( R n )) → K ( Cl τ ( R n +1 )).Seeing these formulas, everyone thinks that “ K −∞ ( R ∞ )” must be isomorphic to Z . This naive idea wasjustified in [HKT]. N. Higson, G. Kasparov and J. Trout rewrote the Bott periodicity map as the inducedmap of a ∗ -homomorphism between the graded-suspended algebras C ( R ) (cid:98) ⊗ Cl τ ( R n ) → C ( R ) (cid:98) ⊗ Cl τ ( R n +1 ),and defined the HKT algebra S C ( R ∞ ) by the C ∗ -algebra inductive limit of this sequence. The HKT algebrawas used to solve the Baum-Connes conjecture for a-T-menable groups in [HK], and generalized to Hilbertbundles [Tro], and to Fredholm manifolds [DT]. Recently, the HKT algebra appeared in the context of thegauge theory [Kat]. We will extensively study the HKT algebras in Section 4.About the loop groups, we should explain something about the representations. Loop groups have therotation symmetry: θ .l ( θ ) = l ( θ + θ ). We would like to deal with representations of LG satisfying thefollowing conditions: they are infinite-dimensional; they reflect the rotation symmetry; and they satisfy acertain finiteness condition. Positive energy representations (PERs for short) are such ones [PS]. It is knownthat PERs must be projective. Let LG τ be an appropriate U (1)-central extension of LG . On an irreduciblePER, the added center U (1) ⊆ LG τ acts on the representation space with a fixed weight, which is called thelevel of the representation. Surprisingly, there are only finitely many equivalence classes of irreducible PERs Even if a commutative C ∗ -algebra can be defined from M , it must be the C -algebra of another locally compact Hausdorffspace. Such an algebra does not carry information on the original manifold M
3t fixed level. The Wess-Zumino-Witten Hilbert space (WZW Hilbert space) is defined by ⊕ λ ∈ (cid:100) LG τ V λ ⊗ V ∗ λ ,where (cid:100) LG τ is the set of isomorphism classes of irreducible PERs of LG at level τ . By an imitation of thePeter-Weyl theorem [KO, Kna], the WZW Hilbert space is regarded as the “ L -space of LG ”. It is used inthe context of the conformal field theory [Gaw].D. Freed, M. Hopkins and C. Teleman defined a “Dirac operator” as an operator acting on the tensorproduct of the WZW Hilbert space and the “Spinor space” which is defined by the complex structure in[FHT]. They used this operator in order to define an isomorphism (called the FHT isomorphism here)between the representation group of LG at level τ and the τ -twisted G -equivariant K -theory of G . Whatthey tried to define was a map R τ ( LG ) → K τ +dim( LG ) LG ( L g ∗ ), although K τ +dim( LG ) LG ( L g ∗ ) does not makesense. There is a “local equivalence” between L g ∗ //LG and G//G as groupoids. They used this equivalenceto define the FHT isomorphism. However, what they actually defined from a PER of LG is an element of K τ +dim( G ) G ( G ).This result is important for us at least for the following two reasons. Firstly, the construction of suchDirac operator is essential. The Dirac operator is an infinite sum, but this infinite sum converges forthe algebraic reason. For details on this technique, see also [Tak1, Tak3]. The most important tech-nical issues on this paper is in some sense a quantitative version of theirs. Secondly, the “FHT iso-morphism” R τ ( LG ) → KK τ +dim( LG ) LG ( L g ∗ ) ∼ = K LGτ +0 ( L g ∗ ) reminds us of the Baum-Connes assembly map RK Γ0 ( E Γ) → K ( C ∗ red (Γ)) [Val, GHT], where E Γ is the universal example for proper actions of Γ. The“isomorphism KK τ +dim( LG ) LG ( L g ∗ ) ∼ = K LGτ +0 ( L g ∗ )” is given by the Poincar´e duality. We expect the followingthings: The τ -twisted LG -equivarint K -homology group K LGτ +0 ( L g ∗ ) makes sense in terms of noncommutativegeometry; L g ∗ is one of the universal examples for the proper actions of LG in the sense of [BCH]; and theFHT isomorphism can be regarded as the inverse of the Baum-Connes assembly map. This conjecture is areformulation of the result of [Loi]: In this paper, Y. Loizides proved that the inverse of the FHT isomorphismis given by a kind of assembly map, where the twisted K -theory is identified with the twisted K -homology.We hope our project provides a good tool to describe his result in the purely infinite-dimensional framework,and in a direct way. We will not study further this topic.Let us explain Hamiltonian loop group spaces from the viewpoint of the geometric quantization, beforegoing to the explanations of the outline of the constructions of the present paper. A Hamiltonian LG -space is an infinite-dimensional symplectic manifold equipped with an LG -action and a proper moment maptaking values in L g ∗ . It was introduced in [MW]. There is a one to one correspondence between (infinite-dimensional symplectic) Hamiltonian LG -spaces and (compact, possibly non- Spin c ) quasi-Hamiltonian G -spaces [AMM]. E. Meinrenken studied the geometric quantization problem of Hamiltonian LG -spaces usingthis correspondence and the FHT isomorphism in [Mei]. Y. Song studied the same problem in a directway: He define a Hilbert space which can be regarded as an “ L -space” of the original Hamiltonian LG -space. In fact, he defined it by the L -space of a Hilbert bundle over the corresponding quasi-Hamiltonian G -space. Then, the method of abelianization was introduced in [LMS]. Our Φ − ( t ) is a special case of theabelianization. Using this method, the quantization problem of Hamiltonian LG -space was studied in [LS].Our problem was very inspired by these researches, but the method is quite different. Our methodis more noncommutative geometrical, and more direct than theirs. However, we have not studied the nonco-mutative group cases, and it seems to be difficult to generalize our work to such general cases without muchefforts.Let us outline the present paper. We can precisely set the problem here. A Hamiltonian LT -space witha proper moment map automatically satisfies the following conditions. One can find the proof of this fact in[Tak3]. Definition 1.5.
An infinite-dimensional manifold M is a “proper LT -space” if it has an LT -action and aproper, equivariant and smooth map Φ : M → L t ∗ . The LT -action on L t ∗ is not the coadjoint action butthe gauge action l.A = A − l − dl for l ∈ LT and A ∈ L t ∗ . We also suppose that Φ − ( t ) is even-dimensionaland T × Π T -equivariantly Spin c , that is, M is “even-dimensional and Spin c ”. In fact, what they defined is an operator on V ⊗ S , wehre V is a PER and S is the Spinor. However, that operator clearlymodeled on the Dirac operator on the L -space. See Section 2 in [FHT] also. roblem 1.6. For a proper LT -space M with a τ -twisted LT -equivariant line bundle L , construct an LT -equivariant Spinor bundle S , and study the KK -theoretical LT -equivariant index theory. More precisely,construct three Kasparov modules corresponding to the index element, the Clifford symbol element and theDirac element, and prove the KK -theoretical index theorem equality. Then, study the assembly map. Remarks . (1) The phrase “infinite-dimensional and even-dimensional” sounds too strange. However,everyone probably thinks that a complex manifold or a symplectic manifold must be “even-dimensional”,even if the manifold is infinite-dimensional. In our case, M is in fact Φ − ( t ) × U as observed in [Tak1, Tak3],where U is an infinite-dimensional complex vector space. In this sense, our manifold is “even-dimensional”.(2) The Spin c -condition for infinite-dimensional manifolds is not clear. We adopt an easy condition:Our manifold is of the form “a finite-dimensional Spin c -manifold × an infinite-dimensional almost complexmanifold”.The core idea of the present paper is to use the HKT algebra as the function algebra of M . Unfor-tunately, the HKT algebra is the substitute of, not the C -algebra, but the Cl τ -algebra. However, for aneven-dimensional G -equivariantly Spin c -manifold X , C ( X ) is G -equivariantly Morita-Rieffel equivalent to Cl τ ( X ), as explained in Lemma 2.30. Using this equivalence and the graded suspension C ( R ) (cid:98) ⊗− , we willreformulate the Kasparov index theorem in Section 2.5. In this reformulated version, only C ( R ) (cid:98) ⊗ Cl τ ( X )appears as function algebras of the manifold. This C ∗ -algebra is the model of the HKT algebra.In Section 3, we will introduce the LT -equivariant KK -theory. We will define the Kasparov product inthe LT -equivariant setting at the level of modules there, but we will not prove that the product is well-definedat the level of KK -theory. However, the following phrase still makes sense: “ An LT -equivariant Kasparovmodule is a Kasparov product of other two Kasparov modules ”. Then, we will set the problem precisely,and we will divide it into two parts just like [Tak1, Tak2, Tak3]. Consequently, we will find that we mayconcentrate on one infinite-dimensional manifold U with the standard U -action.In Section 4, we will study the HKT algebra: its definition, the group action and a representation ona Hilbert S -module. The original HKT algebra inherits a locally compact group (affine) action on theunderling space, as explained in [HKT, HK]. We will prove that the algebra also inherits the U -action.Thanks to this result, we can consider U -equivariant KK -theory of S C ( U ). We will also introduce aHilbert S -module on which S C ( U ) acts, using the WZW model. Although the both of the HKT algebraand the WZW Hilbert space are not new objects, the connection between them is probably a new result.The main part of the present paper is Section 5. We will define three U -equivariant Kasparov mod-ules there. In order to define the index element [ (cid:101) (cid:1) ∂ ] ∈ KK τU ( S C ( U ) , S ) and the Dirac element [ (cid:102) d U ] ∈ KK U ( S C ( U ) , S ), we need to define a Hilbert S -module which is equipped with a left S C ( U )-action, andappropriate operators. In fact, we will use the same module and the same operator. The operator is basedon the Dirac operator in [FHT] or [Son]. Our operator is not actually equivariant, but “almost equivariant”,which can be adopted as an operator defining a Kasparov module. Once a pair of a Hilbert module andan operator which looks like an index element is defined, we only have to check several conditions to bean equivariant Kasparov module. These are purely functional analytical tasks. The story is quite simple,but the calculation is very complicated. We must frequently verify some infinite sums converge, because theinfinite sum defining the Dirac operator becomes an actual infinite sum, after multiplying an element of theHKT algebra with an element of an “ L -section”, unlike [FHT]. This is the biggest difference from theprevious researches.
We will carry out these programs in Section 5.1.For the Dirac element, we use the same operator and the same Hilbert module, but the different U -action.It will be an easy task to modify the group action, and it will be studied in Section 5.2. in this subsection,we will also explain why the model of our Dirac element is appropriate.For the Clifford symbol element, we need a τ -twisted U -equivariant ( S C ( U ) , S C ( U ))-bimodule. Forthis aim, we modify the U -action on S C ( U ), which is τ -twisted. Once these Kasparov modules are defined,it is not difficult to prove the main result: Theorem 5.24.In the final section, we will deal with “assembly maps” for our case. In locally compact cases, assemblymaps played an important role in [Per] to study the cohomological formula of the Kasparov index theorem. not the KK -element! KK -theory, but we will not be ableto introduce the crossed product and the descent homomorphism. This is the reason why our assembly mapis not very satisfying for us. In the final subsection, we will write some observations on the crossed productsand descent homomorphisms.Finally, we add some notational remarks. Remarks . (1) From now on, we use the graded language, without any special notice. We summarize ithere, although we believe that they are standard. • For a Z -graded vector space (Hilbert space, algebra or Hilbert module) A A = A (cid:98) ⊕ A , the gradingis denoted by ∂ : for a ∈ A i , ∂a = i . The grading homomorphism is always denoted by (cid:15) (or (cid:15) A ): (cid:15) ( a ) = ( − ∂a a for a ∈ A ∪ A . • If two algebras (vector spaces, Hilbert modules and so on) A = A (cid:98) ⊕ A and B = B (cid:98) ⊕ B are given, A ⊗ B is graded by, as usual, [( A ⊗ B ) ⊕ ( A ⊗ B )] (cid:98) ⊕ [( A ⊗ B ) ⊕ ( A ⊗ B )]. The product (orthe ring-action on the module) is also twisted: [ a ⊗ b ] · [ a (cid:48) ⊗ b (cid:48) ] = ( − ∂b∂a (cid:48) aa (cid:48) ⊗ bb (cid:48) , if a (cid:48) and b arehomogeneous. Such tensor product is often written as A (cid:98) ⊗ B , but we omit the “hat”. We sometimesremind the reader of this rule to emphasize it. • The commutator is always graded [ F , F ] = F F − ( − ∂F ∂F F F , where F and F can be justoperators, not elements of some algebra. They can be unbounded operators. If the space on whichan operator F acts is Z -graded, we say that F is odd or even, if F reverses or preserves the grading,respectively.(2) We sometimes use the C ∗ -algebra C ( R ) equipped with a non-trivial Z -grading defined by (cid:15) ( f )( t ) := f ( − t ). S denotes this graded algebra.(3) We frequently use some objects which are substitutes of something that we can not define in classicalmethods. Such substitute are denoted by the standard symbol with the underline. For example, our WZWmodel for LT is a substitute of “ L ( LT )”. In such a case, L ( LT ) denotes the WZW model. Here is anotherexample: For a Fredholm operator T : H → H , ind( T ) is a substitute of “dim( H ) − dim( H )”, and so wemay write ind( T ) as dim( H ) − dim( H ).(4) We use the following notations: For a Hilbert B -module E , L B ( E ) or simply L ( E ) is the C ∗ -algebraconsisting of adjointable operators on E , and K B ( E ) or simply K ( E ) is the C ∗ -algebra consisting of B -compactoperators, following [JT].(5) A group action is denoted by g.v , for a group element g and an element v of a set on which the groupacts. A product in an algebra or an algebra action on a module is denoted by a · φ , for an element a of analgebra and an element φ of a module or the algebra on which the algebra acts.(6) Throughout this paper, parentheses ( •|• ) means an inner product which is linear in the second variablesand anti-linear in the forst one. An inner product takes values in some C ∗ -algebra (including C ). On theother hand, angle brackets (cid:104)• , •(cid:105) or (cid:104)(cid:104)• , •(cid:105)(cid:105) is a bilinear form (or pairing) taking values in R or C . In this section, we review the Kasparov index theorem, in order to clarify notations. Then, we will reformulateit in an appropriate form for our problem. This section does not contain many new results: Proposition 2.6and related things are probably new; Section 2.5 is maybe also new, but it is just an exercise for the equivariant KK -theory. The bounded picture is convenient for the general study of the KK -theory. This picture works very well, evenfor equivariant cases. On the other hand, although natural KK -elements appearing in geometry are often6nbounded, general theory for unbounded cycles are quite difficult because of functional analytical issues onHilbert modules.Unbounded Kasparov modules was introduced in [BJ], and the criterion to be a Kasparov product wasstudied by [Kuc] and [Mes]. In this subsection, we prepare some issues on unbounded equivariant Kasparovmodules. First of all, let us recall several definitions. Let A and B be separable C ∗ -algebras. Definition 2.1.
Let E be a countably generated Hilbert B -module equipped with a ∗ -homomorphism A →L B ( E ), and let D be a (possibly) unbounded, densely defined, regular , odd and self-adjoint operator. Thepair ( E, D ) is called an unbounded Kasparov ( A, B ) -module , if the following conditions are satisfied: • The set of a ∈ A such that [ a, D ] extends to a bounded operator, is dense. • a (1 + D ) − belongs to K B ( E ) for each a ∈ A .The first condition means that “ D is of first order”, and the second one means that “ D is an ellipticoperator”.Under the following bounded transformation, an unbounded Kasparov module is transformed into abounded one [BJ, Bla]. Definition 2.2.
Let b be a function on R defined by b ( x ) := x √ x . For an unbounded Kasparov module( E, D ), the bounded transformation of this module is defined by b ( E, D ) := ( E, b ( D )) = (cid:18) E, D √ D (cid:19) . Let us move to the equivariant KK -theory. For details, see [Kas88, Bla]. Let G be a second countable,locally compact and Hausdorff group. Suppose that the C ∗ -algebras A and B are G - C ∗ -algebras, that is, G acts there and the map g (cid:55)→ g ( a ) is continuous for any a ∈ A or B . Definition 2.3.
Let E be a Z -graded, countably generated, G -equivariant Hilbert B -module equipped witha G -equivariant ∗ -homomorphism A → L ( E ), and F ∈ L ( E ) be an odd and self-adjoint operator. Thepair ( E, F ) is called a bounded G -equivariant Kasparov ( A, B ) -module , if the following conditions aresatisfied: • [ a, F ], a (1 − F ), a ( g ( F ) − F ) are B -compact operators for any a ∈ A and g ∈ G . • The map G (cid:51) g (cid:55)→ g ( aF ) ∈ L ( E ) is norm-continuous for any a ∈ A .The set of homotopy classes of G -equivariant Kasparov ( A, B )-modules turns out to be an abelian group withrespect to the direct sum of modules. This group is denoted by KK G ( A, B ). For a G -equivariant Kasparov( A, B )-module (
E, F ), the corresponding KK -element (or the homotopy class represented by ( E, F )) isdenoted by [(
E, F )].The condition to be “equivariant” is not very simple, but the small failure to be “actually equivariant”is useful. See also Section 5.2. On the other hand, that failure makes it difficult to translate the definitionto the unbounded picture. The aim of this subsection is to give a sufficient condition so that an unboundedKasparov module with a group action is transformed into a bounded equivariant Kasparov module by thebounded transformation. One of the easiest sufficient condition is probably to assume the operator to be“actually equivariant”. Obviously such unbounded “actually equivariant” Kasparov module is transformedinto a bounded “actually equivariant” Kasparov module by the bounded transformation. It means that theassumption is too strong. The following is our current answer.
Definition 2.4.
An unbounded Kasparov (
A, B )-module (
E, D ) is an unbounded G -equivariant Kas-parov ( A, B ) -module if the following conditions are satisfied: A densely defined adjointable operator D on E is said to be regular if and only if 1 + D has dense range. This conditionis not automatic. E is G -equivariant ( A, B )-bimodule. • G preserves dom( D ). • g ( D ) − D and [ D, D − g ( D )] · D λ + D are bounded for all g ∈ G and λ ≥ • The map g (cid:55)→ g ( D ) − D is norm-continuous. Remark . This assumption is very relaxed in comparison with being “actually equivariant”, but still toostrong. We hope that some better sufficient condition will be found.However, our conditions have geometrical sense. Suppose that A = C ( M ) for some smooth manifold M , B = C , and D is a first order differential operator on M . If g ( D ) − D is bounded, the symbol must beinvariant, and the potential is at most linear along the direction of the group action. A sufficient condition sothat the map g (cid:55)→ g ( D ) − D is continuous, is the following: The potential function is uniformly continuous.The rest condition is less clear. However, if the manifold is compact, this condition is almost automatic,thanks to the elliptic regularity.Let us prove that our unbounded equivariant Kasparov modules are transformed to bounded ones by thebounded transformation. Proposition 2.6.
For an unbounded G -equivariant Kasparov ( A, B ) -module ( E, D ) , the bounded transfor-mation b ( E, D ) is a bounded G -equivariant Kasparov ( A, B ) -module.Proof. As explained in [Bla], b ( E, D ) is actually a Kasparov module. We need to check the equivariancecondition and the G -continuity condition.The G -continuity is obvious because g ( b ( D )) = b ( g ( D )). For the equivariance, we use the usual technique: b can be rewritten as b ( x ) = x √ x = 1 π (cid:90) ∞ √ λ x x + λ dλ. Let us calculate a ( b ( D ) − g ( b ( D ))) using this formula. a ( b ( D ) − g ( b ( D ))) = a (cid:104) (1 + D ) − D − g ( D )(1 + g ( D ) ) − (cid:105) = a · (cid:20) π (cid:90) ∞ √ λ ·
11 + D + λ · Ddλ − π (cid:90) ∞ √ λ · g ( D ) ·
11 + g ( D ) + λ dλ (cid:21) = a · π (cid:90) ∞ √ λ
11 + D + λ (cid:104) D (1 + g ( D ) + λ ) − (1 + D + λ ) g ( D ) (cid:105)
11 + g ( D ) + λ dλ = a · π (cid:90) ∞ √ λ
11 + D + λ (cid:104) (1 + λ )( D − g ( D )) + D [ g ( D ) − D ] g ( D ) (cid:105)
11 + g ( D ) + λ dλ = 1 π (cid:90) ∞ √ λ a ·
11 + D + λ · ( D − g ( D )) · (cid:18) λ g ( D ) + λ (cid:19) dλ + 1 π (cid:90) ∞ √ λ a ·
11 + D + λ · D [ g ( D ) − D ] g ( D ) · (cid:18)
11 + g ( D ) + λ (cid:19) dλ. The first term is compact, because D − g ( D ) is bounded, λ g ( D ) + λ is uniformly bounded in λ and a · (1 + D + λ ) − is a compact operator whose norm is at most (cid:107) a (cid:107) λ . For the second one, we notice that D [ g ( D ) − D ] g ( D ) = − [ g ( D ) − D ] g ( D ) + [ g ( D ) , g ( D ) − D ] g ( D ) − [ g ( D ) − D ] g ( D ) . Multiplying by g ( D ) + λ from the right, we get bounded operators whose norms are uniformly bounded in λ . More precisely, for the second term,[ g ( D ) , g ( D ) − D ] g ( D ) 11 + g ( D ) + λ = g (cid:18) [ D, D − g − ( D )] 11 + D + λ (cid:19)
8s uniformly bounded, by the assumption. The other terms are easier to handle.We have not mentioned the equivalence relations on unbounded Kasparov modules. In fact, even fornon-equivariant KK -theory, we do not know what relation in the unbounded picture completely correspondsto the homotopy in the bounded picture. See [DGM] for the current state. We only verify that homotopyequivalence is a sufficient condition. G , A and B are as usual. Definition 2.7.
Two unbounded G -equivariant Kasparov ( A, B )-modules ( E , D ) and ( E , D ) are homo-topic if there exists an unbounded G -equivariant Kasparov ( A, BI )-module ( E (cid:48) , D (cid:48) ) such that π i ∗ ( E (cid:48) , D (cid:48) ) ∼ =( E i , D i ) for i = 0 , G -equivariant Kasparov ( A, B )-modules are homotopic by ( E (cid:48) , D (cid:48) ), their boundedtransformations are homotopic, because the bounded transformation of ( E (cid:48) , D (cid:48) ) gives a homotopy betweenthese transformed Kasparov modules.Let us introduce a criterion to be a Kasparov product in the unbounded picture. The following isimmediately proved thanks to [Kuc]. Proposition 2.8 ([Kuc]) . Let ( E , D ) be an unbounded G -equivariant Kasparov ( A, C ) -module, and ( E , D ) be an unbounded G -equivariant Kasparov ( C, B ) -module. x and y are the corresponding KK -elements of ( E , D ) and ( E , D ) , respectively. A G -equivariant Kasparov ( A, B ) -module ( E ⊗ E , D ) is a representa-tive of the Kasparov product of x and y , if the following 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 ) , where T v ( w ) := v ⊗ w : E → E ⊗ E . • dom( D ) is contained in dom( D ⊗ id) . • The (graded) commutator [ D ⊗ id , D ] is bounded below.Proof. In the bounded picture of KK G , the following holds: z = x ⊗ y in KK G if and only if z = x ⊗ y in KK and z is G -equivariant. The conditions in the statement guarantee that ( E ⊗ E , D ) gives a Kasparovproduct of x and y , thanks to [Kuc]. Since we suppose that ( E, D ) itself is G -equivariant, the statement hasbeen proved.Let us define the descent homomorphism in the unbounded picture. Note that this homomorphism hasalready been defined in the bounded picture in [Kas88]. For simplicity, we assume the following, from nowon. For details on the amenability, see [Pie]. Assumption 2.9. G is amenable and unimodular.Thus, we do not have to pay attention to the norm of crossed products or modular functions. Definition-Proposition 2.10.
For an unbounded G -equivariant Kasparov ( A, B ) -module ( E, D ) , consider C c ( G, E ) and the operator [ (cid:101) D ( f )]( g ) := D [ f ( g )] for f ∈ C c ( G, E ) . C c ( G, E ) is a pre-Hilbert C c ( G, B ) -moduleby the following operations: f ∗ b ( g ) := (cid:90) G f ( h ) h. [ b ( h − g )] dh, ( f | f ) B (cid:111) G ( g ) := (cid:90) G h − ( f ( h ) | f ( hg )) B dh for f, f , f ∈ C c ( G, E ) and b ∈ C c ( G, B ) . This module admits a left C c ( G, A ) -action defined by a ∗ f ( g ) := (cid:90) a ( h ) h. [ f ( h − g )] dh. he completion of this module with respect to the above inner product is denoted by E (cid:111) G . The pair ( E (cid:111) G, (cid:101) D ) is an unbounded Kasparov ( A (cid:111) G, B (cid:111) G ) -module, and denoted by j G ( E, D ) .The bounded transformation of the value of the new descent homomorphism coincides with the value ofthe descent homomorphism of the bounded transformed module: j G ( b ( E, D )) = b ( j G ( E, D )) .Proof. Firstly, we note that b ( (cid:101) D ) = (cid:93) b ( D ). Therefore, the pair ( E (cid:111) G, b ( (cid:101) D )) coincides with j G ( E, b ( D )).In order to prove that j G ( E, D ) is a Kasparov module, we must check two conditions: (1) [ a, (cid:101) D ] is boundedfor dense a ∈ A (cid:111) G ; and (2) a (1 + (cid:101) D ) is compact for all a ∈ A (cid:111) G .For (1), let A := { a ∈ A | [ a, D ] is bounded } , and let a ∈ C c ( G, A ). We may assume that a ( h ) is homo-geneous and its degree is independent of h . Then, for e ∈ C c ( G, dom( D )),[ a, (cid:101) D ] e ( g ) = (cid:90) G a ( h ) h. [ D ( e ( h − g ))] dh − ( − ∂a (cid:90) G D [ a ( h ) h.e ( h − g )] dh = (cid:90) G a ( h )[ { h ( D ) − D + D } h. ( e ( h − g )] dh − ( − ∂a (cid:90) G D [ a ( h ) h.e ( h − g )] dh = (cid:90) G { a ( h )( h ( D ) − D ) + [ a ( h ) , D ] } [ h. ( e ( h − g ))] dh. The last form is obviously bounded in e .For (2), we note the following computation:(1 + (cid:101) D ) − = 1 − b ( (cid:101) D ) = 1 − (cid:93) b ( D ) . Since ( E, b ( D )) is a G -equivariant bounded Kasparov module, a (1 − b ( (cid:101) D ) ) is compact for all a ∈ A (cid:111) G . A large advantage of unbounded Kasparov modules is that the exterior tensor product can be explicitlydescribed at the level of modules. This feature is inherited to the equivariant setting. Let G and G belocally compact, second countable and Hausdorff group, and let G := G × G . Lemma 2.11.
Let A and B be G - C ∗ -algebras, and let A and B be G - C ∗ -algebras. Then A ⊗ C A and B ⊗ C B are G - C ∗ -algebras.Suppose that an unbounded G -equivariant Kasparov ( A , B ) -module ( E , D ) and an unbounded G -equivariant Kasparov ( A , B ) -module ( E , D ) are given. Then, ( E, D ) := ( E ⊗ E , D ⊗ id + id ⊗ D ) is an unbounded G -equivariant Kasparov ( A, B ) -module.Proof. We only have to check the equivariance conditions. E is obviously G -equivariant ( A, B )-module by the action ( g , g ) . ( e , e ) := ( g .e , g .e ). Moreover, G preserves dom( D ). Since g i (id) = id,( g , g )( D ) − D = [ g ( D ) − D ] ⊗ id + id ⊗ [ g ( D ) − D ]is a bounded operator. This description also implies that the map ( g , g ) (cid:55)→ [( g , g )( D ) − D ] is continuous,10y the assumption that ( E i , D i ) are equivariant Kasparov modules. Moreover,[ D, D − ( g , g )( D )] · D λ + D = ([ D , D − g ( D )] ⊗ id + id ⊗ [ D , D − g ( D )]) · D ⊗ id + id ⊗ D λ + D ⊗ id + id ⊗ D = [ D , D − g ( D )] · D D ⊗ id · D ⊗ id1 + λ + D ⊗ id + id ⊗ D + [ D , D − g ( D )] ⊗ D D · ⊗ D λ + D ⊗ id + id ⊗ D + id ⊗ [ D , D − g ( D )] · D D · ⊗ D λ + D ⊗ id + id ⊗ D − D D ⊗ [ D , D − g ( D )] · D ⊗ id1 + λ + D ⊗ id + id ⊗ D . The last operator is bounded whose norm is bounded.This operation is compatible with the Kasparov product in the following sense. Let A i , B i and C i be G i - C ∗ -algebras for i = 1 , Lemma 2.12.
Let ( E i , D i ) be unbounded G i -equivariant Kasparov ( A i , C i ) -modules, and ( E (cid:48) i , D (cid:48) i ) be un-bounded G i -equivariant Kasparov ( C i , B i ) -modules for i = 1 , . Suppose that ( E i , D i ) := ( E i ⊗ E (cid:48) i , D i ) satisfythe criterion to be a Kasparov product of ( E i , D i ) and ( E (cid:48) i , D (cid:48) i ) , respectively in i . Then, the exterior productof ( E , D ) and ( E , D ) : ( E ⊗ C E , D ⊗ id + id ⊗ D ) is a Kasparov product of the exterior products ( E , D ) ⊗ C ( E (cid:48) , D (cid:48) ) and ( E , D ) ⊗ C ( E (cid:48) , D (cid:48) ) .Remark . At the level of KK -theory, this equality is obvious from the associativity of the Kasparovproduct, and the commutativity of the exterior tensor products: ( x ⊗ C y ) ⊗ C ( x ⊗ C y ) = ( x ⊗ C x ) ⊗ C ⊗ C ( y ⊗ C y ). However, for our purpose, we need to prove that at the module level. We leave theproof to the reader.The descent homomorphism is also compatible with the formula about the exterior tensor product. Itcan be proved by the same way in Lemma 3.15 in [Tak3]. Lemma 2.14.
Let ( E i , D i ) be unbounded G i -equivariant Kasparov ( A i , B i ) -modules for i = 1 , , respectively.Let D be D ⊗ id + id ⊗ D . Then, j G × G ( E ⊗ E , D ⊗ id + id ⊗ D ) = j G ( E , D ) ⊗ j G ( E , D ) . These results will be used in Section 3 to simplify the problem.
In this subsection, we describe some algebraic features of Clifford algebras and their modules.The Clifford algebra is defined as follows: For a Euclidean space V , let Cliff − ( V ) := T ( V ) ⊗ C / (cid:10) v + | v | (cid:11) ,where T ( V ) is the tensor algebra. We will also encounter Cliff + , whose definition is T ( V ) ⊗ C / (cid:10) v − | v | (cid:11) .In fact, these are isomorphic via the homomorphism generated by a non-canonical isomorphism v (cid:55)→ iv for v ∈ V , but it is better to distinguish them. This is because there are various different choices of isomorphisms: i id V ⊕ ( − i id V ⊥ ) generates an isomorphism between Cliff + and Cliff − , where V is a subspace of V .These two algebras turn out to be C ∗ -algebras. Lemma 2.15.
The metric of V induces metrics on Cliff ± ( V ) . By the left multiplications Cliff ± ( V ) → End(
Cliff ± ( V )) , and the C ∗ -algebra structures on End(
Cliff ± ( V )) , we find that Cliff ± ( V ) are C ∗ -algebras.The involution on Cliff ± ( V ) are given by v ∗ = ± v for v ∈ V , respectively. roof. We deal with only
Cliff − ( V ). Let v ∈ V be a unit vector, and consider a vector v x + y := vv · · · v k + w · · · w l , where v ⊥ v i , v ⊥ w j , v i ⊥ v i (cid:48) for i (cid:54) = i (cid:48) , and w j ⊥ w j (cid:48) for j (cid:54) = j (cid:48) . Similarly for v x (cid:48) + y (cid:48) . Then,( v · ( v x + y ) | v x (cid:48) + y (cid:48) ) = ( − x + v y | v x (cid:48) + y (cid:48) )= − ( x | y (cid:48) ) + ( v y | v x (cid:48) )= − ( v x | v y (cid:48) ) + ( y | x (cid:48) )= − ( v x + y | v · ( v x (cid:48) + y (cid:48) )) . Thanks to this result, we find that C ( X, Cliff + ( T ∗ X )) is a C ∗ -algebra. It is denoted by Cl τ ( X ).In this paper, we will use Cliff − for the Clifford multiplication, and hence the Dirac operator is formallyself-adjoint. Then, Cliff + appears when we describe the Clifford symbol class. This is because a left Cliff − -module automatically admits a right Hilbert Cliff + -module structure, as proved below.A Z -graded Hermite vector space S is called a Clifford module of V , if it is equipped with an even ∗ -homomorphism c : Cliff − ( V ) → End( S ). In other words, it is a Z -graded Hermite vector space equippedwith a linear map c : V → End( S ) such that c ( v ) is a skew-adjoint and odd operator for each v ∈ V ,and satisfies that c ( v ) c ( w ) + c ( w ) c ( v ) = − v | w ) V . A Clifford module of V is called a Spinor if it is anirreducible representation of
Cliff − ( V ). It is known that an even-dimensional vector space V has two Spinorsup to equivalence. Fix one of them, and the fixed Spinor is denoted by ( S, c ). It is known that the Cliffordmultiplication c gives an isomorphism as Z -graded C ∗ -algebras Cliff − ( V ) ∼ = End( S ).We can define an irreducible representation of Cliff + ( V ) from Cliff − ( V ) (cid:8) S as follows. Lemma 2.16.
The dual space S ∗ admits a structure of a ∗ -representation space of Cliff + ( V ) , defined by c ∗ ( v ) · f := ( − ∂f f ◦ c ( v ) for f ∈ S ∗ and v ∈ V .Remark . Let τ : Cliff − ( V ) → Cliff + ( V ) be the linear extension of τ ( v · · · v k ) := ( − (cid:80) k − j =1 j v k · · · v and τ (1) = 1. This is well-defined: τ ( v + (cid:107) v (cid:107) ) = − v + (cid:107) v (cid:107) = 0. Moreover, this is an anti-homomorphism τ ( αβ ) = ( − ∂α · ∂β τ ( β ) τ ( α ). It can be proved as follows: Let α := a · · · a k , β := b · · · b l , and then τ ( αβ ) = ( − (cid:80) k + l − i i b l · · · b a k · · · a = ( − lk τ ( β ) τ ( α ). Using τ , we can rewrite the action as c ∗ ( α ) · f =( − ∂f · ∂α f ◦ c ( τ − ( α )) for f ∈ S ∗ and α ∈ Cliff + ( V ). Proof. c ∗ induces a map from the tensor algebra T ( V ) → End( S ∗ ). We check that c ∗ ( v ) c ∗ ( w ) + c ∗ ( w ) c ∗ ( v ) =2 ( v | w ) V . Let s ∈ S . (cid:2)(cid:0) c ∗ ( v ) c ∗ ( w ) + c ∗ ( w ) c ∗ ( v ) (cid:1) f (cid:3) ( s ) = c ∗ ( v )[ c ∗ ( w ) f ]( s ) + c ∗ ( w )[ c ∗ ( v ) f ]( s )= ( − ∂f +1 [ c ∗ ( w ) f ( c ( v ) s ) + c ∗ ( v ) f ( c ( w ) s )]= − f ( c ( w ) c ( v ) s + c ( v ) c ( w ) s )= 2 ( v | w ) V f ( s ) . Therefore the map c ∗ from Cliff + ( V ) is defined. Let us verify that c ∗ preserves the involution ∗ . Note that c ∗ ( v ) is given by t c ( v ) ◦ (cid:15) S ∗ . [ c ∗ ( v )] ∗ = [ t { c ( v ) } ◦ (cid:15) S ∗ ] ∗ = (cid:15) ∗ S ∗ ◦ ( t c ( v )) ∗ = (cid:15) S ∗ ◦ t ( c ( v ) ∗ ) = − (cid:15) S ∗ ◦ t c ( v ) = t c ( v ) ◦ (cid:15) S ∗ = c ∗ ( v ) , where we used the fact that c ( v ) is an odd operator, and the transpose commutes with the adjoint.Thanks to this result, c ∗ : Cliff + ( V ) → End( S ∗ ) is a ∗ -isomorphism. Using this isomorphism, we definea Hilbert Cliff + ( V )-module structure on S . 12 emma 2.18. A left Cliff − ( V ) -module S equipped with c : V → End( S ) , admits a right Hilbert Cliff + ( V ) -module structure given by the followings: By regarding S as S ∗∗ , we define • For s ∈ S and v ∈ V , s · v := s ◦ c ∗ ( v ) . • ( s | s ) Cliff + := s ∗ ⊗ s ∈ End( S ∗ ) ∼ = Cliff + ( V ) ⊗ C .Proof. Non-trivial thing is the compatibility of these two operations: ( s | s · v ) Cliff + = ( s | s ) Cliff + · v . Letus check that. Let f ∈ S ∗ . ( s | s · v ) Cliff + ( f ) = s ∗ ⊗ ( s ◦ c ∗ ( v ))( f )= s ∗ ⊗ s ( c ∗ ( v ) f )= (cid:16) ( s | s ) Cliff + · v (cid:17) ( f ) . This result is important to describe the Clifford symbol element.
In this subsection, we review the Kasparov index theorem [Kas83, Kas16]. After defining the ingredients, weoutline the proof, in the unbounded picture of equivariant Kasparov modules.
Problem 2.19.
Study the index theory in the following situation: X is an even-dimensional completeRiemannian Spin c -manifold; G is a locally compact, second countable and Hausdorff group; G acts on X isometrically, properly and cocompactly; ( S, c ) is a G -equivariant Spinor bundle of X ; E is a Z -graded G -equivariant Hermite vector bundle on X ; and D is a G -equivariant Dirac operator acting on C ∞ ( X, E ⊗ S ).The following is the typical example: For a compact Riemannian Spin c -manifold M , the universal cover X := (cid:102) M and the fundamental group G := π ( M ) satisfy the above condition. The Spinor and the Diracoperator on X can be defined by the pullback of ones on M .The following is independent of D , S and E . In this sense, they are universal. Definition 2.20. (1) For ξ ∈ Cliff + ( V ) and e k ∈ V , let (cid:98) e k ξ := ( − ∂ξ ξ · e k . Note that ( (cid:98) e k ) = −(cid:107) e k (cid:107) id.Define a differential operator ∂ on L ( X, Cliff + ( T ∗ X )) by ∂ := (cid:88) k (cid:98) e k ∂∂x k at x ∈ X , where { x , x , · · · , x n } is a coordinate around x ∈ X such that (cid:110) ∂∂x ( x ) , ∂∂x ( x ) , · · · , ∂∂x n ( x ) (cid:111) isan orthonomal basis, and e i := dx i . This operator is independent of the choice of such coordinate. The pair( L ( X, Cliff + ( T ∗ X )) , ∂ ) defines a G -equivariant ( Cl τ ( X ) , C )-module. Let [ d X ] := [( L ( X, Cliff + ( T ∗ X )) , ∂ )] ∈ KK G ( Cl τ ( X ) , C ). It is called the Dirac element (Definition 2.2 in [Kas16]).(2) At each x ∈ X , let H x := L ( T ∗ x X, Cliff + ( T ∗ x X )). Consider the pair ( H x , ∂ x ), for ∂ x := i (cid:88) k e k · ∂∂x k , where { e k } is an orthonormal basis of T ∗ x X , and e k · means the multiplication by e k from the left side.Consider the continuous family[ d ξ ] := [( C ( X, ∪ H x ) , { ∂ x } x ∈ X )] ∈ KK G ( C ( T ∗ X ) , Cl τ ( X )) . f ∈ Cl τ ( X ), its action on H x is given by the right multiplication of f ( x ) ∈ Cliff + ( T ∗ x X ). The inner prod-uct is given by ( f ⊗ v | f ⊗ v ) Cl τ ( X ) := ( f | f ) L · v ∗ v . Note that ( f | f ) L is a scalar-valued continuousfunction. This element is called the fiberwise Dirac element (Definition 2.5 in [Kas16]).(3) A cut-off function is a compactly supported continuous function c : X → R ≥ satisfying that (cid:82) G c ( g − .x ) dg = 1 for all x ∈ X . It defines a projection [ c ]( g, x ) := (cid:112) c ( x ) c ( g − .x ) in C ( X ) (cid:111) G . Thecorresponding KK -element is also denoted by [ c ] ∈ KK ( C , C ( X ) (cid:111) G ). It is called the Mishchenko linebundle .Note that [ d ξ ] gives a KK G -equivalence between C ( T ∗ X ) and Cl τ ( X ).From the given data E , S and D , we can define several KK G -elements. Definition 2.21. (1) [ D ] := [( L ( X, E ⊗ S ) , D )] ∈ KK G ( C ( X ) , C ) is called the index element (Lemma3.7 in [Kas16]).(2) Let c E be the Clifford multiplication on E ⊗ S , and let p : T ∗ X → X be the canonical projection. Let[ σ D ] := [( C ( T ∗ X, p ∗ ( E ⊗ S )) , ic E )] ∈ KK G ( C ( X ) , C ( T ∗ X )). It is called the symbol element .(3) Let [ σ ClD ] := [ σ D ] ⊗ C ( T ∗ X ) [ d ξ ] ∈ KK G ( C ( X ) , Cl τ ( X )). It is called the Clifford symbol element .We would like to define the analytic index for this case. The definition is based on the following observa-tion: On a separable Hilbert space H , consider the space of Fredholm operators on H equipped with the normtopology, Fred( H ). The set of connected components of Fred( H ) is isomorphic to Z , and the isomorphism isgiven by the Fredholm index. In this sense, the Fredholm index of an operator is the homotopy class itself ofthe operator . Therefore, the homotopy class of the following Hilbert module should be the analytic index, ifit is a Kasparov module. Definition 2.22. On C c ( X, E ⊗ S ), define the right action of C c ( G ) by s ∗ b ( x ) := (cid:90) G g ( s )( x ) · b ( g − ) dg for b ∈ C c ( G ) and s ∈ C c ( X, E ⊗ S ), and the inner product by( s | s ) C(cid:111) G ( g ) := (cid:90) X ( s ( x ) | g ( s )( x )) ( E ⊗ S ) x dx for s , s ∈ C c ( X, E ⊗ S ). We obtain a Hilbert C (cid:111) G -module by the completion of this pre-Hilbert module.The completion of D is an unbounded, densely defined, regular, odd and self-adjoint operator on the obtainedHilbert C (cid:111) G -module. The pair of the Hilbert C (cid:111) G -module obtained by the completion of C c ( X, E ⊗ S )and the operator D is called the analytic index of D and denoted by ind( D ). Remarks . (1) One must notice that we used the compactly supported sections to define the index.It means that, in order to generalize this procedure to infinite-dimensional manifolds, we need some anothertrick to find some “nice” subspace. In fact, we have used an algebraic technique in [Tak2] to specify the densesubspace.(2) We used the fact that D is actually G -equivariant to define the analytic index. For the general case( D is not G -equivariant, but the symbol is still equivariant), we need an averaging procedure. For the details,see Proposition 5.1 in [Kas16].We call the following the Kasparov index theorem [Kas83, Kas16]. Theorem 2.24. (1) ind( D ) is an unbounded Kasparov ( C , C (cid:111) G ) -module. (2) The analytic index is completely determined by the K -homology element [ D ] ∈ KK G ( C ( X ) , C ) , bythe formula ind( D ) = [ c ] ⊗ C ( X ) (cid:111) G j G ([ D ]) . (3) On the other hand, [ D ] is completely determined by the topological data [ σ ClD ] , by the formula [ D ] =[ σ ClD ] ⊗ [ d X ] .Combining all of them, we find that ind( D ) is completely determined by the homotopy class of the symbol,which is a generalization of the Atiyah-Singer index theorem. emark . In this paper, the correspondence [ D ] (cid:55)→ [ c ] ⊗ j G ([ D ]) is called the assembly map, and denotedby µ G .In order to prove this result in our language without additional efforts, it is convenient to suppose thefollowing differential geometrical condition. See also [Roe], for example. Assumption 2.26.
Let ∇ S be a metric connection on S and let V, W be smooth vector fields on X . Weidentify vector fields with 1-forms by the metric. We impose that ∇ S satisfies the following condition : For s ∈ C ∞ ( X, S ), ∇ SV ( c ( W ) s ) = c ( ∇ LC V ( W ))( s ) + c ( W )( ∇ SV ( s )) , where ∇ LC is the Levi-Civita connection.Take a metric connection on E , denoted by ∇ E . Then, a metric connection on E ⊗ S is induced, denotedby ∇ E ⊗ S . We impose that our Dirac operator must be of the following form: D = (cid:88) k id E ⊗ c ( e k ) ⊗ ∇ E ⊗ Se k + h, where h is a bounded section of End( E ⊗ S ), and commutes with the Clifford multiplication.Let us outline the proof. For the first part, D has a well-defined index, we need to check the criterion sothat the module defined in Definition 2.22 is a Kasparov ( C , C (cid:111) G )-module. For this purpose, we need tocheck that D is regular and (1 + D ) − is C (cid:111) G -compact. They are proved in Theorem 5.8 in [Kas16].For the equality ind( D ) = [ c ] ⊗ j G ([ D ]), note that [ c ] is given by [([ c ] ∗ [ C ( X ) (cid:111) G ] , q : C c ( G, C c ( X, E ⊗ S )) → C c ( X, E ⊗ S ) defined in Section 5 in [Kas16]. Anyelement [ c ] ∗ f ∈ [ c ] ∗ [ C ( X ) (cid:111) G ] can be rewritten as [ c ] ∗ ([ c ] ∗ f ), and hence we find that ([ c ] ∗ f ) ⊗ F canbe identified with q ( f ∗ F ) ∈ C c ( X, E ⊗ S ). Therefore the isomorphism holds at the module level. It sufficesto check the criterion to be the Kasparov product. In fact, the second and the third conditions are obvious.For the first one, one can verify that the commutator contains no differentials, thanks to the Leibniz rule.For the equality [ D ] = [ σ ClD ] ⊗ [ d X ], which is a main interest of the present paper, we use the followingcomputation. This is a more explicit version of Proposition 3.10 in [Kas16] Proposition 2.27. [ σ ClD ] = [( C ( X, E ⊗ S ) , ∈ KK G ( C ( X ) , Cl τ ( X )) . The Hilbert Cl τ ( X ) -module struc-ture is given by the family version of Lemma 2.18.Proof. The module for the Kasparov product is given by the space of continuous sections of the Hilbertbundle (cid:91) x ∈ X E x ⊗ S x ⊗ L ( T ∗ x X ) ⊗ Cliff + ( T ∗ x X ) ∼ = (cid:91) x ∈ X E x ⊗ S x ⊗ L ( T ∗ x X ) ⊗ S ∗ x ⊗ S x . Let H x be the fiber E x ⊗ S x ⊗ L ( T ∗ x X ) ⊗ S ∗ x ⊗ S x . Consider the operator ∂ (cid:48) x on H x defined by ∂ (cid:48) x := id E ⊗ ic ( ξ ) ⊗ id ⊗ id + i (cid:88) id ⊗ id ⊗ ∂∂ξ k ⊗ e k · .ξ ∈ T ∗ x X . One can prove that ( C ( X, ∪ x ∈ X H x ) , { ∂ (cid:48) x } x ∈ X ) is a G -equivariant unbounded ( C ( X ) , Cl τ ( X ))-module. In fact, { ∂ (cid:48) x } x ∈ X is a Cl τ ( X )-module homomorphism , commutes with the C ( X )-action, and { (1 + ( ∂ (cid:48) x ) ) − } x ∈ X is a family of compact operators. That means the pair is a Kasparov module. Moreover,the operator is actually G -equivariant, and hence the pair gives an equivariant cycle.One can check that the above class is a Kasparov product of [ σ D ] and [ d ξ ], by a simple computation usingthe Leibniz rule.We would like to compute { ker( ∂ (cid:48) x ) } x ∈ X . For this aim, we may compute the square of the operator. Inthe following, c ∗ ( e k ) denotes the Clifford multiplication from the left: e k · . This notation is compatible withthe original c , because Cliff + is isomorphic to S ∗ ⊗ S as ( Cliff + , Cliff + )-bimodules. Briefly speaking, c is “flat”. in the graded sense Recall that
Cliff + ( T ∗ x X ) acts on H x by the right multiplication. ∂ (cid:48) x ) = id ⊗ id ⊗ (cid:88) k ξ k ⊗ id ⊗ id − (cid:88) k id ⊗ c ( e k ) ⊗ ξ k · ∂∂ξ k ⊗ c ∗ ( e k ) ⊗ id+ (cid:88) k id ⊗ c ( e k ) ⊗ ∂∂ξ k · ξ k ⊗ c ∗ ( e k ) ⊗ id + id ⊗ id ⊗ (cid:88) k (cid:18) − ∂ ∂ξ k (cid:19) ⊗ id ⊗ id= id ⊗ id ⊗ (cid:88) k (cid:18) − ∂ ∂ξ k + ξ k (cid:19) ⊗ id ⊗ id + (cid:88) k id ⊗ c ( e k ) ⊗ id ⊗ c ∗ ( e k ) ⊗ id . Under the natural identification S x ⊗ S ∗ x ∼ = Cliff − ( T ∗ x X ), the element c ( e k ) ⊗ c ∗ ( e k ) defines a map Cliff − ( T ∗ x X ) (cid:51) X (cid:55)→ e k · ( − ∂X X · e k . In fact, c ( e k ) ⊗ c ∗ ( e k )( s ⊗ f ) = ( − ∂s [ c ( e k )( s ) ⊗ c ∗ ( e k ) f ] = ( − ∂s + ∂f [ e k · s ⊗ f · e k ] . Then, 1 ∈ Cliff − ( T ∗ x X ) is the lowest weight vector of (cid:80) k c ( e k ) ⊗ c ∗ ( e k ), whose weight is − n . Such vectoris unique up to scalar multiplication. On the other hand, (cid:80) k (cid:16) − ∂ ∂ξ k + ξ k (cid:17) has the one-dimensional lowestweight space whose weight is n . Therefore the family of the kernel of ∂ (cid:48) x ’s can be naturally identified with E ⊗ S . Proposition 2.28. [ D ] = [ σ ClD ] ⊗ [ d X ] .Proof. Since S ⊗ Cl ( S ∗ ⊗ S ) ∼ = S , L ( X, E ⊗ S ) ∼ = C ( X, E ⊗ S ) ⊗ Cl τ ( X ) L ( X, Cliff + ( T ∗ X )). Once we noticethis isomorphism, we only have to check three conditions in Proposition 2.8. We leave it to the reader.Thanks to Section 2.2, we obtain a product formula, in our language. See Proposition 2.11 in [Tak2] also. Proposition 2.29.
Suppose that X i , G i , S i , E i , and D i satisfy the conditions in Problem 2.19, for i = 1 , .Let X := X × X , G := G × G , S := S (cid:2) S , E := E (cid:2) E and D := D ⊗ id + id ⊗ D . Then, at themodule level, [ D ] = [ D ] ⊗ [ D ];[ σ ClD ] = [ σ ClD ] ⊗ [ σ ClD ];[ d X ] = [ d X ] ⊗ [ d X ];[ c X ] = [ c X ] ⊗ [ c X ] . Moreover, j G ([ D ]) = j G ([ D ]) ⊗ j G ([ D ]) and ind( D ) = ind( D ) ⊗ ind( D ) . Consequently, the four equalities [ D i ] = [ σ ClD i ] ⊗ [ d X i ] and ind( D i ) = [ c X i ] ⊗ j G i ([ D i ]) for i = 1 ,
2, implythe index theorem on X : [ D ] = [ σ ClD ] ⊗ [ d X ] and ind( D ) = [ c X ] ⊗ j G ([ D ]). We “use” this property to definethe objects on the total space in Section 3. As we have pointed out in Introduction, the C -algebra for infinite-dimensional space is trivial. However,another noncommutative algebra which is related with C , can be generalized to infinite-dimensional situation.In fact, the HKT algebra is such an algebra.In order to use the HKT algebra, we reformulate the Kasparov index theorem. We will avoid using the C -algebra. We recall the following KK G -equivalence. Lemma 2.30.
For an even-dimensional and G -equivariantly Spin c -manifold X , fix a G -equivariant Spinorbundle S . By definition, S x is a left Cliff − ( T ∗ x X ) -module at each x ∈ X . At the same time, it is a right liff + ( T ∗ x X ) -module. Considering the construction Lemma 2.18 fiberwisely, we obtain a right Hilbert Cl τ ( X ) -module C ( X, S ) equipped with a left C ( X ) -action. It determines a KK G -element [ S ] := [( C ( X, S ) , ∈ KK G ( C ( X ) , Cl τ ( X )) . Similarly, we can define [ S ∗ ] = [( C ( X, S ∗ ) , ∈ KK G ( Cl τ ( X ) , C ( X )) .Then, [ S ] ⊗ Cl τ ( X ) [ S ∗ ] = 1 C ( X ) and [ S ∗ ] ⊗ C ( X ) [ S ] = 1 Cl τ ( X ) . Consequently, C ( X ) is KK G -equivalentto Cl τ ( X ) . Recall one more operation. Suppose that C is a Z -graded separable C ∗ -algebra. For simplicity, weassume that C is nuclear. Definition 2.31.
The homomorphism τ C : KK G ( A, B ) → KK G ( C ⊗ A, C ⊗ B ) is given by( E, D ) (cid:55)→ ( C ⊗ E, id ⊗ D ) . In fact, C is always S in this paper, where S is C ( X ) equipped with the grading homomorphism (cid:15) givenby (cid:15) ( f )( x ) := f ( − x ). Let S C ( X ) be the graded tensor product S ⊗ Cl τ ( X ). The same symbol denotes theHKT algebra.Let us reformulate the Kasparov index theorem by using these operations. Put[ (cid:101) D ] := τ S ([ S ∗ ] ⊗ [ D ]) ∈ KK G ( S C ( X ) , S );[ (cid:103) σ ClD ] := τ S (cid:0) [ S ∗ ] ⊗ [ σ ClD ] (cid:1) ∈ KK G ( S C ( X ) , S C ( X ));[ (cid:102) d X ] := τ S ([ d X ]) ∈ KK G ( S C ( X ) , S );[ (cid:101) c ] := τ S (cid:0) [ c ] ⊗ j G ([ S ]) (cid:1) ∈ KK ( S , S C ( X ) (cid:111) G ) . Using these reformulated KK -elements, we can reformulate the index theorem. Proposition 2.32.
The following two equalities hold: τ S (ind( D )) = [ (cid:101) c ] ⊗ S C ( X ) (cid:111) G j G ([ (cid:101) D ]) , [ (cid:101) D ] = [ (cid:103) σ ClD ] ⊗ S C ( X ) [ (cid:102) d X ] . As a result, τ S (ind( D )) is determined by the topological data [ (cid:103) σ ClD ] .Proof. The proof is done by several formal computations in KK G -theory.[ (cid:103) σ ClD ] ⊗ S C ( X ) [ (cid:102) d X ] = τ S (cid:0) [ S ∗ ] ⊗ C ( X ) [ σ ClD ] (cid:1) ⊗ S C ( X ) τ S ([ d X ])= τ S (cid:0) [ S ∗ ] ⊗ C ( X ) [ σ ClD ] ⊗ Cl τ ( X ) [ d X ] (cid:1) = τ S (cid:0) [ S ∗ ] ⊗ C ( X ) [ D ] (cid:1) = [ (cid:101) D ] . [ (cid:101) c ] ⊗ S C ( X ) (cid:111) G j G ([ (cid:101) D ]) = τ S (cid:0) [ c ] ⊗ C ( X ) (cid:111) G j G ([ S ]) (cid:1) ⊗ S C ( X ) (cid:111) G j G (cid:0) τ S (cid:0) [ S ∗ ] ⊗ C ( X ) [ D ] (cid:1)(cid:1) = τ S (cid:0) [ c ] ⊗ C ( X ) (cid:111) G j G ([ S ]) ⊗ Cl τ ( X ) (cid:111) G j G (cid:0) [ S ∗ ] ⊗ C ( X ) [ D ] (cid:1)(cid:1) = τ S (cid:0) [ c ] ⊗ C ( X ) (cid:111) G j G (cid:8) [ S ] ⊗ Cl τ ( X ) [ S ∗ ] ⊗ C ( X ) [ D ] (cid:9)(cid:1) = τ S (cid:0) [ c ] ⊗ C ( X ) (cid:111) G j G ([ D ]) (cid:1) = τ S (ind( D )) .
17e can easily describe the modified Kasparov modules, by using S ∗ ⊗ S ∼ = End( S ∗ ) ∼ = Cliff + . Lemma 2.33.
The algebra bundle Cliff + ( T ∗ X ) acts on itself from the both sides. By this action, [ S ∗ ] ⊗ [ D ] = [( L ( X, E ⊗ Cliff + ( T ∗ X )) , id S ∗ ⊗ D )] ∈ KK G ( Cl τ ( X ) , C );[ S ∗ ] ⊗ [ σ ClD ] = [( C ( X, E ⊗ Cliff + ( T ∗ X )) , ∈ KK G ( Cl τ ( X ) , Cl τ ( X )) . Then, the reformulated KK -elements [ (cid:101) D ] and [ (cid:103) σ ClD ] can be easily described. Our construction in Section5 is modeled on these formulas.
In our problem 1.1, a U (1)-central extension appears. In the finite-dimensional setting, it is possible to regardsomething τ -twisted G -equivariant, as something G τ -equivariant satisfying a certain condition. On the otherhand, in the infinite-dimensional setting, the situation is completely different: We have defined a substituteof the τ -twisted group C ∗ -algebra of LT , but it seems to be too difficult to construct a substitute of untwistedone which should be isomorphic to “ C ( (cid:99) LT )” because of the Pontryagin duality. Therefore, we must study τ -twisted equivariant theory without making reference to G τ -equivariant one.Let us recall the twisted equivariant KK -theory for special cases. For details, consult with [Tak2]. Let1 → U (1) i −→ G τ p −→ G → U (1)-central extension of G . The homomorphisms are supposed to be smooth. Let A and B be G - C ∗ -algebras. Through p , we can regard A and B as G τ -algebras. Definition 2.34. A τ -twisted G -action on a vector space, a Hilbert module or a vector bundle, is a G τ -action ρ satisfying that ρ ( i ( z )) = z id for any z ∈ U (1). Definition 2.35.
Let k ∈ Z . A kτ -twisted G -equivariant Kasparov ( A, B ) -module is a G τ -equivariantKasparov module ( E, F ) such that i ( z ) ∈ i ( U (1)) ⊆ G τ acts on E as z k id E . The set of homotopy classesof such Kasparov modules is an abelian group, and denoted by KK kτG ( A, B ). The case k = 1 is standard:We always assume that k = 1 by replacing τ with its tensor power kτ . This is the reason why we put KK τG ( A, B ) := KK τG ( A, B ).For any G τ -equivariant Kasparov module ( E, F ), we can consider the averaging procedure with respect tothe U (1)-action; we may assume that F is actually U (1)-equivariant. Decompose E with respect to the weightof the U (1)-action as E = (cid:81) n E n , and then F preserves this decomposition. Put F = (cid:81) n F n . Consequently, KK G τ ( A, B ) is decomposed as (cid:76) KK kτG ( A, B ).One can define the unbounded picture version of it in the obvious way. We will use this picture in thefollowing.Needless to say, KK G τ has the Kasparov product: KK G τ ( A, C ) × KK G τ ( C, B ) → KK G τ ( A, B ). Then,how is the restriction to KK kτG ’s? The answer is the following. See [Tak2] also. Lemma 2.36.
The Kasparov product of x ∈ KK kτG ( A, C ) and y ∈ KK lτG ( C, B ) takes value in KK ( k + l ) τG ( A, B ) .In particular, KK τG ( A, C ) ⊗ KK G ( C, B ) → KK τG ( A, B ) is defined. We can define the partial descent homomorphism as follows. For this aim, we need to introduce thetwisted crossed products of algebras and modules.
Definition 2.37 (Definition 2.13 in [Tak2]) . For a G - C ∗ -algbera A , let A (cid:111) kτ G be the completion of thesubset (cid:8) a ∈ C c ( G τ , A ) | a ( zg ) = z k a ( g ) (cid:9) in the C ∗ -algebra A (cid:111) G τ . It is a direct summand as the C ∗ -algebras.Similarly, for a τ -twisted G -equivariant B -Hilbert module E , let E (cid:111) kτ G be the completion of the subset (cid:8) e ∈ C c ( G τ , E ) | e ( zg ) = z k e ( g ) (cid:9) in E (cid:111) G τ with respect to the B (cid:111) G τ -valued inner product.Let ( E, D ) be a kτ -twisted G -equivariant Kasparov ( A, B )-module. As proved in Lemma 2.16 in [Tak2],18 ( A (cid:111) mτ G ) · ( E (cid:111) nτ G ) = 0 unless m = n − k . • ( E (cid:111) nτ G ) · ( B (cid:111) mτ G ) = 0 unless m = n . • For e , e ∈ E (cid:111) nτ G ⊆ E (cid:111) G τ , ( e | e ) B (cid:111) G τ ∈ B (cid:111) nτ G . • E (cid:111) nτ G is orthogonal to E (cid:111) n (cid:48) τ G if n (cid:54) = n (cid:48) .Therefore, the pair ( E (cid:111) nτ G, (cid:101) D | E (cid:111) nτ G ) turns out to be a Kasparov ( A (cid:111) ( n − k ) τ G, B (cid:111) nτ G )-module. Thefollowing is actually defined at the level of modules. Definition 2.38.
The partial descent homomorphism is the correspondence j nτG : KK kτG ( A, B ) (cid:51) [( E, D )] (cid:55)→ [( E (cid:111) nτ G, (cid:101) D | E (cid:111) nτ G )] ∈ KK ( A (cid:111) ( n − k ) τ G, B (cid:111) nτ G ) .j τG denotes j τG .The name “partial descent homomorphism” comes from the fact that j G τ = (cid:80) n j nτG .Let us describe the partial assembly map. Recall that X is a complete Spin c -manifold, and G acts on X isometrically, properly and cocompactly. For this time, we assume that E ⊗ S is a τ -twisted G -equivariantClifford module bundle, and D is a G τ -equivariant Dirac operator on E ⊗ S . Then, ( L ( X, E ⊗ S ) , D ) is a τ -twisted G -equivariant ( C ( X ) , C )-module, and the analytic index is an element of KK ( C , C (cid:111) τ G ).The Mishchenko line bundle [ c ] is an element of KK ( C , C ( X ) (cid:111) G ), but it can be regarded as an element of KK ( C , C ( X ) (cid:111) G τ ) at the same time, by the projection onto the direct summand C ( X ) (cid:111) G τ → C ( X ) (cid:111) G .Then we can define the Kasparov product of [ c ] and j τG ([ D ]) ∈ KK ( C ( X ) (cid:111) G, C (cid:111) τ G ). Let us define thepartial assembly map: µ τG ([ D ]) := [ c ] ⊗ C ( X ) (cid:111) G j τG ([ D ]) : KK τG ( C ( X ) , C ) → KK ( C , C (cid:111) τ G ). In oursituation, this partial assembly map is the same with the assembly map for the whole group G τ . Proposition 2.39.
The following diagram commutes: KK τG ( C ( X ) , C ) µ τG (cid:41) (cid:41) j τG (cid:15) (cid:15) (cid:31) (cid:127) (cid:47) KK G τ ( C ( X ) , C ) j Gτ (cid:15) (cid:15) µ Gτ (cid:117) (cid:117) KK ( C ( X ) (cid:111) G, C (cid:111) τ G ) [ c ] ⊗ C X ) (cid:111) G − (cid:15) (cid:15) KK ( C ( X ) (cid:111) G τ , C (cid:111) G τ ) [ c ] ⊗ C X ) (cid:111) Gτ − (cid:15) (cid:15) KK ( C , C (cid:111) τ G ) (cid:31) (cid:127) (cid:47) KK ( C , C (cid:111) G τ ) . Remark . When the central extension G τ → G is topologically trivial, we can define a U (1)-valued2-cocycle τ on G by fixing a diffeomorphism G τ ∼ = G × U (1), that is, τ : G × G → U (1) satisfying that( g, z ) · ( g , z ) = ( g g , z z τ ( g , g )). For a vector space, a vector bundle or a Hilbert module E , that E is τ -twisted G -equivariant, is equivalent to that a map ρ : G → Aut( E ) satisfying that ρ ( g ) ρ ( g ) = ρ ( g g ) τ ( g , g ), is given. Using this description, we can construct the τ -twisted G -equivariant theory, withoutmaking reference to the extended group G τ . In fact, this new description is more convenient for some cases,and we will mainly use it, although the old description (with the extended group) is useful to compare withthe untwisted cases, just like Proposition 2.39. LT -equivariant K K -theory and the problem
In this short section, we will introduce the LT -equivariant KK -theory and its twisted version. The generalstudy on this topic is far from being enough. See Caution 3.3. Then, we will set the problem and simplify itby dividing the manifold into two parts, just like [Tak1, Tak2, Tak3].19 .1 LT -equivariant KK -theory Our loop group LT = C ∞ ( S , T ) is equipped with the C ∞ -topology. In fact, this topology is regarded asthe inverse limit of the “ L k -topology” in this paper. In order to prove the continuity of a map from LT , wealways check the continuity with respect to the “ L k -topology” for some appropriate k . Definition 3.1.
Let A and B be LT - C ∗ -algebras. Kasparov ( A, B )-modules and LT -equivariant KK -groups are defined, just like Definition 2.3. The LT -equivariant KK -group is denoted by KK LT ( A, B ). Thehomotopy class of a Kasparov module (
E, F ) is denoted by [(
E, F )], as usual.Just like usual cases, KK LT ( A, B ) is a homotopy invariant of A and B , which is covariant and contravari-ant in B and A respectively. Definition 3.2.
Let ( E , F ) and ( E , F ) be an LT -equivariant Kasparov ( A, C )-module and an LT -equivariant Kasparov ( C, B )-module, respectively. An LT -equivariant Kasparov ( A, B )-module ( E ⊗ E , F )is a Kasparov product of ( E , F ) and ( E , F ), if and only if ( E ⊗ E , F ) is a Kasparov product of ( E , F )and ( E , F ) after forgetting the LT -action. Caution 3.3.
We have defined the concept of Kasparov products at the level of Kasparov modules. However,we have not proved the homotopy invariance of the product. Moreover, we have not proved the existence northe uniqueness of the Kasparov product. These are because we have not yet proved the Kasparov technicaltheorem for LT -equivariant cases. At least, the elegant proof in [Kas88] is not valid for our cases.Thus, we can say “a Kasparov module z is a Kasparov product of Kasparov modules x and y ”, but we cannot say “the Kasparov product of KK LT -elements [ x ] and [ y ], is [ z ]”. In this sense, [ z ] = [ x ] ⊗ [ y ] isjust a formal expression. It is perhaps better to write it as z ∈ { x ⊗ y } or something. Such a problem willbe interesting, but we do not study further general theory on KK LT .We can also formulate unbounded Kasparov modules and the bounded transformations for LT -equivariantcases, in the obvious way. The criterion 2.8 is still valid.We can of course study the twisted version of the LT -equivariant KK -theory. We do not repeat thedetails. Definition 3.4.
We can define bounded τ -twisted LT -equivariant Kasparov modules, τ -twisted LT -equivariant KK -theory, unbounded τ -twisted LT -equivariant Kasparov modules, the bounded transformation, Kasparovproducts, and the criterion to be a Kasparov product in the unbounded picture. Let us set the problem precisely here. Firstly, we recall the gauge action of LT on L t ∗ := Ω ( S , t ). Thisaction is defined by l.A = A − l − dl for l ∈ LT and A ∈ L t ∗ , where we identify Ω ( S , t ) with the set ofconnections on the trivial bundle S × T → S . Our manifold differs from L t ∗ by some “compact-part”. Definition 3.5.
An infinite-dimensional manifold M is a “proper LT -space” if it has an LT -action and aproper, equivariant and smooth map Φ : M → L t ∗ . LT has specific subgroups: T is the set of constant loops, Π T is the set of geodesic loops starting fromthe origin. The “rest part” defined below, is denoted by U . Each element g of the identity component of LT , has a lift f ∈ C ∞ ( S , t ) such that g ( θ ) = exp( f ( θ )). U is the set of such g = exp( f ) satisfying that (cid:82) S f ( θ ) dθ = 0. Then, the following canonical decomposition is defined: LT = ( T × Π T ) × U. In this decomposition, the component T is regarded as the set of averages of contractible loops. T is not a vector space, and “ L k ” does not make sense. However, up to some finite-dimensional issue, LT can be identifiedwith C ∞ ( S , t ) where we can define the concept of L k .
20y this decomposition, the group action LT (cid:8) L t ∗ is simplified: T acts there trivially, and U acts therefreely. The constant 1-forms t ⊆ L t ∗ is a global slice of the U -action. Π T preserves t , and acts there bytranslations. Therefore, L t ∗ is isomorphic to t × U as ( T × Π T ) × U -spaces. Moreover, Φ − ( t ) =: (cid:102) M is aglobal slice in M with respect to the U -action; M is, as ( T × Π T ) × U -spaces, isomorphic to (cid:102) M × U . We impose the following assumption on the (cid:102) M -part in order to study the index theory. Assumption 3.6. Φ − ( t ) is even-dimensional and T × Π T -equivariantly Spin c , that is, M is “even-dimensionaland Spin c ”.Our problem is the following. Problem 3.7.
For a proper LT -space M satisfying Assumption 3.6 and equipped with a τ -twisted LT -equivariant line bundle L , construct an LT -equivariant Spinor bundle S , and study the KK -theoretical LT -equivariant index theory. More precisely, construct three Kasparov modules corresponding to the indexelement, the Clifford symbol element and the Dirac element. Then, study the assembly map.Just like [Tak1, Tak2, Tak3], we would like to divide the problem into two parts. For this aim, we recallSection 2.2. Just like Lemma 2.11, we can prove the following. Lemma 3.8.
Let A and B be T × Π T - C ∗ -algebras, and let A and B be U - C ∗ -algebras. Then, A := A ⊗ A and B := B ⊗ B are LT - C ∗ -algebras.Suppose that an unbounded T × Π T -equivariant Kasparov ( A , B ) -module ( E , D ) and unbounded an U -equivariant Kasparov ( A , B ) -module ( E , D ) are given. Then, ( E, D ) := ( E ⊗ E , D ⊗ id + id ⊗ D ) is an LT -equivariant Kasparov ( A, B ) -module. The twisted version of this lemma is not obvious, but it is still valid, thanks to Proposition 2.27 in [FHT](or Proposition 2.8 in [Tak3]). In this proposition, an admissible U (1)-central extension of LT is proved tobe split into two parts: LT τ = ( T × Π T ) τ (cid:2) U (1) U τ . We use the same symbol for the restrictions of τ . Lemma 3.9.
Under the isomorphism
M ∼ = (cid:102) M × U , L = L| (cid:102) M (cid:2) L| U including the group action. The linebundle L| U is given by the standard construction U τ × U (1) C . Thanks to these results, we may construct everything on (cid:102) M and U , separately. Since (cid:102) M is finite-dimensional, we can construct anything without serious problems. For the U -part, we set the following(simplified) problem. We will concentrate on it from now on. Problem 3.10.
For a positive definite central extension τ of U , construct three unbounded Kasparov moduleswhich are representatives of the index element [ (cid:101) (cid:1) ∂ ] ∈ KK τU ( S C ( U ) , S ), the Clifford symbol element [ (cid:103) σ Cl (cid:1) ∂ ] ∈ KK τU ( S C ( U ) , S C ( U )) and the Dirac element [ (cid:102) d U ] ∈ KK U ( S C ( U ) , S ), respectively. Then, prove that theKasparov module representing [ (cid:101) (cid:1) ∂ ] is a Kasparov product of the one representing [ (cid:103) σ Cl (cid:1) ∂ ] and the one representing[ (cid:102) d U ]. Formally, prove the equality [ (cid:101) (cid:1) ∂ ] = [ (cid:103) σ Cl (cid:1) ∂ ] ⊗ S C ( U ) [ (cid:102) d U ] . Then, study the assembly map.If we can solve this problem, we can define the Kasparov modules on the whole manifold M , “using”Section 2.2. Put S C ( M ) := S C ( U ) ⊗ Cl τ ( (cid:102) M ), and let (cid:26)(cid:26) D be the ( T × Π T ) τ -equivariant Dirac operator on (cid:102) M . We define them as follows. The following tensor product is indeed at the level of modules.[ (cid:101) D ] := [ (cid:101) (cid:26)(cid:26) D ] ⊗ [ (cid:101) (cid:1) ∂ ];[ (cid:103) σ Cl D ] := [ (cid:103) σ Cl (cid:26) D ] ⊗ [ (cid:103) σ Cl (cid:1) ∂ ];[ (cid:103) d M ] := [ d (cid:102) M ] ⊗ [ (cid:102) d U ] . In [Tak1, Tak3], we set M := M / Ω T , where Ω T is the set of loops starting from the origin. (cid:102) M is a Π T -covering space of M . (cid:101) D ] = [ (cid:103) σ Cl D ] ⊗ [ (cid:103) d M ].As a final remark of this section, we recall some examples of proper LT -spaces. See [Tak3]. Examples . (1) A Hamiltonian LT -space is automatically a proper LT -space.(2) In particular, a gauge orbit U × Π T ⊆ L t ∗ is a proper LT -space.(3) In general, for an even-dimensional Spin c -manifold M equipped with a T -action, suppose that it hasa T -equivariant Spinor bundle, and a T -equivariant map φ : M → T , where T acts on itself by the conjugateaction, namely trivially. Then, the pullback construction M × T L t ∗ with respect to the holonomy map L t ∗ → T gives a proper LT -space. The condition looks slightly complicated, but a symplectic manifold withan T -action always satisfies these condition, after perturbing the symplectic form. The map taking values in T is the circle valued-moment map studied in [McD]. In this section, we study the HKT algebra: its definition, the symmetry, and the representation. The newpoint is the connection with the the representation of the infinite-dimensional group U . U We begin with studies of U . U is identified with (cid:8) f : S → t | (cid:82) S f ( θ ) dθ = 0 (cid:9) . Fix an inner product of t ,denoted by ( •|• ) t , and then we can take finite-dimensional approximations of U :Lie( U N ) := Span (cid:26) cos θ √ π , sin θ √ π , cos 2 θ √ π , sin 2 θ √ π , · · · , cos N θ √ N π , sin N θ √ N π (cid:27) , where is a unit vector of t . From now on, we fix , and we omit it. Lie( U ) has a symplectic form ω ( f , f ) := (cid:82) (cid:16) f (cid:12)(cid:12)(cid:12) df dθ (cid:17) t dθ . The above is a symplectic bases: ω ( cos kθ √ kπ , cos lθ √ lπ ) = ω ( sin kθ √ kπ , sin lθ √ lπ ) = 0 and ω ( cos kθ √ kπ , sin lθ √ lπ ) = δ k,l .The group U is, as a Lie group, diffeomorphic to “ R ∞ ”. The coordinate is often written as, when U isregarded as a space , exp (cid:18)(cid:88) (cid:18) x k cos kθ √ kπ + y k sin kθ √ kπ (cid:19)(cid:19) (cid:55)→ ( x , y , x , y , · · · )and the coordinate is often written as, when U is regarded as a group ,exp (cid:18)(cid:88) (cid:18) g k cos kθ √ kπ + h k sin kθ √ kπ (cid:19)(cid:19) (cid:55)→ ( g , h , g , h , · · · ) . The completion of U with respect to the norm on the Lie algebra (cid:107) ( g , h , g , h , · · · ) (cid:107) L l := (cid:88) k (1 + k l )( g k + h k )is denoted by U L l . An element of this group is called an “ L l -loop”. In order to define the HKT algebra, weuse the L / -metric . With respect to this norm, ω ( • , • ) is a continuous bilinear form. Definition 4.1.
The central extension U τ is given by U × U (1) equipped with the multiplication(exp( f ) , z ) · (exp( f ) , z ) := (exp( f + f ) , z z e i ω ( f ,f ) ) . Lie( U τ ) ∼ = Lie( U ) ⊕ u (1). Let K be the generator of u (1). For exp( f ) ∈ U , X ∈ Lie( U ) and z ∈ U (1), theadjoint representation is given by Ad exp( f ) ( X ) = X + ω ( f, X ) K, Topology of “ R ∞ ” is highly non-trivial. z ( X ) = X, and hence the Lie algebra structure is given by, for f , f ∈ Lie( U ),[ f , f ] = ω ( f , f ) K, [ K, f ] = 0 .LT has an S -symmetry θ.f ( s ) := f ( s + θ ). When we deal with this action, we write S as T rot . Apositive energy representation is a representation of LT which reflects the T rot -symmetry, and which satisfiesa certain finiteness condition. We define it for U in our language. Consult [Tak3] (or of course [PS, FHT])for the T × Π T -part. Definition 4.2.
Let V be a separable Hilbert space, and let U ( V ) be the unitary group of V , which is topol-ogized by the compact open topology. A continuous map ρ : U → U ( V ) is a positive energy representation(PER for short) at level τ of U if it satisfies the following condition: • ρ (exp( f )) ◦ ρ (exp( f )) = ρ (exp( f + f )) e i ω ( f ,f ) . In other words, ρ is a homomorphism from theextended group U τ satisfying that ρ ( i ( z )) = z id V . • It extends to ρ : U (cid:111) T rot → U ( V ). • When we decompose V by the weight of T rot -action by ρ | T rot as V = ⊕ n V n , each V n is finite-dimensionaland V n = 0 for any sufficiently small n .For the sake of computations, it is more convenient to study the infinitesimal version dρ : Lie( U ) → End( V ), which satisfies the commutation relation [ dρ ( f ) , dρ ( f )] = iω ( f , f )id. Let d be the infinitesimalgenerator of T rot . Then, the Lie bracket Lie( U τ (cid:111) T rot ) is given by [ d, f ] = f (cid:48) and [ d, K ] = 0.We can construct an irreducible PER as follows. It is known that there is only one PER up to isomorphism. Definition 4.3.
By an L -function on R defined by π / e − x which is a unit vector, define an isometricembedding I N : L ( R N ) (cid:51) f (cid:55)→ f ⊗ π / e − x ∈ L ( R N +1 ) , and define L ( R ∞ ) by the Hilbert space inductive limit lim −→ L ( R N ). The “infinite tensor product” π / e − x ⊗ π / e − x ⊗ · · · defines a unit vector denoted by b , where “ b ” comes from “boson”.The “function” π / e − x N ⊗ π / e − x N +12 ⊗ · · · ⊗ π / e − x M is denoted by ( b ) MN , where 1 ≤ N ≤ M ≤ ∞ .Note that M = ∞ is allowed. In this notation, b = ( b ) ∞ .Let L ( R N ) fin be the subspace which is algebraically spanned by functions of the form “polynomial × e − (cid:107) x (cid:107) ”. L ( R ∞ ) fin denotes the algebraic inductive limit lim −→ alg L ( R N ) fin . Needless to say, b is an elementof L ( R ∞ ) fin .In a diagram, our Hilbert spaces are organized in the following commutative diagram L ( R ) I (cid:47) (cid:47) J (cid:41) (cid:41) L ( R ) I (cid:47) (cid:47) J (cid:36) (cid:36) · · · I N − (cid:47) (cid:47) L ( R N ) I N (cid:47) (cid:47) J N (cid:121) (cid:121) L ( R N +1 ) I N +1 (cid:47) (cid:47) J N +1 (cid:116) (cid:116) · · · L ( R ∞ )Every Hilbert space L ( R N ) has a dense subspace L ( R N ) fin . I N ’s and J N ’s preserve these dense subspaces.23 efinition 4.4. On L ( R ∞ ) fin , let dρ (cid:18) cos kθ √ kπ (cid:19) := ∂∂x k ; dρ (cid:18) sin kθ √ kπ (cid:19) := ix k × . Strictly speaking, these operations are defined as follows: For f ∈ L ( R ∞ ) fin , choosing (cid:101) f ∈ L ( R N ) such that J N ( (cid:101) f ) = f and N ≥ k , we define dρ (cid:18) cos kθ √ kπ (cid:19) ( f ) := J N (cid:32) ∂ (cid:101) f∂x k (cid:33) , and similarly for dρ (cid:16) sin kθ √ kπ (cid:17) ( f ).These operators are skew-adjoint operators, and hence they generate a unitary operators. Moreover, theysatisfy the positive energy condition. In order to prove that, we introduce the complex basis: z k := 1 √ (cid:18) cos kθ √ kπ + i sin kθ √ kπ (cid:19) ; z k := 1 √ (cid:18) cos kθ √ kπ − i sin kθ √ kπ (cid:19) . These are unit vectors. The infinitesimal generator of the rotation action is given by dρ ( d ) := − i (cid:88) k kdρ ( z k ) dρ ( z k ) . This is well-defined and satisfies [ dρ ( d ) , f ] = dρ (cid:16) dfdθ (cid:17) , as computed in [Tak1]. Moreover, each eigenvalue of dρ ( d ) /i is a non-negative integer, and its multiplicity is the number of partitions. In fact, L ( R ∞ ) fin is theset of finite linear combinations of eigenvectors of dρ ( d ).We can define the dual representation in the usual way. To clarify the notation, we describe it. Definition 4.5.
Let L ( R ∞ ) ∗ be the dual space of L ( R ∞ ), which is anti-linearly isomorphic to L ( R ∞ ).For g ∈ U , let ρ ∗ ( g ) ∈ U ( L ( R ∞ ) ∗ ) be the operator defined by [ t ρ ( g − )]. Then, ρ ∗ (exp( f )) ◦ ρ ∗ (exp( f )) = ρ ∗ (exp( f + f )) e − i ω ( f ,f ) , that is, ρ ∗ is a “ − τ -twisted” representation. For θ ∈ T rot , let ρ ∗ ( θ ) := t ρ ( − θ ).This operation extends ρ to a homomorphism from U τ (cid:111) T rot11 . It satisfies the “negative energy condition”.The set of finite linear combinations of eigenvectors of dρ ∗ ( d ), is denoted by L ( R ∞ ) ∗ fin . Just like L ( R ∞ ),we use the symbol ∗ b and ( ∗ b ) MN .As proved in Theorem 3.15 of [Tak1], L ( R N ) ⊗ L ( R N ) ∗ is isomorphic to L ( U N ) as twisted representationspaces, by the Peter-Weyl type homomorphism . We only define the regular representations and describethe Peter-Weyl type homomorphism. For the details, see also section 3 in [Tak1]. In the following, we identifyLie( U N ) with U N by the exponential map to simplify the notation. For g = exp( f ) ∈ U , ω ( g, • ) means that ω ( f, • ). Definition 4.6.
For f ∈ L ( U N ) and g ∈ U N , define[ L ( g )( f )]( x ) := f ( x − g ) e i ω ( g,x ) ;[ R ( g )( f )]( x ) := f ( x + g ) e i ω ( g,x ) . The action of i ( z ) ∈ i ( U (1)) ⊆ U τ is given by z − id. It is more natural to regard L ( U N ) as the Hilbert space consisting of L -sections of a line bundle L → U N defined by U τN ⊗ U (1) C . In this picture, several properties can be easily proved: two actions L and R clearly commute, thanks to theassociativity of the group operation. However, in the present paper, it is more convenient to forget the line bundle. L and R are twisted representations at opposite level: L ( g ) ◦ L ( g ) = L ( g + g ) e i ω ( g ,g ) ; R ( g ) ◦ R ( g ) = R ( g + g ) e − i ω ( g ,g ) . Let e k ’s and f k ’s be the unit vectors of Lie( U N ) corresponding to cos kθ √ kπ ’s and sin kθ √ kπ ’s, respectively. Let( x , y , · · · , x N , y N ) be the coordinate described above. Then, the infinitesimal action is computed as follows: dL ( e k ) = − ∂∂x k + i y k ; dL ( f k ) = − ∂∂y k − i x k ; dR ( e k ) = ∂∂x k + i y k ; dR ( f k ) = ∂∂y k − i x k . Proposition 4.7.
The Peter-Weyl type homomorphism
Ψ : L ( R N ) ⊗ L ( R N ) ∗ → L ( U N ) defined by Ψ( v ⊗ f )( x ) := (cid:18) √ π (cid:19) N (cid:104) ρ ( − x )( v ) , f (cid:105) is an isometric isomorphism as representation spaces of U τN × U − τN . Seeing this result, we define the “ L -space of U ” as follows. See also Section 3.2 of [Tak1]. Definition 4.8. L ( U ) is defined by L ( R ∞ ) ⊗ L ( R ∞ ) ∗ . The representations ρ ⊗ id and id ⊗ ρ ∗ are alsodenoted by L and R , respectively.Let L ( U ) fin be the algebraic tensor product L ( R ∞ ) fin ⊗ alg L ( R ∞ ) ∗ fin . Each element of this densesubspace is regarded as “polynomial × (cid:16) √ π (cid:17) ∞ e − | g | ”. We introduce notationsvac := Ψ( b ⊗ ∗ b ); vac MN := Ψ(( b ) MN ⊗ ( ∗ b ) MN ) . S C ( U ) We recall the definition of the HKT algebra which can be regarded as the “suspension of the Clifford algebra-valued function algebra of U ”. In fact, we will slightly modify the description so that it fits with ourrepresentation introduced in the next subsection. This is because our “smooth functions” must be “approx-imately constant”. We will prove that the modified one is isomorphic to the original one as C ∗ -algebrasthorough a “diffeomorphism on U ”.We concentrate on U which has a special property: T rot acts there linearly, and each weight space isreal two-dimensional. These properties allow us to define a natural filtration of finite-dimensional subspacesthanks to the Fourier series theory U ⊆ U ⊆ · · · ⊆ U N ⊆ · · · ⊆ U. We use only this filtration to construct the HKT algebra. This is the first (but small) different point fromthe original one.We also modify the Clifford element. This is the main different point from the original one. Recall thecoordinate on U : exp (cid:16)(cid:80) k (cid:16) x k cos kθ √ kπ + y k sin kθ √ kπ (cid:17)(cid:17) (cid:55)→ ( x , y , x , y , · · · ), and recall that e k and f k denote thecotangent vectors dx k and dy k , respectively. Definition 4.9.
Let l be a positive real number greater than . As a Clifford algebra-valued function on U M , we introduce a Clifford element C MN := M (cid:88) k = N k − l ( x k e k + y k f k ) ∈ C ∞ ( U M , Cliff + ( T ∗ U M ))25or N ≤ M < ∞ .By an abuse of notation, C ∞ N (or simply C N ) means the formal infinite sum (cid:80) ∞ i = N k − l ( x k e k + y k f k ). Wewill prove that this formal infinite sum makes sense as an operator on L ( U ). Remarks . (1) Except for the statement of Lemma 4.15 and its proof, we fix l , and we omit l in thesymbol of the Clifford element.(2) The original Clifford element is (cid:80) Mi = N ( x k e k + y k f k ) ∈ C ∞ ( U M , Cliff + ( T ∗ U M )). If we consider (for-mally) the case when N = 1 and M = ∞ , this original Clifford element is invariant under all the isometriclinear action. On the other hand, our Clifford element is much less symmetric.Let us recall that a self-adjoint unbounded operator can be substituted for a function on the real line.For example, the heat kernel e − tD can be defined for a Dirac operator D . If the operator has compactresolvent and the function vanishes at infinity, the obtained operator is compact. Just like this, we can getan element of a C ∗ -algebra from an “unbounded multiplier with compact resolvent” and a function on thereal line vanishing at infinity. Definition-Proposition 4.11 (See [Wor] or Appendix A.3. of [HKT]) . Let A be a Z -graded C ∗ -algebra.An A -linear operator D from a dense, Z -graded, right A -submodule A , into A is an (odd) unboundedself-adjoint multiplier if the following conditions are fulfilled: • ( Dx ) ∗ y = x ∗ ( Dy ) for all x, y ∈ A . • The operator D ± i id are isomorphisms from A to A . • D reverses the grading.For such D , there is an operator calculus homomorphism taking values in the multiplier algebra M ( A ) of A , S (cid:51) f (cid:55)→ f ( D ) ∈ M ( A ) mapping ( x ± i ) − to ( D ± i id) − . S is graded by [ (cid:15) ( f )]( t ) = f ( − t ) . With respect to this grading, the abovehomomorphism preserves the grading.If ( D ± i id) − belongs to A itself, this homomorphism takes value in A . Such D is said to be with compactresolvent , which is because the set of A -compact operators on the right A -module is A itself, while the setof adjointable operators on A is M ( A ) . On S , the operator X defined by [ Xf ]( x ) := xf ( x ) is an unbounded self-adjoint multiplier with compactresolvent. This is a typical example of unbounded multipliers with compact resolvent.Moreover, the Clifford element C N,M is also an unbounded self-adjoint multiplier on the C ∗ -algebra C ( U M (cid:9) U N − ) := Cl τ ( U M (cid:9) U N − ). It can be proved as follows. Firstly, let N be 1. Then, C ,M is anunbounded multiplier with compact resolvent. This is because ( C M + i )( C M − i ) = 1 + (cid:80) k − l ( x k + y k )is strictly positive. The general case is completely parallel to this one. Consequently, X ⊗ id + id ⊗ C MN isan unbounded self-adjoint multiplier with compact resolvent of S ⊗ C ( U M (cid:9) U N − ). Let S C ( U N ) be thegraded tensor product S ⊗ C ( U N )We are in the position to talk about the homomorphism connecting S C ( U N ) and S C ( U M ). Notice that S C ( U M ) ∼ = S C ( U M (cid:9) U N ) ⊗ C ( U N ). Definition 4.12. A ∗ -homomorphism β N,M : S C ( U N ) → S C ( U M ) for N ≤ M is defined by β N,M ( f ⊗ h ) := f ( X ⊗ id + id ⊗ C MN +1 ) ⊗ h. Thanks to the equality C MN +1 + C LM +1 = C LN +1 and Proposition 3.2 in [HKT], we find that β M,L ◦ β N,M = β N,L . It enable us to take an inductive limit of the sequence
S C ( U ) β , −−→ S C ( U ) β , −−→ · · · . efinition 4.13. By the C ∗ -algebra inductive limit, we define S C ( U ) := lim −→ S C ( U N ) . We call this algebra the
HKT algebra .We define a subalgebra of the HKT algebra consisting of “rapidly decreasing functions”. It is convenientto adopt narrower one than the natural one. Let f e ( x ) := e − x and f o ( x ) := xe − x . They generate S ,thanks to the Stone-Weierstrass theorem. “ e ” and “ o ” come from even and odd, respectively. Definition 4.14.
Let S fin ⊆ S be the ∗ -subalgebra algebraically generated by f e and f o . Let C ( U N ) fin be the dense subalgebra consisting of Schwartz functions. Let S C ( U N ) fin be the algebraic tensor productof S fin and C ( U N ) fin . The Bott map β N,M maps
S C ( U N ) fin into S C ( U M ) fin , and hence we can define adense subalgebra by the algebraic inductive limit S C ( U ) fin := alg lim −→ S C ( U N ) fin . The ∗ -homomorphism defining the inductive limit S C ( U N ) → S C ( U ) is denoted by β N . For the specialone β : S → S C ( U ), we use β .Our HKT algebra is isomorphic to the original one as C ∗ -algebras, as follows. However, our descriptionis much more convenient for (possibly only) our purpose. Lemma 4.15.
Let l β N,M be the ∗ -homomorphism in Definition 4.12 induced by the Clifford element l C MN := (cid:80) Mi = N k − l ( x k e k + y k f k ) . Fix a trivialization T ∗ U M ∼ = U M × Lie( U M ) ∗ , and we regard C N as, not a sectionof the Clifford algebra baundle, but a Clifford algebra-valued function. If we define a ∗ -isomorphism l Φ : C ( U N ) → C ( U N ) by l Φ N ( h )( x , y , x , y , · · · , x N , y N ) := h ( x , y , − l x , − l y , · · · , N − l x N , N − l y N ) , the following diagram commutes: S C ( U N ) id ⊗ l Φ N −−−−−→ S C ( U N ) β N,M (cid:121) (cid:121) l β N,M
S C ( U M ) id ⊗ l Φ M −−−−−→ S C ( U M ) Consequently, our HKT algebra is ∗ -isomorphic to the original one.Remark . Let l φ be the diffeomorphism given by ( x , y , · · · , x N , y N ) (cid:55)→ ( x , y , · · · , N − l x N , N − l y N ).Note that l Φ is
NOT the natural pullback ( l φ ) ∗ , but just l Φ( f ⊗ c ) = [( l φ ) ∗ ( f )] ⊗ c , where f is a scalar -valuedfunction and c ∈ Cliff + ( U N ). In this sense, our isomorphism is not very natural. Proof.
Once we prove the statement when N = 0, the rest part is easy. All the homomorphisms are ∗ -homomorphisms, and hence it suffices to check the commutativity for generators: f e and f o . Let us computethe compositions at f e and f o . See Proposition 3.2 in [HKT] also. For f e , (cid:2) (id ⊗ l Φ M ) ◦ β ,M ( f e ) (cid:3) ( x , y , x , y , · · · , x M , y M )= (id ⊗ l Φ M )[ f e ( X ) ⊗ f e ( l C M )]( x , y , x , y , · · · , x M , y M )= f e ( X ) ⊗ e − (cid:80) Mk =1 k − l ( x k + y k ) . Since l Φ is the identity, l β ,M ◦ (id ⊗ l Φ )( f e )( x , y , x , y , · · · , x M , y M )= f e (cid:32) X ⊗ id + id ⊗ M (cid:88) k =1 k − l ( x k e k + y k f k ) (cid:33) = f e ( X ) ⊗ e − (cid:80) Mi =1 k − l ( x k + y k ) . f o , (id ⊗ l Φ M ) ◦ β ,M ( f o )( x , y , x , y , · · · , x M , y M )= Xf e ( X ) ⊗ f e ( x , y , − l x , − l y , · · · , M − l x M , M − l y M ) + f e ( X ) ⊗ Y f e ( Y ) | Y = l C ,M = Xf e ( X ) ⊗ e − (cid:80) Mk =1 k − l ( x k + y k ) + f e ( X ) ⊗ M (cid:88) k =1 k − l ( x k e k + y k f k ) e − (cid:80) Mb =1 b − l ( x b + y b ) . l β ,M ◦ (id ⊗ l Φ )( f o )( x , y , x , y , · · · , x M , y M )= f o ( X ⊗ id + id ⊗ l C M )= Xf e ( X ) ⊗ e − (cid:80) Mk =1 k − l ( x k + y k ) + f e ( X ) ⊗ M (cid:88) k =1 k − l ( x k e k + y k f k ) e − (cid:80) Mb =1 b − l ( x b + y b ) . Remark . This isomorphism suggests that our “function on U ” is “continuous” even in a weaker topology. In this subsection, we will see that U acts on S C ( U ), and S C ( U ) has a representation on a Hilbert S -module. It looks like the canonical S -module S ⊗ “representation of C ( U )”,although the phrase “representation of C ( U )” does not make sense.These two ingredients the “group action” and the “representation” are compatible. We will put off theproof until the next (main) section, because we will deal with two different U -actions on the representationspace. Theorem 4.18. U acts on S C ( U ) continuously by ∗ -automorphisms.Proof. In this proof, to be precise, σ denotes the group action of U on itself: σ g ( x ) := x + g , where we identify U with Lie( U ), and we regard U as a vector space.Let U fin be the algebraic inductive limit of U N : U fin := ∪ N U N . It acts on S C ( U ) fin in the naturalway: Let g ∈ U N and f ⊗ h ∈ S C ( U M ) fin . When N ≤ M , U N ⊆ U M and hence U N acts on U M . Theaction is defined by g. ( f ⊗ h ) := f ⊗ σ ∗ g − h . When M < N , replace f ⊗ h with β M,N ( f ⊗ h ). This actionis well-defined: β M,L [ g. ( f ⊗ h )] = g. [ β M,L ( f ⊗ h )] for L > M ≥ N . The map induced by g clearly gives anisometric ∗ -automorphism.We must check that the action extends to a U -action on S C ( U ). For this aim, we need some quantitativearguments.Firstly, we check that the operator on S C ( U ) fin induced by g ∈ U fin is continuous with respect to thenorm of S C ( U ). Indeed, for a ∈ S C ( U M ) and g ∈ U N (suppose that M ≥ N ), (cid:107) β M ( g.a ) (cid:107) S C ( U ) = lim L →∞ (cid:107) β M,L ( g.a ) (cid:107) S C ( U L ) = lim L →∞ (cid:107) g. [ β M,L ( a )] (cid:107) S C ( U L ) = lim L →∞ (cid:107) β M,L ( a ) (cid:107) S C ( U L ) = (cid:107) β M ( a ) (cid:107) S C ( U ) , where we used the fact that U N acts on S C ( U L ) as isometric ∗ -automorphisms for any L ≥ N . Consequently, U fin acts on S C ( U ) by ∗ -automorphisms. The precise definition is as follows: For a ∈ S C ( U ) and g ∈ U fin ,choose an approximating sequence { a n } ⊆ S C ( U ) fin , and define g.a by lim n g.a n . Since g ∈ U fin gives anisometry, this limit exists and independent of the choice of { a n } .28econdly, we need to extend the U fin -action to a U -action. For this purpose, it is natural to define theaction by g.a := lim n →∞ ( g n ) .a . In order to justify this definition, we need to check the following: For anyfixed a ∈ S C ( U ) and for any two sufficiently close elements g, g (cid:48) ∈ U fin , g.a − g (cid:48) .a is small. In order todescribe everything precisely, take a positive number ε , and choose a ∈ S C ( U N ) so that (cid:107) a − a (cid:107) < ε .Then (cid:107) g.a − g (cid:48) .a (cid:107) ≤ (cid:107) g. ( a − a ) (cid:107) + (cid:107) g.a − g (cid:48) .a (cid:107) + (cid:107) g (cid:48) . ( a − a ) (cid:107) The first and the third terms are less than ε , because g and g (cid:48) give isometries. For the second term, supposethat g and g (cid:48) belong to U L . We may assume that a is of the form (cid:80) finite sum f i ⊗ a i for some a i ’s which arecompactly supported and smooth, and f i ’s which are monomials of f e and f o . In a similar way to prove theLeibniz rule, we may deal with only f e ⊗ a and f o ⊗ a . In fact, we may assume that f i ⊗ a i is of the form( f e ) n ( f o ) m ( a (cid:48) ) n + m − a (cid:48)(cid:48) fow some suitable n , m , a (cid:48) and a (cid:48)(cid:48) . It can be written as ± ( f e · a (cid:48) ) ·· · · ( f o · a (cid:48) ) · ( f o · a (cid:48)(cid:48) ). Then, if g. ( f e · a (cid:48) ) − g (cid:48) ( f e · a (cid:48) )’s are small enough, (cid:107) g ( f i ⊗ a i ) − g (cid:48) ( f i ⊗ a i ) (cid:107) < ε . Therefore we must prove theboth of g.β N,M ( f e ⊗ a ) − g (cid:48) .β N,M ( f e ⊗ a ) ∈ S C ( U M ) and g.β N,M ( f o ⊗ a ) − g (cid:48) .β N,M ( f o ⊗ a ) ∈ S C ( U M )are arbitrary small uniformly in M , if g is sufficiently close to g (cid:48) . In fact, we need to estimate the limit of thenorm of the above as M tends to infinity, and so we may assume that M is large enough so that g, g (cid:48) ∈ U M .Let us recall, from [HKT], that β N,M ( f e ⊗ a ) = f e ⊗ e − ( C MN +1 ) ⊗ a and β N,M ( f o ⊗ a ) = f o ⊗ e − ( C MN +1 ) ⊗ a + f e ⊗ C MN e − ( C MN ) ⊗ a. Let us estimate the norm of the following functions of the variable x ∈ U M : (cid:104) e − ( C MN +1 ) ⊗ a (cid:105) ( x − g ) − (cid:104) e − ( C MN +1 ) ⊗ a (cid:105) ( x − g (cid:48) ) , (cid:104) C MN +1 e − ( C MN +1 ) ⊗ a (cid:105) ( x − g ) − (cid:104) C MN +1 e − ( C MN +1 ) ⊗ a (cid:105) ( x − g (cid:48) ) . For x = ( x , y , x , y , · · · , x M , y M ), let x | N denote ( x , y , x , y , · · · , x N , y N ) ∈ U N . We use the coordinate x = ( x , y , x , y , · · · , x M , y M ), g = ( g , h , g , h , · · · , g M , h M ) and g (cid:48) = ( g (cid:48) , h (cid:48) , g (cid:48) , h (cid:48) , · · · , g (cid:48) M , h (cid:48) M ). (cid:12)(cid:12)(cid:12)(cid:104) e − ( C MN +1 ) ⊗ a (cid:105) ( x − g ) − (cid:104) e − ( C MN +1 ) ⊗ a (cid:105) ( x − g (cid:48) ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:32) − M (cid:88) N +1 k − l (cid:2) ( x k − g k ) + ( y k − h k ) (cid:3)(cid:33) a ( x − g ) − exp (cid:32) − M (cid:88) N +1 k − l (cid:2) ( x k − g (cid:48) k ) + ( y k − h (cid:48) k ) (cid:3)(cid:33) · a ( x − g (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:32) − M (cid:88) N +1 k − l (cid:2) ( x k − g k ) + ( y k − h k ) (cid:3)(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · | a ( x − g | N ) − a ( x − g (cid:48) | N ) | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:32) − M (cid:88) N +1 k − l (cid:2) ( x k − g k ) + ( y k − h k ) (cid:3)(cid:33) − exp (cid:32) − M (cid:88) N +1 k − l (cid:2) ( x k − g (cid:48) k ) + ( y k − h (cid:48) k ) (cid:3)(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | a ( x − g (cid:48) ) |≤ sup z | grad( a )( z ) | · (cid:107) g − g (cid:48) | N (cid:107) l + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:32) − M (cid:88) N +1 k − l (cid:2) ( x k − g k ) + ( y k − h k ) (cid:3)(cid:33) − exp (cid:32) − M (cid:88) N +1 k − l (cid:2) ( x k − g (cid:48) k ) + ( y k − h (cid:48) k ) (cid:3)(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | a ( x − g (cid:48) ) | . S ⊗ C ( U N ) is the graded tensor product!
29e need to estimate | exp( • ) − exp( • (cid:48) ) | . For this purpose, let x := (( N +1) − l x N +1 , ( N +1) − l y N +1 , · · · , M − l x M , M − l y M )and similarly for g and g (cid:48) . By abuse of notation, let f e ( x N +1 , y N +1 , · · · , x M , y M ) be e − (cid:80) Mk = N +1 ( x k + y k ) untilthe end of this proof. Then, f e ( C MN +1 )( x N +1 , y N +1 , · · · , x M , y M ) = f e ( x ).Let us continue the estimate. Thanks to the mean-value theorem (note that f e is real-valued), | f e ( x − g ) − f e ( x − g (cid:48) ) | ≤ sup w | grad( f e )( w ) |(cid:107) ( x − g ) − ( x − g (cid:48) ) (cid:107) , where w = ( u N +1 , v N +1 , u N +2 , v N +2 , · · · , u M , v M ) runs over U M (cid:9) U N . Since grad( f e ) is bounded, we obtainthe desired result which is independent of M .For the f o -part, we need more subtle argument. In the estimate for f e , we divide the problem into twoparts: the U N -part and the U M (cid:9) U N -part. We use the same technique, and we may assume N = 0. Whatwe would like to compute is the norm of (cid:88) k − l ( x k − g k ) e − (cid:80) Nk =1 b − l [ ( x b − g b ) +( y b − h b ) ] ⊗ e k − (cid:88) k − l ( x k − g (cid:48) k ) e − (cid:80) Nk =1 b − l [ ( x b − g (cid:48) b ) +( y b − h (cid:48) b ) ] ⊗ e k + (cid:88) k − l ( y k − h k ) e − (cid:80) Nk =1 b − l [ ( x b − g b ) +( y b − h b ) ] ⊗ f k − (cid:88) k − l ( y k − h (cid:48) k ) e − (cid:80) Nk =1 b − l [ ( x b − g (cid:48) b ) +( y b − h (cid:48) b ) ] ⊗ f k . For simplicity, we deal with only the former half. Put r := (cid:80) Nk =1 b − l (cid:2) ( x b − g b ) + ( y b − h b ) (cid:3) and ( r (cid:48) ) := (cid:80) Nk =1 b − l (cid:2) ( x b − g (cid:48) b ) + ( y b − h (cid:48) b ) (cid:3) .Then, the square of the pointwise norm is given as follows: (cid:13)(cid:13)(cid:13)(cid:88) k − l ( x k − g k ) e − r ⊗ e k − (cid:88) k − l ( x k − g k ) e − ( r (cid:48) ) ⊗ e k (cid:13)(cid:13)(cid:13) = (cid:88) k − l (cid:13)(cid:13)(cid:13) ( x k − g k ) e − r − ( x k − g (cid:48) k ) e − ( r (cid:48) ) (cid:13)(cid:13)(cid:13) thanks to the C ∗ -condition and the formula in the Clifford algebra. What we need to prove is the following:The sup norm of the function x (cid:55)→ (cid:13)(cid:13)(cid:13)(cid:80) k − l ( x k − g k ) e − r ⊗ e k − (cid:80) k − l ( x k − g k ) e − ( r (cid:48) ) ⊗ e k (cid:13)(cid:13)(cid:13) is arbitrarysmall if g (cid:48) is sufficiently close to g . (cid:88) k − l (cid:107) ( x k − g k ) e − r − ( x k − g (cid:48) k ) e − ( r (cid:48) ) (cid:107) = (cid:88) k − l (cid:107) ( x k − g k ) e − r − ( x k − g (cid:48) k ) e − r + ( x k − g (cid:48) k ) e − r − ( x k − g (cid:48) k ) e − ( r (cid:48) ) (cid:107) ≤ (cid:88) k − l | g (cid:48) k − g k | e − r + 2 (cid:88) k − l | x k − g (cid:48) k | · | e − r − e − ( r (cid:48) ) | ≤ (cid:107) g − g (cid:48) (cid:107) L − l e − r + 2( r (cid:48) ) · | e − r − e − ( r (cid:48) ) | . The first term is, clearly, arbitrary small if g and g (cid:48) are sufficiently close. For the second term, we needmore computations. Let | • | be the “ L − l -norm”. Note that | r − r (cid:48) | ≤ | g − g (cid:48) | . Put δ := | g − g (cid:48) | , and notethat r + r (cid:48) ≤ r (cid:48) + δ . Then,( r (cid:48) ) · | e − r − e − ( r (cid:48) ) | = ( r (cid:48) ) e − r (cid:48) ) | e ( r (cid:48) ) − r − | ≤ (cid:40) ( r (cid:48) ) e − r (cid:48) ) ( e (2 r (cid:48) + δ ) δ − r (cid:48) ≥ r ( r (cid:48) ) e − r (cid:48) ) (1 − e − (2 r (cid:48) + δ ) δ ) r (cid:48) ≤ r. The both are arbitrary small if δ is sufficiently small. We compute only the square root of the former one.Notice the following: r (cid:48) e − ( r (cid:48) ) ( e (2 r (cid:48) + δ ) δ −
1) = ( r (cid:48) − δ ) e − ( r (cid:48) − δ ) · e δ − r (cid:48) e − ( r (cid:48) ) + δe − ( r (cid:48) − δ ) · e δ . Since the function t (cid:55)→ te − t is uniformly continuous and bounded, the last quantity can be arbitrary smallif δ is sufficiently small. 30 emarks . (1) The above action is, in fact, continuous with respect to the L − l -metric. This fact mightsuggest that our HKT algebra is in some sense “ S C ( U L − l )” or something.(2) We will frequently encounter similar arguments in the present paper: Firstly, the correspondingstatement on finite-dimensional objects is obvious. Secondly, to prove the statement for infinite-dimensionalobjects, we discuss some quantitative issues.We would like to define a representation of S C ( U ) on a Hilbert S -module. Seeing Lemma 2.33 andDefinition 2.20, we find that the “natural” representation of “ C ( U )” should be on “ L ( U, Cliff + ( T ∗ U ))”.Let us begin with the construction of the Spinor S . Recall the complex basis of Lie( U ) ∗ ⊗ C : z k := √ ( e k + if k ) and z k := √ ( e k − if k ). Definition 4.20 ([FHT]) . Let S fin := ∧ alg ⊕ alg k C z k be the algebraic exterior algebra of the anti-holomorphicpart of U fin ⊗ C . It is naturally graded by the structure of the exterior algebra. We introduce the Cliffordmultiplication by (cid:40) γ ( z k ) := −√ z k (cid:99) γ ( z k ) := √ z k ∧ . Let S be the completion of S fin with respect to the inner product induced by that of Lie( U ) ∗ ⊗ C . S fin has “1” as a unit vector. In order to distinguish from b ∈ L ( R ∞ ), we write this specific element of S fin as f , where “ f ” comes from “fermion”.We have explained the dual Spinor is automatically defied by S . However, we will use a dense subspaceof it in order to define a “differential” operator, and hence we should concretely describe it. Definition 4.21.
Let S ∗ fin := ∧ alg ⊕ alg k C z k , and let (cid:40) γ ∗ ( z k ) := −√ z k ∧ ◦ (cid:15) S ∗ γ ∗ ( z k ) := √ z k (cid:99) ◦ (cid:15) S ∗ . As usual, (cid:15) S ∗ is the grading homomorphism (cid:15) S ∗ ( z i ∧ z i ∧ · · · ∧ z i n ) = ( − n z i ∧ z i ∧ · · · ∧ z i n . The specificvector 1 is also denoted by ∗ f .The bilinear pairing between S fin and S ∗ fin is given by (cid:104) z i ∧ z i ∧ · · · ∧ z i n , z j ∧ z j ∧ · · · ∧ z j n (cid:105) = δ i ,j δ i ,j · · · δ i n ,j n for i < i < · · · < i n and j < j < · · · < j n , and (cid:104) z i ∧ z i ∧ · · · ∧ z i n , z j ∧ z j ∧ · · · ∧ z j m (cid:105) = 0for n (cid:54) = m .We will introduce the “mixed” actions of the Clifford algebras on S ∗ ⊗ S . Definition 4.22. On S ∗ fin ⊗ alg S fin , define two Clifford multiplications for X ∈ Lie( U ) by c ( X ) := 1 √ ⊗ γ ( X ) − iγ ∗ ( X ) ⊗ id) ; c ∗ ( X ) := 1 √ γ ∗ ( X ) ⊗ id − i id ⊗ γ ( X )) . Note that c is a homomorphism from Cliff − (Lie( U )), while c ∗ is a homomorphism from Cliff + (Lie( U )).One can easily check that c anti-commutes with c ∗ . The Clifford multiplication c will appear as the “symbol”of our Dirac operator, and c ∗ will appear as the “potential”.We will sometimes encounter Spinors of subspaces. S V and S ∗ V denotes the Spinor of a subspace V ⊆ U corresponding to S and S ∗ , respectively. In the following, as a subspace, U N , U M (cid:9) U N , U ⊥ N = U (cid:9) U N willappear.In order to define the representation, we need to introduce a Hilbert space which plays a role of “ L ( U, Cliff + ( T ∗ U ))”.Recall that Cliff + ∼ = End( S ∗ ) in finite-dimensional setting.31 efinition 4.23. Let H be the Hilbert space tensor product L ( U ) ⊗ S ∗ ⊗ S. We sometimes use its dense subspace H fin := L ( U ) fin ⊗ alg S ∗ fin ⊗ alg S fin . The specific vector vac ⊗ ∗ f ⊗ f is denoted by Vac.The actual representation space is a Hilbert S -module S ⊗ H . We also deal with a dense subspace S fin ⊗ alg H fin .We would like to define a ∗ -homomorphism Φ : S C ( U ) → L S ( S ⊗ H ). For this purpose, it is essentialto define an operator Φ( C N )’s corresponding to the Clifford element C N ’s. If such an operator is definedappropriately, we can define the representation. Let us briefly explain it here before the definition of Φ( C N )’s.Firstly, C ( U N ) acts on L ( U N ) ⊗ S ∗ U N ⊗ S U N : For f ∈ C ( U N ) and v ∈ Cliff + ( U N ), [ f ⊗ v ] · ( φ ⊗ s ⊗ s (cid:48) ) :=( f · φ ) ⊗ c ∗ ( v )[ s ⊗ s (cid:48) ]. This action is denoted by a · v for a ∈ C ( U N ) and v ∈ L ( U N ) ⊗ S ∗ U N ⊗ S U N . Then,Φ N : S C ( U N ) → L S ( S ⊗ H ) is defined by Φ N ( f ⊗ h ) := f ( X ⊗ id + id ⊗ Φ( C N +1 )) ⊗ h through S ⊗ L ( U ) ⊗ S ∗ ⊗ S ∼ = (cid:104) S ⊗ L ( U ⊥ N ) ⊗ S ∗ U ⊥ N ⊗ S U ⊥ N (cid:105) (cid:78) (cid:2) L ( U N ) ⊗ S ∗ U N ⊗ S U N (cid:3) . After the following proposition,we will check that Φ N ’s are compatible with β • , • ’s, and hence Φ : S C ( U ) → L S ( S ⊗ H ) is well-defined.In order to define the operator Φ( C N ), notice that the multiplication operator of the function x k isdefined on L ( U M ) fin , for any M ≥ k . This operator commutes with β M,L , and hence acts on L ( U ). Inthe representation theoretical language, it is defined by i [ dL ( f k ) + dR ( f k )]. Moreover, e k acts on S ∗ ⊗ S by √ [ c ∗ ( z k ) + c ∗ ( z k )] = [ γ ∗ ( z k ) ⊗ id − i id ⊗ γ ( z k ) + γ ∗ ( z k ) ⊗ id − i id ⊗ γ ( z k )]. Consequently, each summand x k e k + y k f k is defined. The non-trivial thing is whether the infinite sum converges or not in an appropriatesense. Note that each term x k ⊗ e k + y k ⊗ f k is an unbounded operator. Proposition 4.24.
The infinite sum of operators C N := (cid:80) ∞ k = N k − l ( x k ⊗ c ∗ ( e k ) + y k ⊗ c ∗ ( f k )) defines anessentially self-adjoint operator from L ( U ) fin ⊗ alg S fin ⊗ alg S fin to L ( U ) ⊗ S ⊗ S .Proof. Compare with [Tak2] for the technique of the proof.Since each summand x k ⊗ c ∗ ( e k ) + y k ⊗ c ∗ ( f k ) is symmetric operator, so is the infinite sum, provided theinfinite sum converges. Thus it is closable. To be essentially self-adjoint, we need to check that Φ( C N ) ± i idhas dense range. Therefore we need two steps; (1) The infinite sum Φ( C N ) converges in the strong sense; (2)Φ( C N ) ± i id has dense range.(1) Fix an element φ ⊗ s ⊗ s (cid:48) ∈ L ( U ) fin ⊗ alg S ∗ fin ⊗ alg S fin . We study the infinite sum (cid:80) ∞ k = N k − l ( x k ⊗ c ∗ ( e k ) + y k ⊗ c ∗ ( f k ))( φ ⊗ s ⊗ s (cid:48) ). By an abuse of notation, Φ( C N )( φ ⊗ s ) denotes this infinite sum. Bydefinition of L ( U ) fin ⊗ alg S ∗ fin ⊗ alg S fin , the vector φ ⊗ s ⊗ s (cid:48) is a finite linear combination of ( (cid:101) φ ⊗ vac ∞ M ) ⊗ ( z j ∧ z j ∧· · ·∧ z j m ) ⊗ ( z i ∧ z i ∧· · ·∧ z i n )’s for some (cid:101) φ ∈ L ( U M − ) fin ’s. We may choose M so as to be greaterthan i n and j m . We may write the vector as [ (cid:101) φ ⊗ ( z j ∧ z j ∧ · · · ∧ z j m ) ⊗ ( z i ∧ z i ∧ · · · ∧ z i n )] ⊗ Vac ∞ M . SinceΦ( C M − N ) is clearly a finite sum (it is unbounded, but it is well-defined on rapidly decreasing functions), andsince C N = C M − N + C M , it suffices to check that Φ( C M )(Vac ∞ M ) converges. We may assume that M = 1.On each piece U N +1 (cid:9) U N ∼ = R , the vacuum vector is √ π e − ( x + y ) . With the Gaussian integralformulas, we find that (cid:107) √ π xe − ( x + y ) (cid:107) L = 1. Thus (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (cid:88) k =1 (cid:0) k − l x k ⊗ c ∗ ( e k )Vac + k − l y k ⊗ c ∗ ( f k )Vac (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = L (cid:88) k =1 k − l . The limit as L → ∞ exists, because l is greater than (it is enough to suppose that l > ).(2) To see that Φ( C L ) ± i id has dense range, fix an element (cid:16) (cid:101) φ ⊗ s ⊗ s (cid:48) (cid:17) ⊗ Vac ∞ N ∈ L ( U ) fin ⊗ alg S ∗ fin ⊗ alg S fin .We need to find v ± ⊗ Vac ∞ M ∈ L ( U ) fin ⊗ alg S ∗ fin ⊗ alg S fin such that (Φ( C L ) ± i id)( v ± ⊗ Vac ∞ M ) is close enough32o (cid:16) (cid:101) φ ⊗ s ⊗ s (cid:48) (cid:17) ⊗ Vac ∞ N . Take any real number ε >
0. Since the infinite sum (cid:80) k k − l converges, we canchoose M satisfying that (cid:107) Φ( C M +1 )(Vac ∞ M +1 ) (cid:107) = (cid:80) k ≥ M +1 k − l < ε . We may assume that M is greaterthan N and L . Since (Φ( C ML ) + i id)(Φ( C ML ) − i id) = (Φ( C ML ) − i id)(Φ( C ML ) + i id) = (cid:80) k k − l ( x k + y k ) + 1,we can find v (cid:48)± ∈ L ( U M ) ⊗ S ∗ U M such that (Φ( C ML ) ± i id) v (cid:48)± ⊗ Vac ∞ M +1 = (cid:101) φ ⊗ s ⊗ s (cid:48) . Since L is densein L , and since Φ( C L ) ± i does not reduce the norm, we can find v ± ∈ L ( U M ) fin ⊗ S ∗ U M ⊗ S U M such that (cid:107) (Φ( C ML ± i ) v ± − (Φ( C ML ) ± i ) v (cid:48)± (cid:107) < ε. Therefore, (cid:107) (Φ( C L ) ± i ) v ± − (cid:101) φ ⊗ s ⊗ s (cid:48) ⊗ Vac ∞ L (cid:107) is less than 2 ε .This lemma enables us to define a ∗ -homomorphism S C ( U N ) → L S ( S ⊗ H ) as follows. Lemma 4.25.
The following diagram commutes.
S C ( U N ) β N,M (cid:47) (cid:47) Φ N (cid:39) (cid:39) S C ( U M ) Φ M (cid:119) (cid:119) L S ( S ⊗ H ) Proof.
We may check the statement only for generators: f e ⊗ a and f o ⊗ a . Recall that β N,M ( f e ⊗ a ) = f e ⊗ f e ( C MN +1 ) ⊗ a . Φ M ◦ β N,M ( f e ⊗ a ) = f e ( X ⊗ id + id ⊗ Φ( C ∞ M +1 )) ⊗ e − ( C MN +1 ) ⊗ a = f e ( X ) ⊗ e − Φ( C ∞ M +1 ) ⊗ e − ( C MN +1 ) ⊗ a. On the other hand, Φ N ( f e ⊗ a ) is given by f e ( X ⊗ id + id ⊗ Φ( C ∞ N +1 )) ⊗ a = f e ( X ) ⊗ e − Φ( C ∞ N ) ⊗ a . SinceΦ( C ∞ N +1 ) = C MN +1 + Φ( C ∞ M +1 ), Φ( C ∞ N +1 ) = [ C MN +1 ] + Φ( C ∞ M +1 ) , and [ C MN +1 ] commutes with Φ( C ∞ M +1 ) , f e ( X ) ⊗ e − Φ( C ∞ N ) ⊗ a = f e ( X ) ⊗ e − Φ( C ∞ M +1 ) ⊗ e − ( C MN +1 ) ⊗ a. For f o ⊗ a , use the parallel technique.Consequently, we obtain the following ∗ -homomorphism. Definition-Theorem 4.26.
There uniquely exists a ∗ -homomorphism Φ :
S C ( U ) → L S ( S ⊗ H ) such thatthe following diagram commutes for all N : S C ( U N ) β N (cid:47) (cid:47) Φ N (cid:39) (cid:39) S C ( U ) Φ (cid:120) (cid:120) L S ( S ⊗ H )From now on, we understand S C ( U ) acting on S ⊗ H always by Φ, and we writeΦ( a )( φ ) as a · φ for a ∈ S C ( U ) and φ ∈ S ⊗ H . This section is the main part of the present paper. We will introduce three unbounded U -equivariant Kasparovmodules. One of them is untwisted, and the others are τ -twisted. These modules correspond to the indexelement, the Dirac element and the Clifford symbol element. Then, we will prove the index theorem equality.33 .1 Index element [ (cid:101) (cid:19)(cid:19) ∂ ] Recall that L ( U ) is defined by using the τ -twisted representation theory of U . Thus U naturally acts on S ⊗ H by g. [ f ⊗ v ⊗ v (cid:48) ⊗ s (cid:48) ⊗ s ] := f ⊗ ( g.v ) ⊗ v (cid:48) ⊗ s (cid:48) ⊗ s with cocycle τ . This action is modeled on the left regular representation. Since U preserves the inner productof L ( R ∞ ) and does not touch the S -part, this action preserves the S -valued inner product, and commuteswith the right S -action.We firstly prove that the left S C ( U )-module S ⊗ H is τ -twisted U -equivariant. Lemma 5.1.
For g ∈ U , a ∈ S C ( U ) and φ ∈ S ⊗ H , g. ( a · φ ) = ( g.a ) · ( g.φ ) . Proof.
This equality seems to be obvious at first sight, because of the following (illegal! ) computation: g. ( a · φ )( x ) = ( a · φ )( g − .x ) = a ( g − .x ) · φ ( g − .x ) = ( g.a ) · ( g.φ )( x ). However, we cannot use the languageof functions without any tricks, and our problem is less clear than the first impression. Briefly speaking, weprove this equality by the continuity of the action, and the finite-dimensional approximation.We need to check that (cid:107) g. ( a · φ ) − ( g.a ) · ( g.φ ) (cid:107) < ε for any ε >
0. If g , a and φ come from finite-dimensional objects, the inequality is clear. For example, if g ∈ U N , a = f ( X ⊗ id + id ⊗ C ∞ N +1 ) ⊗ (cid:101) a and φ = f ⊗ Vac ∞ N +1 ⊗ (cid:101) φ ⊗ s , (cid:107) g. ( a · φ ) − ( g.a ) · ( g.φ ) (cid:107) = (cid:13)(cid:13)(cid:13) f ( X ⊗ id + id ⊗ C ∞ N +1 ) · ( f ⊗ Vac ∞ N +1 ) ⊗ (cid:104) g. ( (cid:101) a · (cid:101) φ ) − ( g. (cid:101) a ) · ( g. (cid:101) φ ) (cid:105) ⊗ s (cid:13)(cid:13)(cid:13) , which is clearly zero, because g. ( (cid:101) a · (cid:101) φ ) − ( g. (cid:101) a ) · ( g. (cid:101) φ ) is computed in C ( U N ) as a function.Let us go back to the general case. It suffices to replace general g , a and φ with such (better) ones. Recallthe following things: The U -action on H is unitary; The U -action on S C ( U ) is isometry; and Φ( a ) is abounded operator whose norm is bounded by (cid:107) a (cid:107) . Then, (cid:107) g. ( a · φ ) − g. ( a (cid:48) · φ (cid:48) ) (cid:107) = (cid:107) ( a · φ ) − ( a (cid:48) · φ (cid:48) ) (cid:107) ≤ (cid:107) a − a (cid:48) (cid:107)(cid:107) φ (cid:107) + (cid:107) a (cid:107)(cid:107) φ − φ (cid:48) (cid:107) , (cid:107) ( g.a ) · ( g.φ ) − ( g.a (cid:48) ) · ( g.φ (cid:48) ) (cid:107) ≤ (cid:107) g.a − g.a (cid:48) (cid:107)(cid:107) φ (cid:107) + (cid:107) g.a (cid:48) (cid:107)(cid:107) g.φ − g.φ (cid:48) (cid:107) = (cid:107) a − a (cid:48) (cid:107)(cid:107) φ (cid:107) + (cid:107) a (cid:48) (cid:107)(cid:107) φ − φ (cid:48) (cid:107) . Thus we can find a (cid:48) ∈ S C ( U N ) fin and φ (cid:48) = (cid:101) φ (cid:48) ⊗ s ⊗ Vac ∞ N ∈ (cid:2) L ( U N ) ⊗ alg S ∗ U N ⊗ S U N (cid:3) ⊗ alg [ L ( U (cid:9) U N ) fin ⊗ alg ( S ∗ U (cid:9) U N ) fin ⊗ alg ( S U N ) fin ], such that the both of (cid:107) g. ( a · φ ) − g. ( a (cid:48) · φ (cid:48) ) (cid:107) and (cid:107) ( g.a ) · ( g.φ ) − ( g.a (cid:48) ) · ( g.φ (cid:48) ) (cid:107) are smaller than ε/
4, independently from g . Moreover, we can choose g (cid:48) ∈ U M such that the both of (cid:107) g. ( a (cid:48) · φ (cid:48) ) − g (cid:48) . ( a (cid:48) · φ (cid:48) ) (cid:107) and (cid:107) ( g.a (cid:48) ) · ( g.φ (cid:48) ) − ( g (cid:48) .a (cid:48) ) · ( g (cid:48) .φ (cid:48) ) (cid:107) are smaller than ε/ a (cid:48) and φ (cid:48) , sincethe action is strongly continuous. Thanks to the homomorphism β N,M or the inclusion U M (cid:44) → U N , we mayassume that M = N . Now, the statement is clear by the finite-dimensional case.Thanks to this lemma, we obtain a τ -twisted U -equivariant Hilbert S -module S ⊗H equipped with a left S C ( U )-module structure. We must define an appropriate operator in order to define a Kasparov module.We introduce two operators; one of them is U -equivariant with respect to the action R , and the other is U -equivariant with respect to the action L . The both look natural, but the operator we should deal with isthe “average” of them. It is not actually equivariant, but almost equivariant such that the operator definesan equivariant Kasparov module. In fact, our operator is essentially the perturbed Bott-Dirac operator in[HK]. The value at x of g. ( a · φ ), g. ( a · φ )( x ), does not make sense. Note that we cannot choose the neighborhood of g uniformly in a or φ . This is because the group action is not uniformlycontinuous, but strongly continuous. efinition 5.2. On L ( R ∞ ) fin ⊗ alg L ( R ∞ ) ∗ fin ⊗ alg S ∗ fin ⊗ alg ⊗ S fin , let L (cid:1) ∂ := (cid:88) k √ k ( dρ ( z k ) ⊗ id ⊗ γ ∗ ( z k ) ⊗ id + dρ ( z k ) ⊗ id ⊗ γ ∗ ( z k ) ⊗ id) , R (cid:1) ∂ := (cid:88) k √ k (id ⊗ dρ ∗ ( z k ) ⊗ id ⊗ γ ( z k ) + id ⊗ dρ ( z k ) ⊗ id ⊗ γ ( z k )) . As an unbounded operator on L ( U M (cid:9) U N − ) ⊗ S ∗ U M (cid:9) U N − ⊗ S U M (cid:9) U N − , define L (cid:1) ∂ MN := M (cid:88) k = N √ k ( dρ ( z k ) ⊗ id ⊗ γ ∗ ( z k ) ⊗ id + dρ ( z k ) ⊗ id ⊗ γ ∗ ( z k ) ⊗ id)for 1 ≤ N ≤ M ≤ ∞ , and similarly for R (cid:1) ∂ . Note that M = ∞ is allowed. They are defined on, at least, L ( U M (cid:9) U N − ) fin ⊗ alg S ∗ U M (cid:9) U N − , fin ⊗ alg S U M (cid:9) U N − , fin These are well-defined as proved in [FHT]. For the details, see also Proposition 5.30 in [Tak3]. Clearly L (cid:1) ∂ is U -equivariant with respect to the action R , and R (cid:1) ∂ is U -equivariant with respect to the action L . Theoperator we will study is the following. Note that L (cid:1) ∂ is skew-symmetric, while R (cid:1) ∂ is symmetric. Definition 5.3. On L ( R ∞ ) fin ⊗ alg L ( R ∞ ) ∗ fin ⊗ alg S ∗ fin ⊗ alg ⊗ S fin , define an operator (cid:1) ∂ by (cid:1) ∂ := 1 √ R (cid:1) ∂ + i √ L (cid:1) ∂. Put (cid:1) ∂ MN := √ R (cid:1) ∂ MN + i √ L (cid:1) ∂ MN . Remark . In the previous papers [Tak1, Tak2, Tak3], we essentially studied only R (cid:1) ∂ , although an operatorwhich looks like L (cid:1) ∂ appeared in some related KK -elements. Thus we wanted to prove that the pair [( S ⊗H , R (cid:1) ∂ )] is a τ -twisted U -equivariant Kasparov ( S C ( U ) , S )-module. However, it turns out to be false: theoperator is not with locally compact resolvent. That is essentially because each element of our S C ( U ) istoo “smooth” to work as a “compactly supported function”.In order to overcome this problem, we modify the operator. The new one is, from the beginning, withcompact resolvent. The cost we must pay is giving up the actual equivariance. It made us study the newequivariance condition on equivariant Kasparov modules in Section 2.1. Remark . Let us formally rewrite our Dirac operator using the function language, seeing Definition 4.22.Recall dL ( e k ) = − ∂ x k + i y k and so on. Firstly, note that dL ( z k ) ⊗ γ ∗ ( z k ) ⊗ id + dL ( z k ) ⊗ γ ∗ ( z k ) ⊗ id = dL ( e k ) ⊗ γ ∗ ( e k ) ⊗ id + dL ( f k ) ⊗ γ ∗ ( f k ) ⊗ id , and similarly for R (cid:1) ∂ . Then, the k -th term of (cid:1) ∂ is given by i √ dL ( e k ) ⊗ γ ∗ ( e k ) ⊗ id + dL ( f k ) ⊗ γ ∗ ( f k ) ⊗ id) + 1 √ dR ( e k ) ⊗ id ⊗ γ ( e k ) + dR ( f k ) ⊗ id ⊗ γ ( f k ))= i √ (cid:18)(cid:18) − ∂ x k + i y k (cid:19) ⊗ γ ∗ ( e k ) ⊗ id + (cid:18) − ∂ y k − i x k (cid:19) ⊗ γ ∗ ( f k ) ⊗ id (cid:19) + 1 √ (cid:18)(cid:18) ∂ x k + i y k (cid:19) ⊗ id ⊗ γ ( e k ) + (cid:18) ∂ y k − i x k (cid:19) ⊗ id ⊗ γ ( f k ) (cid:19) = ∂ x k ⊗ c ( e k ) + ∂ y k ⊗ c ( e k ) + x k ⊗ c ∗ ( f k ) + y k ⊗ c ∗ ( − e k ) . By introducing an operator J defined by e (cid:55)→ f and f (cid:55)→ − e , we can formally rewrite the Dirac operator; (cid:1) ∂ = (cid:88) √ k (cid:104) ∂ x k ⊗ c ( e k ) + ∂ y k ⊗ c ( e k ) + x k ⊗ c ∗ ( J ( e k )) + y k ⊗ c ∗ ( J ( e k )) (cid:105) .
35f we identify c as the left multiplication of the Clifford algebra on itself, and c ∗ ◦ J as the right multiplication, (cid:1) ∂ looks like the Bott-Dirac operator.Notice that this formula is illegal: We exchanged the order of operators infinitely many times; Even onfinite energy vectors, the infinite sum does not converges. Remark . In order to reduce symbols, (cid:1) ∂ sometimes stands for the operator id ⊗ (cid:1) ∂ on S ⊗ H .We would like to prove that the pair ( S ⊗ H , (cid:1) ∂ ) is a τ -twisted U -equivariant Kasparov module. For thispurpose, we check the followings: (cid:1) ∂ is self-adjoint and regular; S C ( U ) fin preserves dom( (cid:1) ∂ ); S C ( U ) fin -actioncommutes with (cid:1) ∂ modulo bounded operators; (1 + (cid:1) ∂ ) is compact; U preserves dom( (cid:1) ∂ ); and (cid:1) ∂ satisfies theconditions to be an equivariant Kasparov module. Lemma 5.7. (cid:1) ∂ is self-adjoint and regular on S ⊗ H .Proof. It is easy to see that (cid:1) ∂ is symmetric. It suffices to check that (cid:1) ∂ ± i : H (cid:8) has dense range. Infact, (cid:1) ∂ ± i give unbounded bijections on H fin . We concentrate on (cid:1) ∂ + i in order to simplify the notation.By exchanging the order, see H as [ L ( R ∞ ) ⊗ S ∗ ] ⊗ [ L ( R ∞ ) ∗ ⊗ S ]. Then, (cid:1) ∂ can be written as i √ L (cid:1) ∂ ⊗ id + √ id ⊗ R (cid:1) ∂ . Since L (cid:1) ∂ and R (cid:1) ∂ are anti-commutative (by definition of the graded tensor product), (cid:1) ∂ = (cid:104) − (cid:0) L (cid:1) ∂ (cid:1) ⊗ id + id ⊗ (cid:0) R (cid:1) ∂ (cid:1) (cid:105) . Consider ( (cid:1) ∂ + i )( (cid:1) ∂ − i ) = (cid:1) ∂ + 1. Recall that (cid:0) L (cid:1) ∂ (cid:1) and (cid:0) R (cid:1) ∂ (cid:1) have discretespectrum. Take eigenvectors v ∈ L ( R ∞ ) ⊗ S ∗ and v (cid:48) ∈ L ( R ∞ ) ∗ ⊗ S such that (cid:0) L (cid:1) ∂ (cid:1) ( v ) = λv and (cid:0) R (cid:1) ∂ (cid:1) ( v (cid:48) ) = λ (cid:48) v (cid:48) . Note that λ ≤ λ (cid:48) ≥
0. Then,[ (cid:1) ∂ + 1]( v ⊗ v (cid:48) ) = (cid:18) − λ + λ (cid:48) (cid:19) v ⊗ v (cid:48) . Therefore, ( (cid:1) ∂ + i ) (cid:104) ( (cid:1) ∂ − i ) (cid:16) − λ + λ (cid:48) ( v ⊗ v (cid:48) ) (cid:17)(cid:105) = v ⊗ v (cid:48) .It seems to be easy to prove that [ a, (cid:1) ∂ ] is bounded. This is because (cid:1) ∂ is a combination of first orderderivatives and multiplication operators. The multiplication operator defined by “smooth” a seems to com-mute with the multiplication with scalar-valued functions, and seems to commute, modulo bounded, withdifferential operator of first order. In fact, the formal computation of the commutator is not too difficult. Weneed only one trick to prove the convergence. However, in order to define the commutator, we must provethat (cid:1) ∂ ◦ a is defined on some dense subspace, which is essential for this issue. This is because S C ( U ) fin doesnot preserve H fin . Therefore we actually need to deal with, not H fin , but dom( (cid:1) ∂ ). This is one of the mostcomplicated and the most creative point of the present paper. Before that, we prepare a lemma to represent e − C Vac explicitly
Lemma 5.8. f ( C N ) strongly converges to f ( C ) on S fin ⊗ alg H fin as N → ∞ , for f = f e or f o .Proof. This proof is inspired by the argument in Appendix A.4 in [HKT].Define an orthogonal projection P k on L ( U k (cid:9) U k − ) by the multiplication operator of the function givenby χ k ( x, y ) := (cid:40) x + y ≤ k )0 ( x + y > k ) . We would like to define an infinite tensor product P M :=“id ⊗ · · · ⊗ id ⊗ P M ⊗ P M +1 ⊗ · · · ”. For this aim,we define P NM := id ⊗ · · · ⊗ id ⊗ P M ⊗ P M +1 ⊗ · · · ⊗ P N ⊗ id ⊗ · · · , and we will check that P NM strongly converges to some projection as N → ∞ . Then, we will define P M bythe the strong limit “lim N →∞ P NM ”.Take φ ⊗ s (cid:48) ⊗ s ∈ L ( U ) fin ⊗ alg S ∗ fin ⊗ alg S fin . We may assume that φ ∈ L ( U L ) fin , s (cid:48) ∈ S ∗ U L and s ∈ S U L .In order to prove that P NM [ φ ⊗ s (cid:48) ⊗ s ] converges as N → ∞ , we may assume that N > L . In order to simplify See the proof of Theorem 5.33 in [Tak3] for the detail of the computation.
M < L . It is always possible, because of the embeddings L ( U • ) (cid:44) → L ( U • +1 ).Then, P NM [ φ ⊗ s (cid:48) ⊗ s ] = P LM [ φ ⊗ s (cid:48) ⊗ s ] ⊗ P NL +1 Vac ∞ L +1 . Seeing this formula, we find that it suffices to prove that P NL +1 [Vac ∞ L +1 ] converges as N → ∞ . For this aim,we may ignore the U L -part. Let us regard P NM Vac − P N (cid:48) M Vac for
N < N (cid:48) , as (cid:0) P NM Vac − P N +1 M Vac (cid:1) + (cid:0) P N +1 M Vac − P N +2 M Vac (cid:1) + · · · + (cid:16) P N (cid:48) − M Vac − P N (cid:48) M Vac (cid:17) . Let us estimate (cid:107) P N + kM Vac − P N + k +1 M Vac (cid:107) ’s. (cid:107) P N + kM Vac − P N + k +1 M Vac (cid:107) = (cid:107) P N + kM Vac N + k (cid:107) · (cid:107) Vac N + kN + k − P N + k Vac N + kN + k (cid:107) · (cid:107) Vac ∞ N + k +1 (cid:107) = (cid:107) P N + kM Vac N + k (cid:107) · (cid:115) π (cid:90) r ≥√ N + k e − r dx N + k dy N + k = (cid:107) P N + kM Vac N + k (cid:107) · e − N + k . The coefficient (cid:107) P N + kM Vac N + k (cid:107) cannot be greater than 1, because P is just an orthogonal projection. Thereforethe sum we are estimating cannot exceed (cid:80) N (cid:48) − N − k =0 e − N + k ≤ e − N/ / (1 − e − / ). Thus the sequence P NM [ φ ⊗ s (cid:48) ⊗ s ] is a Cauchy sequence, and hence the limit P M [ φ ⊗ s (cid:48) ⊗ s ] := lim N →∞ P NM [ φ ⊗ s (cid:48) ⊗ s ] exists.As the next step, we check that P M strongly converges to id as M → ∞ on L ( U ) fin ⊗ alg S ∗ fin ⊗ alg S fin .In order to prove this, we may check that (cid:107) vac − P M vac (cid:107) → M → ∞ . Note that (cid:107) vac − P M vac (cid:107) ≤ (cid:107) vac − P NM vac (cid:107) + (cid:107) P NM vac − P M vac (cid:107) . For any ε >
0, there exists M satisfying the following: (cid:107) vac − P NM vac (cid:107) < ε/ any N ≥ M , thanks tothe estimate in the previous paragraph. After that, we can find N satisfying that (cid:107) P NM vac − P M vac (cid:107) < ε/ P M → id strongly, as M → ∞ .Let us prove that e − ( C N ) → e − C as N → ∞ . As usual, we may prove it for vac. (cid:107) e − C vac − e − ( C N ) vac (cid:107)≤ (cid:107) e − C (vac − P M vac) (cid:107) + (cid:107) e − C P M vac − e − ( C N ) P M vac (cid:107) + (cid:107) e − ( C N ) ( P M vac − vac) (cid:107) . One can choose M such that the first and the third terms are small, because P M vac → vac as M → ∞ , and e − C and e − ( C N ) are bounded operators whose norm is at most 1. We would like to find N such that thesecond term is small, and so we may assume that N > M . The second term can be rewritten as (cid:107) e − ( C N ) P NM vac (cid:107) · (cid:107) e − ( C ∞ N +1 ) P N +1 vac ∞ N +1 − P N +1 vac ∞ N +1 (cid:107) . (cid:107) e − ( C N ) P NM vac (cid:107) is at most 1. For (cid:107) e − ( C ∞ N +1 ) P N +1 vac ∞ N +1 − P N +1 vac ∞ N +1 (cid:107) , we firstly recall that C ∞ N +1 commutes with P N +1 . Let h be a function on R defined by h ( X ) := e − X −
1. Then, e − ( C ∞ N +1 ) P N +1 − P N +1 = h ( C ∞ N +1 P N +1 ) P N +1 . We prove that C ∞ N +1 P N +1 is a bounded operator whose operator norm is arbitrary small if N is large enough.On the k -th component, the operator norm of the multiplication operator by the function( x k ⊗ e k + y k ⊗ f k ) χ k is precisely √ k . Therefore, (cid:107) C ∞ N +1 P N +1 (cid:107) ≤ (cid:88) k>N k − l .
37t means that (cid:107) C ∞ N +1 P N +1 (cid:107) can be arbitrary small when N is large enough, since l is greater than . By theproperty of the operator calculus, (cid:107) h ( C ∞ N +1 P N +1 ) (cid:107) ≤ sup | X |≤(cid:107) C ∞ N +1 P N +1 (cid:107) h ( X ). Since h is continuous and h (0) = 0, we obtain the necessary estimate.For f o , we follow the same story. We would like to prove that C N e − ( C N ) → Ce − C as N → ∞ strongly.As usual, we may prove it for vac. Firstly, for sufficiently large M , (cid:107) Ce − C vac − C N e − ( C N ) vac (cid:107)≤ (cid:107) Ce − C (vac − P M vac) (cid:107) + (cid:107) Ce − C P M vac − C N e − ( C N ) P M vac (cid:107) + (cid:107) C N e − ( C N ) ( P M vac − vac) (cid:107) . It suffices to estimate the second term. (cid:107) Ce − C P M vac − C N e − ( C N ) P M vac (cid:107) = (cid:13)(cid:13)(cid:13) C ∞ N +1 e − C P M vac + C N e − ( C N ) P NM vac ⊗ (cid:104) e − ( C ∞ N +1 ) P N +1 vac ∞ N +1 − P N +1 vac ∞ N +1 (cid:105)(cid:13)(cid:13)(cid:13) ≤ (cid:107) C ∞ N +1 e − C P M vac (cid:107) + (cid:107) C N e − ( C N ) P NM vac (cid:107) · (cid:107) e − ( C ∞ N +1 ) P N +1 vac ∞ N +1 − P N +1 vac ∞ N +1 (cid:107) . We check that the both terms are small.For the second one, (cid:107) C N e − ( C N ) P NM vac (cid:107) is bounded in N , because C N e − ( C N ) ’s are given by the operatorcalculus of the same function f o , and so they are bounded operators with the same operator norm. Moreover, (cid:107) e − ( C ∞ N +1 ) P N +1 vac ∞ N +1 − P N +1 vac ∞ N +1 (cid:107) is small if N is large enough thanks to the argument for f e .For (cid:107) C ∞ N +1 e − C P M vac (cid:107) , we rewrite it as (cid:107) e − ( C N ) P M vac (cid:107) · (cid:107) C ∞ N +1 e − ( C ∞ N +1 ) P M vac (cid:107) . The coefficient (cid:107) e − ( C N ) P NM vac (cid:107) is bounded. Moreover, (cid:107) C ∞ N +1 e − ( C ∞ N +1 ) P N +1 vac (cid:107) can be rewritten as f o ( C ∞ N +1 P N +1 )vac . We have proved the following thing: for any ε >
0, if N is large enough, (cid:107) C ∞ N +1 P N +1 (cid:107) op ≤ ε . Since f o (0) = 0and f o is continuous, f o ( C ∞ N +1 P N +1 )vac is arbitrary small if N is large enough. Proposition 5.9.
S C ( U ) fin give bounded operators on dom( (cid:1) ∂ ) with respect to the graph norm.Proof. It is enough to prove the statement for the generators f e ⊗ (cid:101) a and f o ⊗ (cid:101) a . Let us consider[ f e ⊗ (cid:101) a ] · f ⊗ φ ⊗ s ⊗ s (cid:48) and [ f o ⊗ (cid:101) a ] · f ⊗ φ ⊗ s ⊗ s (cid:48) for given f ⊗ φ ⊗ s ⊗ s (cid:48) ∈ S fin ⊗ alg L ( U ) fin ⊗ alg S ∗ fin ⊗ alg ⊗ S fin . Suppose that (cid:101) a ∈ C ( U N ) fin and φ ⊗ s ⊗ s (cid:48) ∈ L ( U N ) fin ⊗ alg S ∗ U N ⊗ alg S U N .Let us recall that[ f e ⊗ (cid:101) a ] · f ⊗ φ ⊗ s ⊗ s (cid:48) = ± f e ( X ) f ⊗ [ f e ( C ∞ N +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] . Thus we may consider only [ f e ( C ∞ N +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ]. Thanks to the previous lemma, [ f e ( C ∞ N +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] = lim M →∞ [ f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ]. Since [ f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] ∈ dom( (cid:1) ∂ ), for ourpurpose, it suffices to check that (cid:1) ∂ (cid:0) [ f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] (cid:1) converges. Let us verify that. Note that (cid:1) ∂ ∞ M +1 (cid:0) [ f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] (cid:1) = 0. (cid:1) ∂ (cid:0) [ f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] (cid:1) = (cid:1) ∂ M (cid:0) [ f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] (cid:1) = [ (cid:1) ∂ M , f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] + ( − ∂ (cid:101) a [ f e ( C MN +1 ) ⊗ (cid:101) a ] · (cid:0) (cid:1) ∂ M [ φ ⊗ s ⊗ s (cid:48) ] (cid:1) = [ (cid:1) ∂ M , f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] + ( − ∂ (cid:101) a [ f e ( C MN +1 ) ⊗ (cid:101) a ] · (cid:0) (cid:1) ∂ N [ φ ⊗ s ⊗ s (cid:48) ] (cid:1) . (cid:1) ∂ M , f e ( C MN +1 ) ⊗ (cid:101) a ]. We may use the function language: (cid:1) ∂ M = M (cid:88) k =1 √ k (cid:16) ∂ x k ⊗ c ( e k ) + ∂ y k ⊗ c ( f k ) + x k c ∗ ( Je k ) + y k c ∗ ( Jf k ) (cid:17) . We may assume that (cid:101) a is of the form A ⊗ w for A ∈ C ∞ ( U N ) and w ∈ Cliff + (Lie( U N )) with (cid:107) w (cid:107) = 1.[ (cid:1) ∂ M , f e ( C MN +1 ) ⊗ (cid:101) a ]= M (cid:88) k =1 √ k (cid:104) ∂ x k ⊗ c ( e k ) + ∂ y k ⊗ c ( f k ) + x k ⊗ c ∗ ( Je k ) + y k ⊗ c ∗ ( Jf k ) , e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ { A ⊗ c ∗ ( w ) } (cid:105) = M (cid:88) k =1 √ k (cid:16)(cid:104) ∂ x k , e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ A (cid:105) ⊗ c ( e k ) c ∗ ( w ) + (cid:104) ∂ y k , e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ A (cid:105) ⊗ c ( f k ) c ∗ ( w ) (cid:17) + M (cid:88) k =1 √ k (cid:16) x k e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ A [ c ∗ ( Je k ) , c ∗ ( w )] + y k e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ A [ c ∗ ( Jf k ) , c ∗ ( w )] (cid:17) = N (cid:88) k =1 √ k (cid:18) e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ ∂A∂x k ⊗ c ( e k ) c ∗ ( w ) + e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ ∂A∂y k ⊗ c ( f k ) c ∗ ( w ) (cid:19) + M (cid:88) k = N +1 √ k (cid:16) − k − l x k e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ A ⊗ c ( e k ) c ∗ ( w ) − k − l y k e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ A ⊗ c ( f k ) c ∗ ( w ) (cid:17) + N (cid:88) k =1 √ k (cid:16) e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ x k A [ c ∗ ( Je k ) , c ∗ ( w )] + e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ y k A [ c ∗ ( Jf k ) , c ∗ ( w )] (cid:17) =: I M + I M + I M . Note that I M can be written as e − ( C MN +1 ) ⊗ ( · · · ), where ( · · · ) is a bounded operator preserving dom( (cid:1) ∂ N ) whichis independent of M . For I M , note that the operator norm of the operator − k − l x k e − (cid:80) Mb = N +1 b − l ( x b + y b ) ⊗ A ⊗ c ( e k ) c ∗ ( w ) is at most 2 k − l · √ e − · (cid:107) A (cid:107) L ∞ , which one can check by considering the maximum value ofthe function k − l x k e − (cid:80) Mb = N +1 b − l ( x b + y b ) . Thus the norm of I M is at most4 √ e − · (cid:107) A (cid:107) L ∞ M (cid:88) k = N +1 k − l . Note that I M is not (cid:80) M ( · · · ) but (cid:80) N ( · · · ). This is because w ∈ Cliff + (Lie( U N )). Since e k and f k areorthogonal to Lie( U N ) if k > N , c ∗ ( Je k ) and c ∗ ( Jf k ) anti-commute with c ∗ ( w ). Consequently, I M is of theform e − ( C MN +1 ) ⊗ ( · · · ), where ( · · · ) is a bounded operator preserving dom( (cid:1) ∂ N ) which is independent of M .Let us prove that the sequence { (cid:1) ∂ [ e − ( C MN +1 ) ⊗ (cid:101) a · φ ⊗ s ⊗ s (cid:48) ] } M converges. (cid:1) ∂ (cid:0) [ f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] (cid:1) − (cid:1) ∂ (cid:16) [ f e ( C M (cid:48) N +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] (cid:17) = [ (cid:1) ∂ M , f e ( C MN +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ] − [ (cid:1) ∂ M (cid:48) , f e ( C M (cid:48) N +1 ) ⊗ (cid:101) a ] · [ φ ⊗ s ⊗ s (cid:48) ]+ ( − ∂ (cid:101) a [ f e ( C MN +1 ) ⊗ (cid:101) a ] · (cid:0) (cid:1) ∂ N [ φ ⊗ s ⊗ s (cid:48) ] (cid:1) − ( − ∂ (cid:101) a [ f e ( C M (cid:48) N +1 ) ⊗ (cid:101) a ] · (cid:0) (cid:1) ∂ N [ φ ⊗ s ⊗ s (cid:48) ] (cid:1) = ( I M − I M (cid:48) )[ φ ⊗ s ⊗ s (cid:48) ] + ( I M − I M (cid:48) )[ φ ⊗ s ⊗ s (cid:48) ] + ( I M − I M (cid:48) )[ φ ⊗ s ⊗ s (cid:48) ]+ ( − ∂ (cid:101) a (cid:104)(cid:110) f e ( C MN +1 ) − f e ( C M (cid:48) N +1 ) (cid:111) ⊗ (cid:101) a (cid:105) (cid:1) ∂ N [ φ ⊗ s ⊗ s (cid:48) ] . The first, the third and the fourth terms are of the form ( e − ( C MN +1 ) − e − ( C M (cid:48) N +1 ) )( · · · ). Thus the sequenceconverges. 39f M (cid:48) > M , I M − I M (cid:48) can be rewritten as I M − I M (cid:48) = I M − e − ( C M (cid:48) M +1 ) I M + e − ( C M (cid:48) M +1 ) I M − I M (cid:48) .e − ( C M (cid:48) M +1 ) I M − I M (cid:48) is given by M (cid:48) (cid:88) k = M +1 √ k (cid:16) e − (cid:80) M (cid:48) b = N +1 b − l ( x b + y b ) ⊗ x k A [ c ∗ ( Je k ) , c ∗ ( w )] + e − (cid:80) M (cid:48) b = N +1 b − l ( x b + y b ) ⊗ y k A [ c ∗ ( Jf k ) , c ∗ ( w )] (cid:17) . This is arbitrary small, if M and M (cid:48) are large enough. Moreover, I M − e − ( C M (cid:48) M +1 ) I M = [1 − e − ( C M (cid:48) M +1 ) ] I M converges strongly to zero, which is because 1 − e − ( C M (cid:48) M +1 ) → M, M (cid:48) → ∞ . More precisely, the norm of I M − e − ( C M (cid:48) M +1 ) I M does not exceed4 √ e − · (cid:107) A (cid:107) L ∞ M (cid:88) k = N +1 k − l + 4 √ e − · (cid:107) A (cid:107) L ∞ M (cid:48) (cid:88) k = N +1 k − l . It is small if M and M (cid:48) are large enough.Combining all of these estimates, we find that (cid:107) (cid:1) ∂ [ f e ( C ) ⊗ (cid:101) a · φ ⊗ s ⊗ s (cid:48) ] (cid:107) ≤ (cid:107) [ f e ( C ) ⊗ (cid:101) a ] · (cid:1) ∂ ( φ ⊗ s ⊗ s (cid:48) ) (cid:107) + c (cid:107) φ ⊗ s ⊗ s (cid:48) (cid:107) for some positive number c . Therefore f e ( C ) ⊗ (cid:101) a gives a bounded operator on dom( (cid:1) ∂ ).For the f o -case, we may study C ∞ N +1 f e ( C ∞ N +1 ) ⊗ (cid:101) a . The difference from the f e -case lies only on the explicitcomputation of the commutator, and we leave it to the reader.We essentially computed the following, in the previous lemma. Proposition 5.10.
For any element a ∈ S C ( U ) fin , [ a, (cid:1) ∂ ] is bounded on S fin ⊗ alg H fin . As a result, thisoperator extends to S ⊗ H .Proof. Just like the beginning of the above proof, we may assume that a ∈ S C ( U ) fin is of the form f ⊗ (cid:101) a for f = f e or f o and (cid:101) a ∈ S C ( U N ) fin , and we may study [ (cid:1) ∂, e − ( C ∞ N +1 ) ⊗ (cid:101) a ] and [ (cid:1) ∂, C ∞ N +1 e − ( C ∞ N +1 ) ⊗ (cid:101) a ].Since (cid:1) ∂ is a closed operator, for φ ⊗ s ⊗ s (cid:48) ∈ H fin ⊗ alg S ∗ fin ⊗ alg ⊗ S fin ,[ (cid:1) ∂, e − ( C ∞ N +1 ) ⊗ (cid:101) a ]( φ ⊗ s ⊗ s (cid:48) ) = lim M →∞ [ (cid:1) ∂, e − ( C MN +1 ) ⊗ (cid:101) a ]( φ ⊗ s ⊗ s (cid:48) )= lim M →∞ [ (cid:1) ∂ M , e − ( C MN +1 ) ⊗ (cid:101) a ]( φ ⊗ s ⊗ s (cid:48) )= lim M →∞ I M + lim M →∞ I M + lim M →∞ I M . As proved above, all of lim M →∞ I Mi ’s (for i = 1 , ,
3) are bounded by c ·(cid:107) φ ⊗ s ⊗ s (cid:48) (cid:107) for some positive constant c . For f o , use the same argument. Proposition 5.11.
As an operator on H , (1 + (cid:1) ∂ ) − is a compact operator.Proof. Since L (cid:1) ∂ and R (cid:1) ∂ are anti-commutative, (cid:1) ∂ = −
12 ( L (cid:1) ∂ ) + 12 ( R (cid:1) ∂ ) . These two operators have the same spectrum, which consists of non-negative integers with finite multiplicity.Therefore (cid:1) ∂ + 1 has discrete and positive spectrum. Its inverse is a compact operator.40his fact tells us the following thing: On S ⊗ H , the operator [ f ⊗ a ] · (1 + id ⊗ (cid:1) ∂ ) − is an S -compactoperator. Thus we have proved that the pair ( S ⊗ H , (cid:1) ∂ ) gives a Kasparov ( S C ( U ) , S )-module.Let us move to the equivariance issues. The following lemma looks obvious, at first sight. However, thereare some difficulties: U does not preserve H fin ; and (cid:1) ∂ is an infinite sum. Lemma 5.12. U preserves the domain of (cid:1) ∂ .Proof. Recall that L ( U ) ⊗ S ∗ ⊗ S ∼ = L ( R ∞ ) ⊗ S ∗ ⊗ L ( R ∞ ) ∗ ⊗ S . Since R (cid:1) ∂ is actually U -equivariant, wemay ignore this part. Moreover, (cid:1) ∂ is trivial on the S -part. As a result, we may concentrate on the operator L (cid:1) ∂ acting on L ( R ∞ ) ⊗ S ∗ .We divide the problem into two parts: (1) g ∈ U fin preserves dom( L (cid:1) ∂ ) and (2) the general case. The formeris much easier than the latter, but it contains the key computation, which has essentially been mentioned in[FHT].(1) What we need to check is the following: For any v ∈ dom( L (cid:1) ∂ ) whose approximating sequence is { v n } ⊆ L ( R ∞ ) fin ⊗ alg S ∗ fin , the sequence { g ( v n ) } converges in the graph norm of L (cid:1) ∂ . Since the U fin -actionon L ( R ∞ ) is unitary, the sequence converges in the norm of L ( R ∞ ) ⊗ S ∗ . Thus, it suffices to check that { L (cid:1) ∂ ( gv n ) } gives a Cauchy sequence.Let us notice an elementary equality: L (cid:1) ∂ ( gv n ) = g. (cid:2) g − ◦ L (cid:1) ∂ ◦ g ( v n ) (cid:3) = (cid:0) g ◦ [ g − ( L (cid:1) ∂ )] (cid:1) ( v ) . In the strong sense on L ( R ∞ ) fin ⊗ alg S ∗ fin , the operator g − ( L (cid:1) ∂ ) can be written as g − ◦ L (cid:1) ∂ ◦ g = (cid:88) k (cid:2) ρ ( g − ) dρ ( z k ) ρ ( g ) ⊗ γ ∗ ( z k ) + ρ ( g − ) dρ ( z k ) ρ ( g ) ⊗ γ ∗ ( z k ) (cid:3) . This equality is proved as follows: Since g ∈ U fin , it preserves ker( dρ ( z k ) ⊗ γ ∗ ( z k ) + dρ ( z k ) ⊗ γ ∗ ( z k )) forsufficiently large k . Hence the infinite sum above is, vectorwisely, a finite sum. Let g = exp( f ). Then, asexplained in Definition 4.1, ρ ( g − ) dρ ( z k ) ρ ( g ) = dρ (Ad g − ( z k )) = dρ ( z k + ω ( − f, z k ) K ) . Since K gives i id on the representation space, g − ◦ L (cid:1) ∂ ◦ g = L (cid:1) ∂ − iγ ∗ ( Jf ). This formula can be proved byconsidering the Fourier expansion f = (cid:80) k ( f k z k + f k z k ) and ω ( z k , z k ) = − i . Seeing this formula, we estimate (cid:107) L (cid:1) ∂ ( gv n ) − L (cid:1) ∂ ( gv m ) (cid:107) . (cid:107) L (cid:1) ∂ ( gv n ) − L (cid:1) ∂ ( gv m ) (cid:107) = (cid:107) g − ( L (cid:1) ∂ )( v n ) − g − ( L (cid:1) ∂ )( v m ) (cid:107) = (cid:107) L (cid:1) ∂ ( v n − v m ) − iγ ∗ ( Jf )( v n − v m ) (cid:107)≤ (cid:107) L (cid:1) ∂ ( v n − v m ) (cid:107) + (cid:107) γ ∗ ( Jf )( v n − v m ) (cid:107) . The first term is small, since { v n } converges in the graph norm of L (cid:1) ∂ . The second one is also small, because γ ∗ ( Jf ) gives a bounded operator.We need to add a comment before going to (2). The above estimate enables us to generalize the formula g − ( L (cid:1) ∂ ) = L (cid:1) ∂ − iγ ∗ ( Jf ) on the L ( R ∞ ) fin ⊗ alg S ∗ fin , to the same formula on dom( L (cid:1) ∂ ). Let v ∈ dom( L (cid:1) ∂ ), and g ∈ U fin . Suppose that { v n } ⊆ L ( R ∞ ) ⊗ S ∗ fin converges to v in the graph norm of L (cid:1) ∂ . Then { g.v n } convergesto gv in the graph norm of L (cid:1) ∂ . g − ◦ L (cid:1) ∂ ◦ g ( v ) = g − . lim k L (cid:1) ∂ ( g.v k ) = lim k g − ◦ L (cid:1) ∂ ◦ g ( v k )= lim k [ L (cid:1) ∂ − iγ ∗ ( Jf )]( v k ) = L (cid:1) ∂ ( v ) − iγ ∗ ( Jf )( v ) . (2) Let us consider the general case. Assume that g ∈ U . Let { g n } ⊆ U fin be an approximating sequenceof g . Take any v ∈ dom( L (cid:1) ∂ ). Then, { g n v } converges to g.v . We need to prove that { L (cid:1) ∂ ( g n v ) } is a Cauchy41equence. Put g n = exp( f n ) and g = exp( f ). L (cid:1) ∂ ( g n v ) − L (cid:1) ∂ ( g m v ) = g n .g − n ( L (cid:1) ∂ )( v ) − g m .g − m ( L (cid:1) ∂ )( v )= g n . (cid:2) L (cid:1) ∂ − iγ ∗ ( Jf n ) (cid:3) ( v ) − g m . (cid:2) L (cid:1) ∂ − iγ ∗ ( Jf m ) (cid:3) ( v )= (cid:104) g n . L (cid:1) ∂ ( v ) − g m . L (cid:1) ∂ ( v ) (cid:105) − (cid:104) g n . [ iγ ∗ ( Jf n − Jf m )] ( v ) (cid:105) − (cid:104) g n .iγ ∗ ( Jf m )( v ) − g m .iγ ∗ ( Jf m )( v ) (cid:105) . Let us prove that all of the three terms are arbitrary small, if n and m are large enough. The first one issmall, because the representation is continuous. The norm of the second term is given by (cid:107) γ ∗ ( Jf n − Jf m )( v ) (cid:107) .Since [ γ ∗ ( X )] ∗ γ ∗ ( X ) = (cid:107) X (cid:107) id by definition of the Clifford multiplication, and { g n } converges, the norm ofthe second term is small.For the third one, we regard g n and g m as operators, and we write that as ( g n − g m ) .iγ ∗ ( Jf m )( v ). Wecompute the square of the norm, noticing that g n − g m commutes with γ ∗ ( Jf m ). (cid:107) ( g n − g m ) .iγ ∗ ( Jf m )( v ) (cid:107) = (( g n − g m ) .iγ ∗ ( Jf m )( v ) | ( g n − g m ) .iγ ∗ ( Jf m )( v ))= (( g n − g m ) . [ γ ∗ ( Jf m )] ∗ γ ∗ ( Jf m )( v ) | ( g n − g m ) .v )= (cid:107) Jf m (cid:107) L (( g n − g m ) .v | ( g n − g m ) .v ) . Since (cid:107) Jf m (cid:107) L converges to (cid:107) Jf (cid:107) L , and since (( g n − g m ) .v | ( g n − g m ) .v ) converges to 0, the quantity we areestimating is small.The following is almost a corollary of the above proof. Proposition 5.13.
For any g ∈ U , g ( (cid:1) ∂ ) − (cid:1) ∂ is bounded, [ (cid:1) ∂, (cid:1) ∂ − g ( (cid:1) ∂ )] (cid:1) ∂ (cid:1) ∂ + λ is uniformly bounded in λ , andthe map g (cid:55)→ g ( (cid:1) ∂ ) − (cid:1) ∂ is continuous on U .Proof. Let g = exp( f ) ∈ U . We firstly extend the formula g ( L (cid:1) ∂ ) − L (cid:1) ∂ = iγ ∗ ( Jf ) for U fin to the general case g ∈ U . Take { g n } ⊆ U fin , such that g n = exp( f n ) and g n → g . For v ∈ dom( (cid:1) ∂ ), g ( L (cid:1) ∂ )( v ) = g. lim n (cid:2) L (cid:1) ∂ ( g − n .v ) (cid:3) = g. lim n g − n . (cid:2) g n ◦ L (cid:1) ∂ ◦ g − n ( v ) (cid:3) = g. lim n g − n . L (cid:1) ∂ ( v ) + g. lim n g − n .iγ ∗ ( Jf n )( v ) . The first term converges to L (cid:1) ∂ ( v ), clearly. The second one converges to iγ ∗ ( Jf )( v ), as follows: (cid:107) g − n iγ ( Jf n )( v ) − g − iγ ( Jf )( v ) (cid:107) ≤ (cid:107) g − n . [ iγ ( Jf n )( v ) − iγ ( Jf )( v )] (cid:107) + (cid:107) ( g − n − g − ) iγ ∗ ( Jf )( v ) (cid:107) . The first term tends to zero, since g − n gives an isometry, and Jf n converges to Jf . The second one alsotends to zero, because g n converges to g .Since iγ ∗ ( Jf ) is a bounded operator, g ( (cid:1) ∂ ) − (cid:1) ∂ is also a bounded operator.We would like to prove that [ (cid:1) ∂, (cid:1) ∂ − g ( (cid:1) ∂ )] (cid:1) ∂ (cid:1) ∂ + λ is uniformly bounded in λ . Firstly, we note that (cid:1) ∂ − g ( (cid:1) ∂ )is a bounded operator, but we have not proved that (cid:1) ∂ − g ( (cid:1) ∂ ) preserves the domain of (cid:1) ∂ for general g ∈ U . However, it is not a serious problem in our case. This is because (cid:1) ∂ − g ( (cid:1) ∂ ) sends H fin to dom( (cid:1) ∂ ), which wewill prove soon, and (cid:1) ∂ preserves H fin . Therefore the operator we are considering is well-defined on at least H fin . Once the operator is proved to be defined on H fin , and if it is bounded, it extends to H itself uniquely.We check that (cid:1) ∂ − g ( (cid:1) ∂ ) maps H fin into dom( (cid:1) ∂ ) here. Let { g n } ⊆ U fin be an approximating sequenceof g , and let v ∈ H fin . We check that { γ ∗ ( Jf n )( v ) } converges in the graph norm of (cid:1) ∂ . It means that[ (cid:1) ∂ − g ( (cid:1) ∂ )]( v ) ∈ dom( (cid:1) ∂ ). Since g n converges to g , the sequence converges in “ L ”. Hence we need to provethat { (cid:1) ∂ [ γ ∗ ( Jf n )( v )] } is a Cuachy sequence. If g ∈ U fin , g ( (cid:1) ∂ ) − (cid:1) ∂ preserves dom( (cid:1) ∂ ). Jf n = (cid:80) l ( a l z l + b l z l ). Since g n ∈ U fin , there is no problem to exchange the order of sum. In thefollowing, we omit ( v ) in order to simplify the notations. (cid:1) ∂ ◦ γ ∗ ( Jf n ) = i √ L (cid:1) ∂, γ ∗ ( Jf n )] − γ ∗ ( Jf n ) ◦ (cid:1) ∂ = i √ (cid:34)(cid:88) k √ k ( dρ ( z k ) ⊗ γ ∗ ( z k ) + dρ ( z k ) ⊗ γ ∗ ( z k )) , (cid:88) l ( a l γ ∗ ( z l ) + b l γ ∗ ( z l )) (cid:35) − γ ∗ ( Jf n ) ◦ (cid:1) ∂ = i √ (cid:88) k,l √ k [ dρ ( z k ) ⊗ γ ∗ ( z k ) + dρ ( z k ) ⊗ γ ∗ ( z k ) , a l γ ∗ ( z l ) + b l γ ∗ ( z l )] − γ ∗ ( Jf n ) ◦ (cid:1) ∂ = i √ (cid:88) k √ k ( a k dρ ( z k ) ⊗ [ γ ∗ ( z k ) , γ ∗ ( z k )] + b k dρ ( z k ) ⊗ [ γ ∗ ( z k ) , γ ∗ ( z k )]) − γ ∗ ( Jf n ) ◦ (cid:1) ∂ = √ i (cid:88) k √ k ( a k dρ ( z k ) + b k dρ ( z k )) ⊗ id − γ ∗ ( Jf n ) ◦ (cid:1) ∂ = √ idρ ( | d | / Jf n ) ⊗ id − γ ∗ ( Jf n ) ◦ (cid:1) ∂, where we put | d | / Jf n := (cid:80) l √ l ( a l z l + b l z l ). We need to check that (cid:1) ∂ ◦ γ ∗ ( Jf n )( v ) converges. Thanks tothe above computation, (cid:1) ∂ ◦ γ ∗ ( Jf n )( v ) = √ i [ dρ ( | d | / Jf n ) ⊗ id]( v ) − γ ∗ ( Jf n ) ◦ (cid:1) ∂ ( v ) . The right hand side converges, if g n converges to g . Moreover, for any g ∈ U , we find that [ (cid:1) ∂, γ ∗ ( Jf )] = √ idρ ( | d | / Jf ) ⊗ id . We are in the position to prove that the operator norm of [ (cid:1) ∂, (cid:1) ∂ − g ( (cid:1) ∂ )] (cid:1) ∂ (cid:1) ∂ + λ is uniformly bounded in λ .Formally, this is a kind of elliptic regularity: [ (cid:1) ∂, (cid:1) ∂ − g ( (cid:1) ∂ )] and (cid:1) ∂ are first order, and (cid:1) ∂ + λ is the inverse ofan elliptic operator of second order; the product is of “0-th order”.Let us prove that (cid:1) ∂ (cid:1) ∂ + λ maps H into dom( (cid:1) ∂ ). Take v ∈ H . Choose an approximation { v n } ⊆ H fin ,of v . Let b ( x ) := x x + λ . Notice that b is a bounded function, and hence b ( (cid:1) ∂ ) is a bounded operator.Consequently, b ( (cid:1) ∂ )( v n ) = (cid:1) ∂ (cid:1) ∂ + λ ( v n ) converges to b ( (cid:1) ∂ ) v . Put b ( x ) = x x + λ . Then, b ( (cid:1) ∂ ) = (cid:1) ∂b ( (cid:1) ∂ ).Since b is also a bounded function, b ( (cid:1) ∂ )( v n ) = (cid:1) ∂ (cid:2) b ( (cid:1) ∂ )( v n ) (cid:3) converges again. Since (cid:1) ∂ is a closed operator, b ( (cid:1) ∂ )( v ) ∈ dom( (cid:1) ∂ ) and (cid:1) ∂ [ b ( (cid:1) ∂ )( v )] = b ( (cid:1) ∂ )( v ).Let us prove that dρ ( X ) maps dom( (cid:1) ∂ ) to H continuously, for all X ∈ Lie( U ). For this aim, it is sufficientto prove the following inequality: (cid:107) dρ ( X ) ⊗ id( v ) (cid:107) ≤ C (cid:0) (cid:107) v (cid:107) + (cid:107) (cid:1) ∂ ( v ) (cid:107) (cid:1) , for some C which is independent of v . Suppose that X = (cid:80) ( a k z k + b k z k ). Note that( dρ ( z k ) v | dρ ( z k ) v ) = ( − dρ ( z k ) dρ ( z k ) v | v ) ≤ k (cid:0) (cid:1) ∂ v (cid:12)(cid:12) v (cid:1) = 2 k (cid:107) (cid:1) ∂ ( v ) (cid:107) , and similar for dρ ( z k ). Then, we find that (cid:107) dρ ( X ) ⊗ id( v ) (cid:107) ≤ (cid:88) (cid:18) | a k |√ k (cid:107) (cid:1) ∂ ( v ) (cid:107) + | b k |√ k (cid:107) (cid:1) ∂ ( v ) (cid:107) (cid:19) ≤ C (cid:0) (cid:107) v (cid:107) + (cid:107) (cid:1) ∂ ( v ) (cid:107) (cid:1) where C can be taken as “the l / norm of X ” times a constant. Let us prove that using the Cauchy-Schwarzinequality. (cid:20)(cid:88) (cid:18) | a k |√ k + | b k |√ k (cid:19)(cid:21) = (cid:88) (cid:18) k · √ k | a k | + 1 k · √ k | b k | (cid:19) ≤ (cid:114) (cid:88) k · (cid:113)(cid:88) k ( | a k | + | b k | ) . U -continuity. It is clear by the explicit description: g ( (cid:1) ∂ ) − (cid:1) ∂ = i √ γ ∗ ( Jf ).As a result of these propositions, we obtain the following index element. Definition-Theorem 5.14.
The pair ( S ⊗H , (cid:1) ∂ ) is an unbounded τ -twisted U -equivariant Kasparov ( S C ( U ) , S ) -module. The corresponding KK -element is denoted by [ (cid:101) (cid:1) ∂ ] ∈ KK τU ( S C ( U ) , S ) , and is called the indexelement .Remark . The operator R (cid:1) ∂ N + (cid:104) √ R (cid:1) ∂ ∞ N +1 + i √ L (cid:1) ∂ ∞ N +1 (cid:105) defines the same KK -element. This operator isactually U N -equivariant. [ (cid:102) d U ] The original Dirac element [ d X ] ∈ KK ( Cl τ ( X ) , C ) for a Riemannian manifold is given by the pair ( L ( X, ∧ ∗ T ∗ X ) , d + d ∗ ). However, it is possible to add a potential term. Let us check that using the definition of unboundedequivariant Kasparov modules.Let X be a complete Riemannian manifold, and consider the above Kasparov module [ d X ]. We write d + d ∗ as D in the following two propositions, in order to simplify the notation. Proposition 5.16.
Let h be a smooth bundle homomorphism of ∧ ∗ T ∗ X satisfying the following conditions: h ( x ) is odd, self-adjoint, commutes with the Cliff − ( T ∗ x X ) -action (or the symbol of D ); and ∇ ( h ) is a boundedsection, where ∇ is induced by the Levi-Civita connection. We also impose that D + th is self-adjoint for t ∈ [0 , . Then, ( L ( X, ∧ ∗ T ∗ X ) , D + h ) defines a K -homology class of Cl τ ( X ) , and it is the same with [ d X ] as KK -elements. More precisely, they are operator homotopic to each other through unbounded Kasparovmodules, by ( L ( X, ∧ ∗ T ∗ X ) , D + th ) .Proof. It suffices to prove the following: ( L ( X, ∧ ∗ T ∗ X ) , D + th ) is unbounded Kasparov modules for each t . Let D t := D + th .Firstly, we check that [ a, D t ] is bounded for each smooth and compactly supported a and each t . Since Cliff + ( T ∗ X )-action commutes with the symbol of D , the commutator [ a, D ] is bounded. Moreover, since[ a, h ] is a compactly supported smooth section of the bundle Cliff + ( T ∗ X ), it gives a bounded operator. a (1 + D ) − is supposed to be compact. In order to prove that a (1 + D t ) − is compact, we compare itwith a (1 + D ) − . a (1 + D t ) − − a (1 + D ) − = a (1 + D ) − (1 + D − − D t )(1 + D t ) − = − a (1 + D ) − ( t ( hD + Dh ) + t h )(1 + D t ) − . It is sufficient to prove that ( t ( hD + Dh ) + t h )(1 + D t ) − is a bounded operator. We note that the operator(1 + D t ) − maps L to dom( D t ). The graph norm of D t , which is denoted by (cid:107) • (cid:107) D t , is given by (cid:107) • (cid:107) D t = (cid:107) • (cid:107) L + (cid:107) D t • (cid:107) L = (cid:107) • (cid:107) L + (cid:107) D • (cid:107) L + t ([ D, h ] •|• ) L + t (cid:107) h • (cid:107) L . Since [
D, h ] is bounded (as proved soon), the norm (cid:107)•(cid:107) D t is equivalent to the norm (cid:107)•(cid:107) L + (cid:107) D •(cid:107) L + t (cid:107) h •(cid:107) L .Therefore h maps dom( D t ) continuously to L . Consequently, ( t ( hD + Dh ) + t h )(1 + D t ) − is a boundedoperator.Finally, we must check that [ D, h ] is bounded. Dh + hD = (cid:88) ∂ x i ⊗ ( dx i ∧ − dx i (cid:99) ) ◦ h + h ◦ (cid:88) ∂ x i ⊗ ( dx i ∧ − dx i (cid:99) )= − (cid:88) ∂h∂x i ⊗ ( dx i ∧ − dx i (cid:99) ) . This is a bounded operator by the assumption that ∇ ( h ) is bounded.44or the equivariant KK -theory version, we need extra conditions. Proposition 5.17.
Suppose that a locally compact, second countable and Hausdorff group Γ acts on X ,isometrically, properly and cocompactly. Suppose that h satisfies the following additional assumptions: • g ( h ) − h is a bounded operator for each g ∈ Γ . • The function Γ (cid:51) g → g ( h ) − h ∈ B ( L ) is norm continuous.Then, two unbounded Γ -equivariant Kasparov modules ( L ( X, S ) , D + h ) and ( L ( X, S ) , D ) are operatorhomotopic.Proof. We will follow the same story: We will verify that D + th gives an unbounded Γ-equivariant Kasparovmodule. For this aim, we need to check the followings: (1) g ( D + th ) − ( D + th ) is bounded for any g ; (2)[ D + th − g ( D + th ) , D + th ] D + th D + th ) is bounded for any g ; and (3) g (cid:55)→ g ( D + th ) − ( D + th ) is continuous.(1) and (3) are clear from the assumptions. Recall that D is actually Γ-equivariant: g ( D ) = D . For(2), we need some computation. Put D t := D + th , and L k,t := dom( D kt ). L = L ,t denotes the original L -space. The operator D + th D + th ) maps L into L ,t continuously. Thus it suffices to check that [ D + th − g ( D + th ) , D + th ] maps L ,t into L continuously.[ D + th − g ( D + th ) , D + th ]= [ D − g ( D ) , D ] + t ([ D − g ( D ) , h ] + [ h − g ( h ) , D ]) + t [ h − g ( h ) , h ]= [ D − g ( D ) , D ] + t (cid:0) [ D, h ] − [ g ( D ) , h ] + [ h, D ] − g [ h, g − ( D )] (cid:1) + t { ( h − g ( h )) h + h ( h − g ( h )) } . Note that D is G -equivariant. Then, first two terms themselves turn out to be bounded operators on L .For the third one, we note that L ,t = dom( D ) ∩ dom( th ), which will be proved in the next paragraph. Bythe assumption, h − g ( h ) is a bounded function, and hence it preserves dom( th ). In particular, the operator { ( h − g ( h )) h + h ( h − g ( h )) } maps dom( th ) ⊇ L ,t into L continuously.Let us prove that L ,t = dom( D ) ∩ dom( th ). There is a common domain C ∞ c . Take a vector v ∈ L ,t and anapproximating sequence { v n } from C ∞ c in the L ,t -norm. Then, v n tends to v in the L -norm, and { D t ( v n ) } gives a Cauchy sequence. Hence, (cid:107) D t ( v n ) − D t ( v m ) (cid:107) = (cid:107) D ( v n − v m ) (cid:107) + t (( Dh + hD )( v n − v m ) | v n − v m ) + t (cid:107) h ( v n − v m ) (cid:107) goes to zero as n, m → ∞ . The second one can be negative, but it goes to zero. This isbecause Dh + hD is a bounded operator. Therefore the both of { Dv n } and { hv n } are Cauchy sequences, andhence v ∈ dom( D ) ∩ dom( h ).These results justify the name of the following KK -element. Definition 5.18 (See [HK]) . On a finite-dimensional Euclidean space W , take an orthonormal basis { e i } .The corresponding coordinate is denoted by { x i } . Let us consider two Clifford multiplications on (cid:86) ∗ T ∗ W : c ( v ) = v ∧ − v (cid:99) and c ∗ ( v ) = v ∧ + v (cid:99) . Define the Bott-Dirac operator as D W := (cid:88) i (cid:18) c ( e i ) ∂∂x i + 12 c ∗ ( e i ) x i (cid:19) .W acts on itself by the parallel transformation. The pair ( L ( W, (cid:86) ∗ T ∗ W ) , D W ) determines an unbounded W -equivariant Kasparov ( Cl τ ( W ) , C )-module. [ d W ] denotes the corresponding KK W -element, and it is calledthe Dirac element . To distinguish it from the original Dirac element, we sometimes call it the
Bott-Diracelement .What we can generalize is not the original Dirac element, but a perturbed version of the Bott-Diracelement. It has a strong advantage: The operator itself is Fradholm. The cost we must pay is that the operatoris not actually equivariant. However, thanks to the flexibility of the definition of unbounded equivariantKasparov modules, it it not a big problem. 45e have pointed out that our Dirac operator is in fact the Bott-Dirac operator in Remark 5.5. The pair( S ⊗ H , (cid:1) ∂ ) defines a τ -twisted U -equivariant Kasparov ( S C ( U ) , S )-module. We would like to define theDirac element by the same module and operator with the different action .The natural ( τ -twisted) action on our “ L -space” looks a combination of the translation and the mul-tiplication of the 2-cocycle. On the other hand, the action to define the Dirac element must look like thetranslation without any cocycles. Such new action is the following. Definition-Theorem 5.19.
The group action A is given by the following: For g = exp( f ) ∈ U , A g := L exp ( f ) R exp ( − f ) . The pair ( S ⊗ H , (cid:1) ∂ ) with the action A gives an unbounded U -equivariant Kasparov ( S C ( U ) , S ) -module. The corresponding KK -element is called the Dirac element and denoted by [ d U ] ∈ KK U ( S C ( U ) , S ) .Remarks . Before going to the proof, we give several remarks.(1) Recall that R (cid:1) ∂ is given by (cid:80) k √ k ( · · · ). Thus it has already been “perturbed” from the originaldefinition, unlike the Bott-Dirac operator of [HK].(2) The Bott-Dirac operator in [HK] is Fredholm, but it is not with compact resolvent. To overcomethis serious problem, N. Higson and G. Kasparov introduced a perturbation family, whose “boundary” givesthe desired “Kasparov module”. This family gives an equivariant E -theory element by the central invariant.Through the isomorphism between KK and E for their cases, they obtained the desired KK -element.(3) Comparing their construction, our definition seems to be too simple. Why is this OK? The reasonis our group action is very special: The linear part of the action is trivial and hence it has already been“diagonalized”. On the other hand, in Lemma 5.7 in [HK], they “weakly diagonalized” the linear part ofa group action, in order to perturb the Bott-Dirac operator. Their action is not actually diagonalizable,and hence the perturbed operator does not give an equivariant Kasparov module; The family is merely“asymptotically equivariant”.In order to generalize our result (to the LG -case for non-commutative G , for example), we may need tofollow Higson-Kasparov’s method. However, in the present paper, we concentrate on our special case. Proof.
We have already proved everything, except for the group action issues.We check the following things: (0) A is actually a group action which is compatible with the action U (cid:8) S C ( U ) and the S C ( U )-module structure on S ⊗ H ; (1) U preserves the domain of (cid:1) ∂ ; (2) g ( (cid:1) ∂ ) − (cid:1) ∂ isbounded; (3) [ (cid:1) ∂, (cid:1) ∂ − g ( (cid:1) ∂ )] · (cid:1) ∂ λ + (cid:1) ∂ is bounded; and (4) the map g (cid:55)→ g ( (cid:1) ∂ ) − (cid:1) ∂ is continuous.(0) Let √ g be exp( f /
2) for g = exp( f ). Note that L g ◦ L g = L g · g e i ω ( f ,f ) and R g ◦ R g = R g · g e − i ω ( f ,f ) . Then, since L and R are commutative, A g ◦ A g = L √ g ◦ R √ g − ◦ L √ g ◦ R √ g − = L √ g ◦ L √ g ◦ R √ g − ◦ R √ g − = L √ g ·√ g e i ω ( f / ,f / L √ g − ·√ g − e − i ω ( f / ,f / = A g · g . In order to prove the compatibility with the
S C ( U )-module structure, we need to recall the definition of U (cid:8) S C ( U ). Formally, this action is given by g.a ( x ) = a ( x − g ). Just like Theorem 4.18, we can check that R g ( a · φ ) = [ g − .a ] · R g ( φ ) for a ∈ S C ( U ) and φ ∈ S ⊗ H . Thus, A g ( a · φ ) = L √ g ◦ R √ g − ( a · φ )= L √ g [ √ g.a · R √ g − ( φ )]= [ √ g. √ g.a ] · L √ g ◦ R √ g − ( φ )= [ g.a ] · A g ( φ ) . Not τ -twisted! (cid:1) ∂ ) = S ⊗ dom( L (cid:1) ∂ ) ⊗ dom( R (cid:1) ∂ ). As Lemma 5.12, the U -action L preservesdom( L (cid:1) ∂ ). In the same way, one can see that the action R preserves dom( R (cid:1) ∂ ). Since A is a combination of L and R , the action A preserves dom( (cid:1) ∂ ).(2) Let us compute g ( (cid:1) ∂ ) − (cid:1) ∂ for the action A . In the following, A g ( (cid:1) ∂ ) means A g ◦ (cid:1) ∂ ◦ A g − , and similarlyfor L and R . A g ( (cid:1) ∂ ) − (cid:1) ∂ = R √ g − L √ g (cid:18) i √ L (cid:1) ∂ + 1 √ R (cid:1) ∂ (cid:19) − i √ L (cid:1) ∂ − √ R (cid:1) ∂ = (cid:20) L √ g (cid:18) i √ L (cid:1) ∂ (cid:19) − i √ L (cid:1) ∂ (cid:21) + (cid:20) R √ g − (cid:18) √ R (cid:1) ∂ (cid:19) − √ R (cid:1) ∂ (cid:21) , where we used the R - and L -invariance of L (cid:1) ∂ and R (cid:1) ∂ , respectively. The first term is bounded, thanks toProposition 4.19. For the second term, repeat the same argument for the action R .(3) Let us compute the commutator [ (cid:1) ∂, (cid:1) ∂ − g ( (cid:1) ∂ )].[ (cid:1) ∂, (cid:1) ∂ − g ( (cid:1) ∂ )] = (cid:20) i √ L (cid:1) ∂ + 1 √ R (cid:1) ∂ , i √ L (cid:1) ∂ + 1 √ R (cid:1) ∂ − R √ g − L √ g (cid:18) i √ L (cid:1) ∂ + 1 √ R (cid:1) ∂ (cid:19)(cid:21) = 12 [ i L (cid:1) ∂ + R (cid:1) ∂ , i L (cid:1) ∂ + R (cid:1) ∂ − R √ g − L √ g ( i L (cid:1) ∂ + R (cid:1) ∂ )]= 12 (cid:110) − [ L (cid:1) ∂, L (cid:1) ∂ − L √ g ( L (cid:1) ∂ )] + [ R (cid:1) ∂, R (cid:1) ∂ − R √ g − ( R (cid:1) ∂ )] (cid:111) . Since R (cid:1) ∂ is L -invariant, L (cid:1) ∂ − L √ g ( L (cid:1) ∂ ) = (cid:1) ∂ − L √ g ( (cid:1) ∂ ). Moreover, since R (cid:1) ∂ anti-commutes with L (cid:1) ∂ and L √ g ( L (cid:1) ∂ ), we find that [ L (cid:1) ∂, L (cid:1) ∂ − L √ g ( L (cid:1) ∂ )] = [ (cid:1) ∂, (cid:1) ∂ − L √ g ( (cid:1) ∂ )]. Similarly, [ R (cid:1) ∂, R (cid:1) ∂ − R √ g − ( R (cid:1) ∂ )] = [ (cid:1) ∂, (cid:1) ∂ − R √ g − ( (cid:1) ∂ )]. Thanks to Proposition 5.13 and the R -action version of it, we obtain the desired boundedness.(4) It is clear from the same proposition and the same computation in (2). [ (cid:102) σ Cl (cid:1) ∂ ] We will define the reformulated Clifford symbol element for our case. For this aim, we recall the formula inLemma 2.33. We would like to define the Clifford symbol element by (
S C ( U ) , τ -twistedaction which looks like the left regular representation on the line bundle L . Formally, our Clifford symbolelement is “( S ⊗ C ( U, L ⊗
Cliff + ( T ∗ U )) , g.a ( • ) = a ( • − g ) e i ω ( f, • ) , for g = exp( f ) ∈ U . In order to justify this definition, we use the finite-dimensional approximation, as usual.Let g ∈ U N and a ∈ S C ( U M ) fin . It is possible to assume that N = M , by β M,N : S C ( U M ) → S C ( U N )(when M < N ) or the inclusion U N ⊆ U M (when N < M ). Since a is of the form (cid:80) f i ⊗ (cid:101) a i , g.a should bedefined by (cid:80) f i ⊗ g. (cid:101) a i , where g. (cid:101) a i is defined by (cid:101) a i ( • − g ) e i ω ( f, • ) . The problem is whether this U fin -actionextends to U or not, as usual. Lemma 5.21.
This action extends to a continuous U -action on S C ( U ) . It is compatible with the originaluntwisted U -action on the C ∗ -algebra S C ( U ) . That is, the pair ( S C ( U ) , equipped with the new action isa τ -twisted U -equivariant Kasparov ( S C ( U ) , S C ( U )) -module.Remark . The above “compatibility” means the following. To be precise, we introduce some symbols usedonly here: σ is the original untwisted action, and ρ is the introduced twisted action. For a, a (cid:48) , a , a ∈ S C ( U ),we will prove that ρ g ( a · a · a (cid:48) ) = σ g ( a ) · ρ g ( a ) · σ g ( a (cid:48) ), and [ ρ g ( a )] ∗ ρ g ( a ) = σ g ( a ∗ a ).47 roof. Once the action is extended, the properties listed above are clear from the formula for U fin . We checkthat the action is well-defined and continuous. Just like the proof of Theorem 4.18, the above action extendsto the action on S C ( U ), since each g gives an isometry. It is enough to check that the action is continuous.To prove the continuity, we will use the usual technique. Let a ∈ S C ( U ) and g, g (cid:48) ∈ U fin . For any ε > (cid:80) f i ⊗ (cid:101) a i ∈ S C ( U ) fin such that (cid:107) a − (cid:80) f i ⊗ (cid:101) a i (cid:107) < ε . We may assume that a can be approximatedby a single element f ⊗ (cid:101) a . Moreover, we may assume that the both of f and (cid:101) a are real-scalar-valued, sincethe U -action is trivial on the fiber Cliff + . We may prove that (cid:107) g. ( f ⊗ (cid:101) a ) − g (cid:48) . ( f ⊗ (cid:101) a ) (cid:107) is less than ε when g and g (cid:48) are close enough. We prove this estimate when f = f e and f o . For other functions, one can use thetechnique of the proof of Theorem 4.18.Suppose that g, g (cid:48) ∈ U N and (cid:101) a ∈ S C ( U M ). Using β M,L for L ≥ N , we identify f e ⊗ (cid:101) a with f e ⊗ e − ( C LM +1 ) ⊗ (cid:101) a. Since the above twisted action on
Cliff τ ( U N ) is continuous, we may ignore the U N -part, just like the proofof Theorem 4.18. Thus, we may assume that M = 0. Let us estimate (cid:12)(cid:12)(cid:12) e − (cid:80) b [ b − l ( x b − g b ) +( y b − h b ) ]+ i (cid:80) ( g b y b − h b x b ) − e − (cid:80) b b − l [( x b − g (cid:48) b ) +( y b − h (cid:48) b ) ]+ i (cid:80) ( g (cid:48) b y b − h (cid:48) b x b ) (cid:12)(cid:12)(cid:12) . Put r := (cid:80) b b − l [( x b − g b ) +( y b − h b ) ], ( r (cid:48) ) := (cid:80) b b − l [( x b − g (cid:48) b ) +( y b − h (cid:48) b ) ], ω ( g, x ) := (cid:80) ( g b y b − h b x b )and ω ( g (cid:48) , x ) := (cid:80) ( g (cid:48) b y b − h (cid:48) b x b ). Then, | e − r + i ω ( g,x ) − e − ( r (cid:48) ) + i ω ( g (cid:48) ,x ) | ≤ | e − r − e − ( r (cid:48) ) | + e − r | e i ω ( g,x ) − e i ω ( g (cid:48) ,x ) | . The first term is small, thanks to Theorem 4.18. The second one can be written as e − r | e i ω ( g − g (cid:48) ,x ) − | .Roughly speaking, this is small, because the growth of ω ( g − g (cid:48) , x ) as x → ∞ is slow enough. The rigorousproof is the following.Take a positive real number ε . Since e − r converges to 0 as r → ∞ , we can find a large number K satisfying the following: if r ≥ K , e − r < ε/
2. Note that the condition is satisfied, if (cid:80) b b − l ( x b + y b ) ≥ (cid:16) K + (cid:112)(cid:80) b b − l ( g b + h b ) (cid:17) . Consequently, if (cid:80) b b − l ( x b + y b ) ≥ (cid:16) K + (cid:112)(cid:80) b b − l ( g b + h b ) (cid:17) , then e − r (cid:48) ) | e i ω ( g − g (cid:48) ,x ) − | < ε .Next, we must estimate the quantity for “small” x . Firstly, choose a positive real number δ such that | e i t − | < ε if | t | < δ . Then, from the following computation using the Cauchy Schwarz type inequality,we can find positive number δ (cid:48) such that | ω ( g − g (cid:48) , x ) | < δ for all x satsfying that (cid:80) b b − l ( x b + y b ) ≤ (cid:0) K + (cid:80) b b − l ( g b + h b ) (cid:1) + 1, if (cid:80) k k l [( g k − g (cid:48) k ) + ( h k − h (cid:48) k ) ] < δ (cid:48) . | ω ( g − g (cid:48) , x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) b [( g b − g (cid:48) b ) y b − ( h b − h (cid:48) b ) x b ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) b b l [( g b − g (cid:48) b ) + ( h b − h (cid:48) b ) ] × (cid:88) b b − l ( x b + y b ) . Combining the technique for the twisted action on f e and the untwisted action on f o , one can deal withthe twisted action on f o . We leave it to the reader.Thanks to this lemma, the ( S C ( U ) , S C ( U ))-bimodule S C ( U ) admits a τ -twisted U -action which iscompatible with the original untwisted U -action on S C ( U ). We have reached the following definition-theorem. Definition-Theorem 5.23.
The pair ( S C ( U ) , is a τ -twisted U -equivariant Kasparov ( S C ( U ) , S C ( U )) -module. The corresponding KK -element is called the Clifford symbol element of (cid:1) ∂ , and denoted by [ (cid:103) σ Cl (cid:1) ∂ ] := [( S C ( U ) , ∈ KK τU ( S C ( U ) , S C ( U )) . .4 The KK -theoretical index theorem The following is the main theorem of the present paper, but it is almost a corollary of three definition-theoremsin this section.
Theorem 5.24.
The Kasparov module defined in Definition-Theorem 5.14 is a Kasparov product of theone defined in Definition-Theorem 5.23 and the one defined in Definition-Theorem 5.19. Formally, in KK τU ( S C ( U ) , S ) , [ (cid:101) (cid:1) ∂ ] = [ (cid:103) σ Cl (cid:1) ∂ ] ⊗ S C ( U ) [ (cid:102) d U ] . Remark . As explained in Section 3, the LT -equivariant KK -theory has not been extensively studied,and we have not proved that the following thing: For two given LT -equivariant KK -elements, there is aKasparov product of them, and such modules are unique up to homotopy . This theorem means, however,there is an example of two Kasparov modules which has a Kasparov product. Proof.
If we forget the group action, we find that [ (cid:103) σ Cl (cid:1) ∂ ] = [id] and [ (cid:101) (cid:1) ∂ ] = [ (cid:102) d U ], and hence the statement isobvious. The non-trivial thing is only the following: The isomorphism S C ( U ) ⊗ S C ( U ) [ S ⊗ H ] → S ⊗ H is τ -twisted U -equivariant, that is, the following diagram commutes: S C ( U ) ⊗ S [ S ⊗ H ] −−−−→ S ⊗ H g ⊗ g (cid:121) (cid:121) g S C ( U ) ⊗ S [ S ⊗ H ] −−−−→ S ⊗ H . In order to prove this commutativity, recall the precise definition of the group actions and the left
S C ( U )-action on S ⊗ H : All of them are defined by the limit of the finite-dimensional approximations. Therefore,we only have to prove the equivariance on U N for all N . Remember that the U -actions on the index elementand the Clifford symbol element have the same formula, and the U -action on the HKT algebra S C ( U )and the Dirac element have the same formula. Once noticing these things, one can prove the statementimmediately. We should study the connection between the main result of the present paper and the previous one of[Tak2]. In this paper, we studied the assembly map and the analytic index without the index element: Wedefined the “crossed product of C ( U ) by U ” C ( U ) (cid:111) U , the “ τ -twisted group C ∗ -algebra of U ” C (cid:111) τ U ,the “image of the index element along the descent homomorphism” j τU ([ (cid:1) ∂ ]) ∈ KK ( C ( U ) (cid:111) U ,
C (cid:111) τ U ),the “Mishchenko line bundle” [ c ] ∈ KK ( C , C ( U ) (cid:111) U ), and the “analytic index in the sense of [Kas16]”ind( (cid:1) ∂ ) ∈ KK ( C , C (cid:111) τ U ). These KK -elements satisfy the relation j τU ([ (cid:1) ∂ ]) = [ c ] ⊗ C ( U ) (cid:111) U j τU ([ (cid:1) ∂ ]). On the other hand, we have studied the index element, the Clifford symbol element and the Dirac elementin the present paper, without index.In this section, we try constructing “an assembly map”, whose value is the same with the analytic indexin [Tak2]. This construction is not elegant , but it might suggest the existence of a (much better) assemblymap which is compatible with the latter half of the assembly map constructed in [Tak2]. In the latter half ofthis section, we discuss the satisfactory form of the assembly maps and the crossed products. In this subsection, we define “an assembly map” in an indirect way. We begin with the definitions of severalingredients: the Bott elements and the τ -twisted group C ∗ -algebras of U N ’s. Unlike the main result of this paper, this Kasparov product makes rigorous sense. This is the reason why we add the quotation mark. efinition 6.1. For any finite-dimensional Euclidean space V , Cl τ ( V ) is KK -equivalent to C via the KK -element [ b V ] := [ Cl τ ( V ) , c ] ∈ KK ( C , Cl τ ( V )) , where c is the Clifford multiplication. At v ∈ V , c acts on the fiber Cliff + ( T ∗ v V ) via c ( v ).The followings are well-known: [ b V ] ⊗ Cl τ ( V ) [ d V ] = 1 C and [ d V ] ⊗ C [ b V ] = 1 Cl τ ( V ) . The direct proof isnot difficult, thanks to Proposition 2.8. The Kasparov product of the former one is given by the harmonicoscillator, whose index is 1. We will prove an infinite-dimensional analogue of the former one, in our language.The reformulated Bott element is just τ S ([ b V ]). It is truly a module, but it has been realized as a ∗ -homomorphism β V : S → S C ( V ) in [HKT]. This reformulated one has an infinite-dimensional analogue[ β U ⊥ N ] and so on. For the special one [ β U ], we use the symbol [ β ]. In the following, we forget the equivariantstructure on [ (cid:102) d U ]. Definition 6.2.
For a real Hilbert space V , the ∗ -homomorphism β V : S → S C ( V ) defines a KK -element[ β V ] ∈ KK ( S , S C ( V )). In particular, when V is given by our infinite-dimensional Lie group U , the Bottelement is denoted by [ β ] ∈ KK ( S , S C ( U )). Proposition 6.3. [ β ] ⊗ S C ( U ) [ (cid:102) d U ] = [1 S ] in KK ( S , S ) .Proof. Since β is a ∗ -homomorphism, [ β ] ⊗ S C ( U ) [ (cid:102) d U ] is given by [( S ⊗ H , (cid:1) ∂ )], where the left action of S isdefined by the composition φ : S β −→ S C ( U ) Φ −→ End S ( S ⊗ H ); φ ( f ) = f ( X ⊗ id + id ⊗ C ), where C is the Clifford operator on H . In order to find much simpler description,we deform φ . Let φ t ( f ) := f ( X ⊗ id + id ⊗ tC ). Note that φ = φ , and φ ( f ) = f ⊗ id. Then, two Kasparovmodules ( S ⊗ H , φ, (cid:1) ∂ ) and ( S ⊗ H , φ , (cid:1) ∂ ) are homotopic by the homotopy( S ⊗ H [0 , , { φ t } t ∈ [0 , , (cid:1) ∂ ) . We need to prove that the homotopy ( S ⊗ H [0 , , { φ t } t ∈ [0 , , (cid:1) ∂ ) is actually a Kasparov ( S , S [0 , S ⊗ H , φ , (cid:1) ∂ ) coincides with the tensor product of the Kasparov ( S , S )-module[( S , C , C )-module [( H , (cid:1) ∂ )]. The latter is 1 C , because (cid:1) ∂ is a Fredholm operator withindex 1.Let us recall the structure of the τ -twisted group C ∗ -algebra C (cid:111) τ U N from [Tak1]. U N has only one τ -twisted irreducible unitary representation on L ( R N ), and so C (cid:111) τ U N is isomorphic to K ( L ( R N ) ∗ ). Therepresentation space L ( R N ) ∗ has a specific vector denoted by ∗ b which is of the highest weight with respectto the infinitesimal rotation action dρ ∗ ( d ). Definition 6.4 (Definition 4.12 in [Tak1]) . The ∗ -homomorphism i N from C (cid:111) τ U N to C (cid:111) τ U N +1 is definedas follows: Let P be the rank one projection onto C ∗ b ∈ L ( R N +1 (cid:9) R N ) ∗ , and let i N ( k ) be k ⊗ P for k ∈ K ( L ( R N ) ∗ ). Using these homomorphisms, define the C ∗ -algebra C (cid:111) τ U by C (cid:111) τ U := lim −→ C (cid:111) τ U N .j N : C (cid:111) τ U N → C (cid:111) τ U denotes the canonical ∗ -homomorphism to define the inductive limit. As a result, C (cid:111) τ U is isomorphic to K ( L ( R ∞ ) ∗ ).Let us define assembly maps, parametrized by N ∈ N . Recall the following isomorphisms: S C ( U ) ∼ = S C ( U ⊥ N ) ⊗ C ( U N ); S C ( U ) (cid:111) U N ∼ = S C ( U ⊥ N ) ⊗ ( C ( U N ) (cid:111) U N ).50 efinition 6.5. The N -assembly map N µ τU : KK τU ( S C ( U ) , S ) → KK ( S , S (cid:111) τ U ) is the composition ofthe following sequence: KK τU ( S C ( U ) , S ) forget −−−→ KK τU N ( S C ( U ) , S ) j τUN −−−→ KK ( S C ( U ) (cid:111) U N , S (cid:111) τ U N ) ([ β U ⊥ N ] ⊗ C [ (cid:101) c UN ]) ⊗− −−−−−−−−−−−−→ KK ( S , S (cid:111) τ U N ) −⊗ [ j N ] −−−−−→ KK ( S , S (cid:111) τ U ) . For an element [ D ] ∈ KK τU ( S C ( U ) , S ), suppose that N µ τU ([ D ]) = N +1 µ τU ([ D ]) = · · · for sufficientlylarge N . We say that such [ D ] is assemblable . The value of the total assembly map is defined by µ τU ([ D ]) := N µ τU ([ D ]) for sufficient large N .The main theorem of this section is that our index element is assemblable. Firstly, we recall Remark 5.15.Our index element [ (cid:101) (cid:1) ∂ ] has another representative defined by the operator M (cid:1) ∂ := R (cid:1) ∂ M + 1 √ R (cid:1) ∂ ∞ M +1 + i √ L (cid:1) ∂ ∞ M +1 , which is actually U N -equivariant for N ≤ M . The new representative is denoted by [ M (cid:101) (cid:1) ∂ ].Note that [ M (cid:101) (cid:1) ∂ ] = [ M (cid:101) (cid:1) ∂ ∞ N +1 ] ⊗ C [ M (cid:1) ∂ N ] ∈ KK τU ⊥ N ( S C ( U ⊥ N ) , S ) ⊗ KK τU N ( Cl τ ( U N ) , C ) . Let us compute j τU N ([ M (cid:101) (cid:1) ∂ ]), after forgetting the U ⊥ N -action. It is easy to see that j τU N ([ M (cid:101) (cid:1) ∂ ]) = [ M (cid:101) (cid:1) ∂ ∞ N +1 ] ⊗ C j τU N [ M (cid:1) ∂ N ] . We have computed this element. See [Tak2] for the details.
Proposition 6.6 (Theorem 3.7 in [Tak2]) . j τU N ([ M (cid:101) (cid:1) ∂ N ]) is given by [( L ( U N ) ⊗ S ⊗ [ C (cid:111) τ U N ] , D ⊗ ⊗ id + id ⊗ L (cid:1) ∂ N )] ∈ KK ( C ( U N ) (cid:111) U N , C (cid:111) τ U N ) . Proposition 6.7 (Proposition 3.9 in [Tak2]) . µ τU N ([ M (cid:1) ∂ N ]) is given by [ c U N ] ⊗ C ( U N ) (cid:111) U N j τU N ([ M (cid:101) (cid:1) ∂ ]) = [( S U N ⊗ ( C (cid:111) τ U N ) , L (cid:1) ∂ N )] . This element is also represented as [( L ( R N ) (cid:98) ⊕ , . Let us compute the rest part of the N -assembly map in our language. Since the KK -element [ (cid:94) M (cid:1) ∂ ∞ N +1 ] doesnot depend on M in KK ( S C ( U ) , S ), we put M = N , and we remove M in the symbol. What we shouldcompute is [ b U ⊥ N ] ⊗ S C ( U ⊥ N ) [ (cid:94) (cid:1) ∂ ∞ N +1 ]. If we forget the group action, [ (cid:102) d U ] coincides with [ (cid:101) (cid:1) ∂ ]. Thus, [ b U ⊥ N ] ⊗ S C ( U ⊥ N ) [ (cid:101) (cid:1) ∂ ∞ N +1 ] = 1. Moreover, since j N maps [( L ( R N ) (cid:98) ⊕ , ∈ KK ( C , C(cid:111) τ U N ) to [( L ( R ∞ ) (cid:98) ⊕ , ∈ KK ( C , C(cid:111) τ U ), we obtain the following theorem. Theorem 6.8. N µ τU ([ M (cid:1) ∂ ]) = [( L ( R ∞ ) (cid:98) ⊕ , for all N . The RHS is independent of N , and our indexelement is assemblable; µ τU ([ (cid:1) ∂ ]) = [( L ( R ∞ ) (cid:98) ⊕ , .This value coincides with ind( R (cid:1) ∂ ) defined in [Tak2]. The previous subsection might suggest that we can define the total assembly map without making referenceto the finite-dimensional approximations: j τU ([ (cid:101) (cid:1) ∂ ]) should be “lim ←− j τU N ([ (cid:101) (cid:1) ∂ ])” in some sense. We discuss suchthings in this subsection. There are no rigorous results here, but we will write several conjectures andobservations on them. 51 onjecture 6.9. We can define two C ∗ -algebras which play roles of “ S C ( U ) (cid:111) U ” and “ S C ( U ) (cid:111) τ U ”,denoted by S C ( U ) (cid:111) U and S C ( U ) (cid:111) τ U , respectively. The C ∗ -algebra S ⊗ K ( L ( U ) ⊗ S ∗ ) is a strong candidate. We have defined a C ∗ -algebra C ( U ) (cid:111) U in [Tak2], by “believing” that the well-known isomorphism C ( G ) (cid:111) G ∼ = K ( L ( G )) for any locally compactgroup G , is valid even for infinite-dimensional cases. Just like this, for even-dimensional Spin c -Lie group G , Cl τ ( G ) (cid:111) G ∼ = K ( L ( G, S G )) for the trivial Spinor bundle S G . This is because Cl τ ( G ) ∼ = C ( G ) ⊗ Cliff + ( T ∗ e G )and Cliff + ( T ∗ e G ) ∼ = End( S e ).On the other hand, for an N -dimensional parallelizable manifolrd X , we may regard Cl τ ( X ) as C ( X ) ⊗ Cliff + ( R N ). Thus the C ∗ -algebra K ( L ( U )) ⊗ CAR is also a strong candidate, where CAR is the canonicalanti-commutation relation algebra. These two algebras are truly different: K (CAR) ∼ = Z [ ] while K ( K ) ∼ = Z . We must choose the mostappropriate C ∗ -algebra for the crossed products.Suppose that the above conjecture is true. We need the descent homomorphisms. Conjecture 6.10.
We can also define the following homomorphisms: • j τU : KK τU ( S C ( U ) , S ) → KK ( S C ( U ) (cid:111) U , S (cid:111) τ U ) . • j τU : KK τU ( S C ( U ) , S C ( U )) → KK ( S C ( U ) (cid:111) U ,
S C ( U ) (cid:111) τ U ) . • j U : KK U ( S C ( U ) , S ) → KK ( S C ( U ) (cid:111) τ U , S (cid:111) τ U ) .These descent homomorphisms are related in the following sense: for [ x ] ∈ KK τU ( S C ( U ) , S ) , [ y ] ∈ KK τU ( S C ( U ) , S C ( U )) and [ z ] ∈ KK U ( S C ( U ) , S ) satisfying that [ x ] = [ y ] ⊗ [ z ] , j τU ([ x ]) = j τU ([ y ]) ⊗ j U ([ z ]) . Moreover, the reformulated Mishchenko line bundle [ (cid:101) c ] ∈ KK ( S , S C ( U ) (cid:111) U ) can be defined, and τ S (ind( (cid:1) ∂ )) = [ (cid:101) c U ] ⊗ j τU ([ (cid:101) (cid:1) ∂ ]) . We have defined the KK -element j τU ([ (cid:1) ∂ ]) in [Tak2]. However, the above conjecture is still difficult. Thisis because we must define the images for all Kasparov modules. However, the construction of the “crossedproducts” will definitely give us some nice hints.If all of these conjectures are correct, we will obtain the index formula using the symbol element. τ S (ind( (cid:1) ∂ )) = [ (cid:101) c U ] ⊗ j τU ([ (cid:103) σ Cl (cid:1) ∂ ]) ⊗ j U ([ (cid:102) d U ]) . We hope to solve these conjectures in the immediate future!
Acknowledgements
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