The index of leafwise G-transversally elliptic operators on foliations
aa r X i v : . [ m a t h . K T ] D ec The index of leafwise G -transversallyelliptic operators on foliations Dedicated to Varghese Mathai on the occasion of his sixtieth birthday
Alexandre Baldare and Moulay-Tahar Benameur Institut für Analysis, Welfengarten 1, 30167 Hannover, Germany, [email protected] IMAG, UMR 5149 du CNRS, Université de Montpellier, France, [email protected]
Abstract
We introduce and study the index morphism for leafwise G -transversally elliptic operators on smoothclosed foliated manifolds. We prove the usual axioms of excision, multiplicativity and induction for closedsubgroups. In the case of free actions, we relate our index class with the Connes-Skandalis index class ofthe corresponding leafwise elliptic operator on the quotient foliation. Finally we prove the compatibilityof our index morphism with the Gysin morphism and reduce its computation to the case of tori actions.We also construct a topological candidate for an index theorem using the Kasparov Dirac element foreuclidean G -representations. Keywords:
Transversally elliptic, foliation, Fredholm index, KK -theory, K -homology, group action. Contents G -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The moment map and some standard G -operators . . . . . . . . . . . . . . . . . . . . . . . . 9 G -transversally elliptic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 The index class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 The index map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 The K-theory multiplicity of a representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 MSC2010 classification: Unbounded version of the index class 37B The technical proposition in the non-compact case 42
Introduction
The present paper is devoted to the index theory for leafwise pseudodifferential operators on smooth foli-ations, which are G -invariant and leafwise G -transversally elliptic, for a given leafwise action of a compactLie group G . Since G -invariant elliptic operators are G -transversally elliptic, our results encompass theequivariant Connes-Skandalis index theory for leafwise elliptic operators [26, 9] as well as the classical indextheory for G -transversally elliptic operators [1, 20] with its fibered version obtained in [6, 7]. Due to the lackof full ellipticity, the kernel of such operators is infinite dimensional in general. However, the G -invariance ofthe operator along the orbits together with its ellipticity in the directions transverse to these orbits ensuresthat this kernel contains irreducible representations only with finite multiplicities.In [1], Atiyah showed that the index of a G -transversally elliptic operator is well defined as a centraldistribution which actually lives in the − dim( G ) Sobolev space of G . He also proved a list of importantaxioms and reduced the computation of the distributional index to the case of linear actions of tori. An im-portant observation is that even in the elliptic case, by embedding the operators in the wider G -transversallyelliptic class, one can benefit from their better functoriality properties and it is often easier to deform whenonly G -transversal ellipticity is prescribed. This idea proved fruitful in many situations explained below, inparticular in some recent approaches to the differential [ Q, R ] = 0 problem, see for instance [53]. In the caseof locally free actions of tori, Atiyah succeded proving a signature formula for the singular quotient, whichwas the starting point for the famous Kawasaki work on orbifolds [42]. Later on, N. Berline and M. Vergneproved a delocalized cohomological formula for G -invariant G -transversally elliptic operators, see [18, 19, 20].This formula computes the Atiyah index distribution around a given s ∈ G as an integral of equivariantcharacteristic classes, by using the Kirillov localization principle. It is worthpointing out the close relationof these delocalized index formulae with the Duistermaat-Heckman theorem for tori actions on compactsymplectic manifolds [27] which actually motivated these index formulae, see also [2]. More recently in [41],Kasparov applied the classical ”Bott ↔ Dirac” proof of the Atiyah-Singer theorem for elliptic operators, toinvestigate the index problem now for G -transversally elliptic operators. His approach is new and computesthe K-homology index class [37] in terms of the stable homotopy class of the principal symbol. Kasparovactually considered the more general setting of locally compact Lie groups acting properly and cocompactlyon smooth manifolds and succeeded in this wide generality to compute the index class as a cup product ofthe symbol class by a fundamental Dirac element. This is an important progress and it increased interestin G -transversally elliptic operators. Previous results were also obtained for bounded geometry manifolds in[43, 44]. Here are some well known constructions where G -transversally elliptic operators play a significantpart:• the Mathai-Melrose-Singer fractional index [47, 48] is the evaluation of the distributional index ofa G -transversally elliptic operator on a specific test function localizing at the neutral element, here G = SU( N ), see as well [52];• projective elliptic genera as introduced by Han-Mathai [30] turn out to also be related with a G -transversally elliptic operator, here G = Spin( N );• the basic index problem for riemannian foliations [39], see the recent developements [29, 21], reducesvia Molino’s theory to the computation of the index of a G -transversally elliptic operator. Here G = SO( N );• For locally free G -actions, one recovers a first affordable index problem for transversally elliptic op-erators on foliations [25]. Then our more general setting gives a first example towards the harder2ndex problem for, F -leafwise elliptic and F -transversally elliptic operators, as studied in [33] for abifoliation F ⊂ F .• as explained above, the Kawasaki index formula for orbifolds [42] can be recovered as a corollary ofthe Berline-Vergne cohomological formula for G -transversally elliptic operators, see [60];• the Guillemin-Sternberg principle [49] and more specifically its spin c version as studied in [53] relieson ideas from the index theory of G -transversally elliptic operators. See also the interesting extensionto proper actions by Hochs-Mathai in [34];• etc...The first observation for the foliation case treated in the present paper is that, using a holonomy invarianttransverse measure, leafwise G -transversally elliptic operators do have well defined measured distributionalindices, similar to the Atiyah distribution, and defined using Connes’ machinary [23] and the Murray-vonNeumann dimension theory. Exactly as G -transversally elliptic operators on closed manifolds provide typeI spectral triples, the holonomy invariant measure allows here to see any leafwise G -transversally ellipticoperator as a semi-finite spectral triple on the convolution algebra C ∞ ( G ), in the sense of [12]. Now whensuch holonomy invariant measure does not exist, one is naturally led to the construction of an index theorytaking place in appropriate bivariant K-theory groups. More precisely, given a smooth foliation F of aclosed riemannian manifold M together with a smooth isometric action of a compact Lie group G by leaf-preserving diffeomorphisms, any leafwise pseudodifferential operator P which is G -invariant and leafwise G -transversally elliptic is shown to have an index class living in the bivariant group KK( C ∗ G, C ∗ ( M, F )),where C ∗ ( M, F ) is Connes’ C ∗ -algebra. Therefore, when evaluated at irreducible representations of G , theindex of P embodies a K-multiplicity map m P : b G −→ K( C ∗ ( M, F )) . When F is top dimensional, this is the usual integer valued multiplicity map [1]. In fact and as expected, ifwe denote by F G the space of leafwise tangent vectors which are orthogonal to the orbits of the G -action,then our index construction induces the index morphismInd F : K G ( F G ) −→ KK( C ∗ G, C ∗ ( M, F )) . We prove here a list of axioms which reduce, as in the classical case, the computation of the index class tothe simplest case of linear tori actions.Let us explain more in detail some results. Due to the high complexity of the transverse geometry offoliations, the expected axioms are surprisingly hard to formulate and to prove, without exploiting Kasparov’stheory and the powerful tool of the cup product. First notice that when the G -invariant operator is leafwiseelliptic, its symbol still defines a class in K G ( F G ) which is the restriction of the usual class in K G ( F ),and its index as a G -transversally elliptic operator can then be deduced from the classical elliptic indexclass in KK G ( C , C ∗ ( M, F )) using composition with the trivial representation of G . Moreover, assume that( M, F ) → ( B, F B ) is a principal G -equivariant bundle so that G preserves the leaves of F and induces thefoliation F B downstairs in B , and that P is a G -invariant leafwise G -transversally elliptic operator on ( M, F )which corresponds through its symbol to a leafwise elliptic operator P on ( B, F B ) then we show that theConnes-Skandalis leafwise index of P can be recovered from the index class of P upstairs by evaluation atthe trivial representation, modulo composition with a standard Morita extension morphism. These resultsexplain the compatibility of the leafwise G -transversally elliptic theory with the elliptic one. We prove heremany axioms for our index morphism, such as multiplicativity and excision, see Theorems 3.8 and 3.14. Letus state now the compatibility of our index map with the Gysin morphisms of foliations.3 heorem 0.1. Let ( M ′ , F ′ ) be a smooth G -foliation. Let ι : M ֒ → M ′ be a G -equivariant embedding of aclosed G -manifold M which is transverse to the foliation F ′ and denote by F = ι ∗ F ′ the pull back foliation.Then for any j ∈ Z , the following diagram commutes: K j G ( F G ) ι ! / / Ind F (cid:15) (cid:15) K j G ( F ′ G ) Ind F′ (cid:15) (cid:15) KK j ( C ∗ G, C ∗ ( M, F )) ⊗ C ∗ ( M, F ) ǫ ι / / KK j ( C ∗ G, C ∗ ( M ′ , F ′ )) . The class ǫ ι is a quasi-trivial Morita extension, see Section 5. Another important feature of the indexmorphism is the generalized reciprocity formula for closed subgroups as well as its good behaviour withrespect to the restriction to a maximal torus. Assuming G connected with a maximal torus T , we obtain forinstance the following theorem which allows to reduce the computation of the index morphism to the caseof tori actions. Theorem 0.2.
Denote by r G T : K j G ( F G ) → K j T ( F T ) the map defined in Section 4 using the Dolbeault operatorassociated with the complex G-structure on G/ T . Then for j ∈ Z the following diagram commutes: K j G ( F G ) r G T / / Ind F (cid:15) (cid:15) K j T ( F T ) Ind F (cid:15) (cid:15) KK j ( C ∗ G, C ∗ ( M, F )) KK j ( C ∗ T , C ∗ ( M, F )) [ i ] ⊗ C ∗ T • o o where [ i ] ∈ KK( C ∗ G, C ∗ T ) is the induction class. In the last section, we provide a topological candidate for an index theorem in our setting. Given a G -embedding of M in a euclidean G -representation E , we show that there exists a topological transversal N G for the foliated space ( M × T G ( E ) , F ×
0) together a K-oriented G -map ι : F G ֒ → N G . Hence we use theGysin morphism ι ! : K j G ( F G ) −→ K j G ( N G ) . composed with the Morita extension morphism K G ( N G ) → K G j ( C ∗ ( M × T G ( E ) , F )) to define the R ( G )-morphism K j G ( F G ) −→ K G j ( C ( T G ( E )) ⊗ C ∗ ( M, F )) . Now, the topological index morphism is obtained by composition of this morphism with the Dirac morphismdefined in [41] on E , this latter step is a replacement for the Bott periodicity in the elliptic case. Even inthe case of closed fibrations, this topological construction is new and completes the bivariant approach tothe families Atiyah problem that was investigated in [6].In order to keep this paper in a reasonnable size, we have chosen to restrict ourselves to compact Liegroup actions although the interested reader can easily check that most of the constructions are immediatelyextendable to the setting of proper cocompact actions, as carried out in [41] for the top-dimensional case.Also, the higher distributional approach will be dealt with in a forthcoming paper where we also develop thecohomological viewpoint in the spirit of [19, 20], using Haefliger cohomology and results from [15, 17, 13].We point out that these latter cohomological results have already been carried out by the first author in thecase of closed fibrations in [7], where the family Berline-Vergne formula was obtained.4 cknowledgements.
The authors wish to thank A. Carey, P. Carrillo-Rouse, T. Fack, J. Heitsch, M.Hilsum, P. Hochs, Y. Kordyukov, V. Mathai, H. Oyono-Oyono, V. Nistor, S. Paycha, M. Puschnigg, A.Rennie and G. Skandalis for many helpful discussions. Part of this work was realized during the postdoctoralposition of the first author in the
Institut Elie Cartan de Lorraine at Metz, he is indebted to the membersof the noncommutative geometry team for the warm hospitality. The second author would like to thank hiscolleagues in Montpellier, and more specifically P.-E. Paradan, for several interesting conversations aroundtransversally elliptic operators and their applications. Both authors thank the French National ResearchAgency for the financial support via the ANR-14-CE25-0012-01 (SINGSTAR).
We gather in this first section many classical results about group actions on foliations that will be used inthe sequel. Most of them are well known to experts. G -modules This first paragraph is devoted to a brief review of some standard results. For most of the classical propertiesof Hilbert C ∗ -modules and regular operators between them, we refer the reader to [45] and [57]. Theconstructions given below extend the standard ones, see for instance [6, 7, 28, 31, 38]. Our hermitian scalarproducts will always be linear in the second variable and anti-linear in the first. Let G be a compact groupwith a fixed bi-invariant Haar measure dg . The convolution ∗ -algebra L ( G ) is defined as usual with respectto the rules ( ϕψ )( g ) := ˆ g ∈ G ϕ ( g ) ψ ( g − g ) dg and ϕ ∗ ( g ) := ϕ ( g − ) . We denote by C ∗ G the C ∗ -algebra associated with G , which is the operator-norm closure of the range of L ( G ) in the bounded operators on L ( G ). A classical construction shows that any finite-dimensional unitaryrepresentation of G naturally identifies with a finitely generated projective module on C ∗ G [36]. There is awell defined action of G by automorphisms of the C ∗ -algebra C ∗ G which is induced by the adjoint actionon C ( G ) given by ( Ad g ϕ )( k ) := ϕ ( g − kg ) , for ϕ ∈ C ( G ) and g, k ∈ G. Let now M be a smooth compact manifold and let F be a given smooth foliation of M . We assumethat G acts smoothly on M by leaf-preserving diffeomorphisms, so any element g ∈ G preserves each leafof ( M, F ). We denote by F the subbundle of T M composed of all the vectors tangent to the leaves of F ,this is the tangent bundle of our foliation and its dual bundle is the cotangent bundle of the foliation and isdenoted as usual by F ∗ . We fix a G -invariant riemannian metric on M so that G acts by isometries of M ,and so that we can identify F ∗ with a G -subbundle of T ∗ M when needed. We denote by G the holonomygroupoid that will be confused with the manifold of its arrows. We assume for simplicity that G is Hausdorffso that M = G (0) can be identified with a closed subspace (and a submanifold) of G . We denote as usual by r and s respectively the range and source maps of G and by G x := s − ( x ) and G x := r − ( x ). The compactgroup G acts obviously on G by groupoid diffeomorphisms, hence r and s are G -equivariant submersions.The G -invariant riemannian metric induces a G -invariant riemannian metric on the leaves, which in turninduces a G -invariant leafwise Lebesgue measure. This allows to define our G -invariant Haar system ν on G . More precisely, on each holonomy cover s : G x := r − ( x ) → L x of the leaf L x through x ∈ M , we havethe well defined ”pull-back” measure ν x , see for instance [23]. The family ν • := ( ν x ) x ∈ M is then easilyseen to be a (continuous and even smooth) Haar system for G in the sense of [54]. Similarly we may definethe measure ν x on the holonomy cover r : G x → L x but this latter can also be seen as the image of ν • under the diffeomorphism γ γ − . The G -invariance of the Haar system means that g ∗ ν x = ν gx for any5 g, x ) ∈ G × M or said differently, that for any f ∈ C c ( G ) one has ˆ G gx f ( γ ) dν gx ( γ ) = ˆ G x f ( gγ ) dν x ( γ ) . The space C c ( G ) of compactly supported continuous functions on G is endowed with the usual structure ofan involutive convolution algebra for the rules( f f )( γ ) := ˆ γ ∈G r ( γ ) f ( γ ) f ( γ − γ ) dν r ( γ ) ( γ ) and f ∗ ( γ ) := f ( γ − ) . Moreover, for any given x ∈ M , we have a ∗ -representation λ x : C c ( G ) → L ( L ( G x )) given by λ x ( f )( ξ )( γ ) := ˆ γ ∈G x f ( γγ − ) ξ ( γ ) dν x ( γ ) . The completion of C c ( G ) in the direct sum representation ⊕ x ∈ M λ x is then a well defined C ∗ -algebra calledthe Connes algebra of the foliation ( M, F ) and denoted C ∗ ( M, F ), see [23] for more details.Let π : E = E + ⊕ E − → M be a Z -graded hermitian vector bundle on M which is assumed tobe G -equivariant with a G -invariant hermitian structure. Then, there is a classical G -equivariant Hilbert C ∗ ( M, F )-module E associated with E , which is composed of sections of the G -equivariant continuous fieldof Hilbert spaces ( L ( G x , r ∗ E )) x ∈ M and that we now recall for the sake of completeness.Setting for η ∈ C c ( G , r ∗ E ) and f ∈ C c ( G ) η · f ( γ ) = ˆ G r ( γ ) η ( γ ) f ( γ − γ ) dν r ( γ ) ( γ ) , we get a right C c ( G )-module structure on C c ( G , r ∗ E ). The prehilbertian structure of this module is obtainedby using the C c ( G )-valued scalar product given by h η ′ , η i ( γ ) = ˆ G r ( γ ) h η ′ ( γ ) , η ( γ γ ) i E r ( γ dν r ( γ ) ( γ ) , for η, η ′ ∈ C c ( G , r ∗ E ) and γ ∈ G . That h η, η i is a non-negative element of the C ∗ -algebra is standard. Moreover, all the axioms for a pre-hilbertian module are satisfied. The completion of C c ( G , r ∗ E ) for k • k E := kh• , •ik / C ∗ ( M, F ) is our Hilbert C ∗ ( M, F )-module E .Our goal now is to use the G -action on ( M, F ) and E in order to define a representation π of the C ∗ -algebra C ∗ G in adjointable operators of the Hilbert module E . An easy inspection of the case of simplefoliations shows that an extra compatibility condition between the action of G and the foliation F needs tobe imposed. Roughly speaking, we need an action of G which preserves each Hilbert space L ( G x , r ∗ E ) sothat the average representation of C ∗ G would make sense. We proceed now to explain this action which istaken from [16]. Recall the action groupoid M ⋊ G , which as a space of arrows is just M × G , with the rules s ( x, g ) = x, r ( x, g ) = gx and ( gx, k ) ◦ ( x, g ) = ( x, kg ) . Definition 1.1. [16]• A leafwise diffeomorphism f is called a holonomy diffeomorphism if there exists a smooth map θ f : M → G so that for any x ∈ M , s ( θ f ( x )) = x and r ( θ f ( x )) = f ( x ) , and the holonomy along θ f ( x ) coincides with the induced action of f on transversals.6 The action of the compact group G is a holonomy action if any g ∈ G is a holonomy diffeomorphismand the induced groupoid morphism θ : M ⋊ G −→ G , given by θ ( x, g ) = θ g ( x ) is smooth.Notice that when θ f exists, it is unique. From the very definition, if f is a holonomy diffeomorphism,then for any γ ∈ G , one has θ f ( r ( γ )) γ θ f ( s ( γ )) − = f ( γ ) . The holonomy diffeomorphisms form a subgroup of the group of leaf-preserving diffeomorphisms of M . Iffor instance f is a holonomy diffeomorphism, then so is f − and we have θ f − ( x ) = θ f ( f − ( x )) − . Whenthe G -action is a holonomy action, we have gθ k ( x ) = θ gkg − ( gx ) and θ k ( gx ) θ g ( x ) = θ kg ( x ) . The following lemma is proved in [16]:
Lemma 1.1. [16] The leaf-preserving diffeomorphism f is a holonomy diffeomorphism in the following cases:1. When the holonomy is trivial, and the foliation is tame in the sense of [22]. See also [16].2. When the foliation is Riemannian.3. When f belongs to a connected (Lie) group which acts on V by leaf-preserving diffeomorphisms. Moregenerally, if f belongs to the path connected component of a holonomy diffeomorphism g in the groupof leaf-preserving diffeomorphisms.4. When restricted to the saturation sat ( V f ) of the fixed point submanifold V f , that is the union of theleaves that intersect V f . As an obvious corollary for instance, we see that when the compact Lie group G is connected, then itsleaf-preserving action is automatically a holonomy action. As for the non-foliated case, we are mainly inter-ested, especially for the cohomological index formula, in the case of the action of a compact connected Liegroup G . However, this assumption is not needed yet and only the holonomy assumption will be necessary. From now on, we shall assume that the leafwise G -action is a holonomy action. Using the groupoid morphism θ , we get for any x ∈ M an action of G on the manifold G x by settingΦ : G × G x −→ G x given by Φ( g, γ ) := θ g ( r ( γ )) γ. Indeed, one hasΦ( g, Φ( k, γ )) = θ g ( kr ( γ ))Φ( k, γ ) = θ g ( kr ( γ )) θ k ( r ( γ )) γ = θ gk ( r ( γ )) γ = Φ( gk, γ ) . The holonomy covering map r : G x → L x is then G -equivariant, so that Φ can be understood as an r -lift ofthe original G action which fixes the source map s . Using the G -invariance of the leafwise Lebesgue-classmeasure, it is then easy to check using the definition of the measure ν x that this latter is Φ-invariant, i.e. ˆ G x f (Φ( g, γ )) dν x ( γ ) = ˆ G x f ( γ ) dν x ( γ ) . Indeed, this follows from the relation Φ( g, γ ) = ( gγ ) θ g ( s ( γ )). We can now define our unitary G -action U x on the Hilbert space L ( G x , r ∗ E ) by setting( U x,g η )( γ ) := g η (Φ( g − , γ )) , η ∈ C c ( G x , r ∗ E ) , g ∈ G, and γ ∈ G x . The family U = ( U x ) x ∈ M actually represents the group G in the unitary adjointable operators on the Hilbertmodule E . More precisely: 7 emma 1.2. For the trivial action of G on C ∗ ( M, F ) , the Hilbert module E is a G -Hilbert module. Indeed,for any η, η ′ ∈ C c ( G , r ∗ E ) and g ∈ G we have: h U g η, η ′ i = h η, U g − η ′ i , so in particular, the operator U g extends to an adjointable (unitary) operator on the Hilbert module E .Proof. For η ∈ E , f ∈ C ∗ ( M, F ) and g ∈ G , the relation ( U g η ) · f = U g ( η · f ) can be easily verified bydirect computation, however this will be automatically satisfied since the operator U g is adjointable. Moreprecisely, we have h U g η, η ′ i ( γ ) = ˆ G r ( γ ) D gη (cid:16) θ g − ( r ( γ ′ )) γ ′ (cid:17) , η ′ ( γ ′ γ ) E E r ( γ ′ ) dν r ( γ ) ( γ ′ ) . Setting γ = θ g − ( r ( γ ′ )) γ ′ and using the G -invariance of the metric on E as well as the Φ-invariance, weget: h U g η, η ′ i ( γ ) = ˆ G r ( γ ) (cid:10) η ( γ ) , g − η ′ ( θ g ( r ( γ )) γ γ ) (cid:11) E r ( γ dν r ( γ ) ( γ )= h η, U g − η ′ i ( γ ) . Corollary 1.3.
The G -action on the foliation C ∗ -algebra C ∗ ( M, F ) is inner, i.e. it is implemented byunitary multipliers and we have λ ( gf ) = U g ◦ λ ( f ) ◦ U g − for any f ∈ C c ( G ) .Proof. When E is the trivial line bundle, the previous lemma shows that G acts by unitary multipliers( U g ) g ∈ G of the foliation C ∗ -algebra C ∗ ( M, F ). We now compute (cid:0) U g ◦ λ ( f ) ◦ U g − (cid:1) ( ξ )( γ ) = g (cid:2) λ ( f )( U g − ξ ) (cid:3) (cid:16) θ g − ( r ( γ )) γ (cid:17) = ˆ G s ( γ ) f (cid:16) θ g − ( r ( γ )) γγ − (cid:17) ξ ( θ g ( r ( γ ) γ ) dν s ( γ ) ( γ )= ˆ G s ( γ ) f (cid:16) θ g − ( r ( γ )) γγ − θ g ( g − r ( γ )) (cid:17) ξ ( γ ) dν s ( γ ) ( γ ) , where the last equality is obtained by using again the Φ-invariance of the Haar system. The result followssince θ g − ( r ( γ )) γγ − θ g ( g − r ( γ )) = g − γθ g − ( s ( γ )) θ g ( g − s ( γ )) g − γ − = g − ( γγ − ) . Proposition 1.4.
We set for η ∈ C c ( G , r ∗ E ) and ϕ ∈ C ( G ) : π ( ϕ )( η ) := ˆ G ϕ ( g ) ( U g η ) dg. (1) Then π extends to an involutive representation π of C ∗ G in the Hilbert module E . More precisely, π is acontinuous ∗ -homomorphism into the C ∗ -algebra of adjointable operators.Proof. Since U g is adjointable with U ∗ g = U g − , we obtain that π ( ϕ ) is also adjointable with π ( ϕ ∗ ) = π ( ϕ ) ∗ .The relation π ( ϕ ⋆ ψ ) = π ( ϕ ) ◦ π ( ψ ) is also immediately verified. It follows that π is a ∗ -homomorphismwhich satisfies, by its very definition, the estimate k π ( ϕ ) k ≤ k ϕ k L G for ϕ ∈ L G . Hence we get a welldefined continuous ∗ -representation of the C ∗ -algebra C ∗ G .8 emark 1.5. If we endow C ∗ G with the conjugation action Ad of G , then it is easy to check that therepresentation π is G -equivariant, i.e. π ( Ad g ϕ )( η ) = U g ◦ π ( ϕ ) ◦ U g − for ϕ ∈ C ∗ ( G ), η ∈ E and g ∈ G . Definition 1.6. [23] Let E = E + ⊕ E − be a Z -graded vector bundle over M . A (classical) pseudodifferential G -operator P of order m acting from E + to E − is a smooth family ( P x ) x ∈ M , where P x : C ∞ c ( G x , r ∗ E + ) −→ C ∞ c ( G x , r ∗ E − ) , is a (uniformly supported and classical) pseudodifferential operator of order m , with the right G -invarianceproperty: P r ( γ ) R γ = R γ P s ( γ ) . The uniform support is assumed here for simplicity and proper support would suffice in order to preservethe space of compactly supported sections, see [51]. We shall denote by Ψ m ( M, F ; E + , E − ) the space of(classical) pseudodifferential G -operators on M of order m . So such pseudodifferential G operators correspondto longitudinal pseudodifferential operators on the graph manifold G with respect to the foliation r ∗ F , butwhich are G -invariant so that they induce operators downstairs acting over the leaves of ( M, F ). We shallalso sometimes call the elements of Ψ m ( M, F ; E + , E − ) longitudinal or leafwise pseudodifferential operatorson ( M, F ) since no confusion can occur.The principal symbol of such a longitudinal operator P of order m is defined as usual by the formula: σ m ( P )( x, ξ ) = σ pr ( P x )( x, ξ ) , for ( x, ξ ) ∈ T ∗ x G x ≃ F ∗ x , where σ pr ( P x ) is the principal symbol of the m -th order classical pseudodifferential operator P x acting onthe manifold G x . When E is a G -equivariant Z -graded hermitian vector bundle, the longitudinal operator P will be G -invariant if for any g ∈ G , the family P commutes with the family of unitaries U g, • , i.e. U g,x ◦ P x = P x ◦ U g,x , ∀ g, x. In this case, the principal symbol of P is G -invariant with the action on the leafwise cotangent bundle F ∗ obtained as usual by codifferentiating the original G -action.A zero-th order longitudinal pseudodifferential operator P : C ∞ c ( G , r ∗ E + ) → C ∞ ( G , r ∗ E − ) extends intoan adjointable operator, still denoted P , between the Hilbert modules E + and E − corresponding to the vectorbundles E + and E − respectively [26, 23]. The formal adjoint of P defined over each G x , with respect tothe hermitian structures and the Haar system, is then again a zero-th order longitudinal pseudodifferentialoperator acting from E − to E + . Moreover, its extension to an adjointable operator from E − to E + isjust the adjoint of P with respect to the Hilbert module structures. So, if we denote by P the operator P := (cid:18) P ∗ P (cid:19) , then P is an adjointable operator on E = E + ⊕ E − which is by construction odd for the Z -grading. Lemma 1.7.
With the previous notations, if we assume in addition that P is G -invariant, then for any ϕ ∈ C ∗ G , we have [ π ( ϕ ) , P ] = 0 .Proof. As can be checked easily, the operator P is G -invariant in the usual sense if and only if P commuteswith the unitary U of E corresponding to the family of unitaries ( U g,x ) ( g,x ) ∈ G × M . Now, let ϕ ∈ L ( G ), thenby definition of π ( ϕ ) we deduce that π ( ϕ ) ◦ P = P ◦ π ( ϕ ). Therefore this commutation relation also holdsfor any ϕ ∈ C ∗ G by continuity. G -operators Assume now that G is a compact Lie group with Lie algebra g , and that the action of G on M preserves theleaves and is through holonomy diffeomorphisms as explained in the previous section. This is for instancethe case for any compact connected Lie group. We start by extending some results from [41, Section 6] to9ur foliation setting, and for the convenience of the reader we shall use Kasparov’s notations from there. For x ∈ M , we hence denote by f x : G → M the map given by f x ( g ) = g x and by f ′ x : g → T x M its tangentmap at the neutral element of G . The dual map of f ′ x is denoted by f ′ ∗ x : T ∗ x M → g ∗ . So, any X ∈ g definesas usual the vector field X ∗ given by X ∗ x := f ′ x ( X ) which, under our assumptions, is tangent to the leaves,i.e. X ∗ x ∈ F x for any x ∈ M . Notice also that g · f ′ x ( X ) = f ′ g x (Ad( g ) X ), for g ∈ G , x ∈ M and X ∈ g [41].Let g M := M × g be the G -equivariant trivial bundle of Lie algebras on M , associated to g for the action g · ( x, v ) = ( g x, Ad( g ) v ). The map f ′ : g M → T M defined by f ′ ( x, v ) = f ′ x ( v ) is a G -equivariant vectorbundle morphism. We endow g M with a G -invariant metric and we denote by k · k x the associated family ofEuclidean norms. Up to normalization, we can always assume that ∀ v ∈ g , k f ′ x ( v ) k ≤ k v k x . Here k f ′ x ( v ) k isthe norm given by the riemannian metric at x . We thus assume from now on that k f ′ x k ≤ ∀ x ∈ M . Thesemetrics on g M and T M are also used to identify g M with g ∗ M and T M with T ∗ M . Then we can define themap φ : T ∗ M → T ∗ M by setting φ x = f ′ x f ′ ∗ x . Again according to Kasparov’s notations [41], we introducethe quadratic form q = ( q x ) x ∈ M on the fibers of T ∗ M by setting: q x ( ξ ) = |h f ′ x f ′ ∗ x ( ξ ) , ξ i| = k f ′ ∗ x ( ξ ) k x , ∀ ( x, ξ ) ∈ T ∗ M. (2)If ξ ∈ T ∗ x M , then it is easy to see that ξ is orthogonal to the G -orbit of m if and only if q x ( ξ ) = 0. Noticealso that we have q x ( ξ ) ≤ k ξ k .As in the seminal book [1], we introduce a second order G -invariant longitudinal differential operator∆ G whose symbol coincides with q . This is achieved for instance by using an orthonormal basis of g for abi-invariant metric on the compact Lie group G and by considering the first order differential operators whichare the Lie derivatives of the G -action, see again [1, page 12]. Recall that if X ∈ g and η ∈ C ∞ ( G , r ∗ E ),then the Lie derivative L ( X )( η ) is defined as L ( X )( η )( γ ) := ddt | t =0 ( e tX · η )( γ ) = ddt | t =0 e tX (cid:0) η (Φ( e − tX , γ ) (cid:1) . So, L ( X ) preserves each space C ∞ c ( G x , r ∗ E ) and the corresponding family of first order differential operatorsis clearly right G -invariant. Note indeed that Φ( g, γ ′ γ ) = Φ( g, γ ′ ) γ. Therefore, R γ ( L ( X ) s ( γ ) η )( γ ′ ) = L ( X ) s ( γ ) η ( γ ′ γ )= ddt | t =0 e tX ( η (Φ( e − tX , γ ′ γ ))= ddt | t =0 e tX ( η (Φ( e − tX , γ ′ ) γ )= ddt | t =0 e tX (cid:0) R γ ( η )(Φ( e − tX , γ ′ )) (cid:1) = L ( X ) r ( γ ) R γ ( η )( γ ′ ) . Then, for any orthonormal basis { V k } of g with dual basis { v k } , we define a longitudinal differential operator d G by considering the right G -invariant family d G = ( d G,x : C ∞ c ( G x , r ∗ E ) −→ C ∞ c ( G x , r ∗ ( E ⊗ g ∗ M ))) x ∈ M of differential operators between E and E ⊗ g ∗ M given by d G,x ( η ) := X k L ( V k ) η ⊗ v k , ∀ η ∈ C ∞ c ( G x , r ∗ E ) . This definition is independent of the choice of the orthonormal basis of g . Remark 1.8.
We may take for ∆ G the operator d ∗ G d G . Indeed, it is easy to see that the symbol of d G at( x, ξ ) ∈ F is given by √− f ′∗ x ( ξ )). So the symbol of d ∗ G is given by −√− f ′∗ x ( ξ )) and hence theprincipal symbol of d ∗ G d G is given by h f ′∗ x ( ξ ) , f ′∗ x ( ξ ) i = q x ( ξ ).10n the same way and working on the manifold G itself with its G -action by left translations, the or-thonormal basis { V k } of g (with dual basis { v k } ) allows to define the exterior differential of the manifold G as follows. For any ϕ ∈ C ∞ ( G ), let ∂ϕ∂V be the derivative along the one-parameter subgroup of G corre-sponding to the vector V ∈ g , then we get the first-order differential operator d acting on smooth functionson G and valued in g ∗ -valued smooth functions on G , by setting dϕ = X k ∂ϕ∂V k ⊗ v k . We may tensor the representation π : C ∗ G → L C ∗ ( M, F ) ( E ) with the identity of the vector space g ∗ and getthe extended map π : C ∗ G ⊗ g ∗ −→ L C ∗ ( M, F ) ( E , E ⊗ g ∗ ) ≃ L C ∗ ( M, F ) ( E ) ⊗ g ∗ . Said differently, we simply set for ψ ∈ L ( G ) and v ∈ g ∗ : π ( ψ ⊗ v ) η = π ( ψ ) η ⊗ v = ˆ G ψ ( g )( U g η ⊗ v ) dg, ∀ η ∈ E . Here again the map π corresponds to a family ( π x ) x ∈ M of maps π x : C ∗ G ⊗ g ∗ −→ L ( L ( G x , r ∗ E )) ⊗ g ∗ . Proposition 1.9. [ ] For ϕ ∈ C ∞ ( G ) and V ∈ g , we have L ( V ) ◦ π ( ϕ ) = π (cid:18) ∂ϕ∂V (cid:19) . In particular, d G ( π ( ϕ ) η ) = π ( dϕ ) η for any η ∈ C ∞ c ( G , r ∗ E ) , or equivalently ( d G,x [ π x ( ϕ )] = π x ( dϕ )) x ∈ M .Proof. We only need to check the first relation with the Lie derivatives. But we have for V ∈ g , ϕ ∈ C ∞ ( G )and η ∈ C ∞ c ( G , r ∗ E ): L ( V ) π ( ϕ ) η ( γ ) = ddt | t =0 e tV (cid:0) π ( ϕ ) η (Φ( e − tV , γ )) (cid:1) = ddt | t =0 ˆ G ϕ ( g )( e tV g ) (cid:0) η (Φ( g − , Φ( e − tV , γ ))) (cid:1) dg = ddt | t =0 ˆ G ϕ ( g )( e tV g ) (cid:0) η (Φ( g − e − tV , γ )) (cid:1) dg = ˆ G ddt | t =0 ϕ ( e − tV h ) h (cid:0) η (Φ( h − , γ )) (cid:1) dh. In the second to third line we have used the relation Φ( g − , Φ( e − tV , γ )) = Φ( g − e − tV , γ ), and in the lastequality, we have substituted e tV k g = h and used the G -invariance of the Haar measure on G . Therefore, weget L ( V ) π ( ϕ ) η ( γ ) = ˆ G ∂ϕ∂V ( h ) U h η ( γ ) dh = (cid:20) π (cid:18) ∂ϕ∂V (cid:19) η (cid:21) ( γ ) . Recall that we are given for any g ∈ G and any x ∈ M a holonomy class θ g ( x ) ∈ G gxx with the naturalproperties recalled in the previous section. So, for any X ∈ g , we have θ e − tX ( x ) ∈ G e − tX xx t θ e − tX ( x ) is a smooth path in G x which starts at x viewed in G x . Therefore, we defined a vector˜ X x ∈ T x G x and hence in F x by setting ˜ X ( x ) := ddt | t =0 θ e − tX ( x ) . An easy inspection in a local chart allows to see that the vector field ˜ X coincides with the vector field X ∗ M . In this section we define the index class of a G -invariant leafwise G -transversally elliptic operator and wealso introduce the K-multiplicity of any unitary irreductible representation in the index class. G -transversally elliptic operators Recall the bundle map f ′ : M × g → F and its fiberwise transpose f ′∗ . Definition 2.1.
Denote by F ∗ G ⊂ F ∗ the kernel of f ′∗ . So, F ∗ G is the subspace of F ∗ composed of leafwisecovectors which are transverse to the G -orbits, or equivalently: F ∗ G := { ( x, ξ ) ∈ F ∗ such that q x ( ξ ) = 0 } . This definition extends the classical one from [1] where T ∗ G M ⊂ T ∗ M is defined similarly. We shall useour Riemannian metric to identify T M with T ∗ M and also F ∗ with F . With these identifications, T ∗ G M canbe identified with the subspace T G M of T ( M ) which is the orthogonal of the G -orbits, and F ∗ G can also beidentified with the subspace of F which is the leafwise orthogonal to the G -orbits, so F G := F ∩ T G M .Recall that any zero-th order longitudinal pseudodifferential operator P gives rise to the self-adjointoperator that we have denoted by P and which is defined following the usual convention, see [41]. Moreprecisely, in the even case, say when P acts from the sections of the hermitian vector bundle E + to thesections of the hermitian vector bundle E − , we consider the Z -graded Hilbert module E = E + ⊕E − associatedwith the Z -grading E = E + ⊕ E − , and the operator P is odd for the grading and given by P = (cid:18) P ∗ P (cid:19) .In the ungraded case, E + = E − = E and P = P is assumed to be a selfadjoint operator, say the boundedextension of a leafwise (formally) selfadjoint operator P : C ∞ c ( G , r ∗ E ) → C ∞ c ( G , r ∗ E ). We shall refer to thisconvention as convention (K). The notion of G -invariant G -transversally elliptic operator was introducedand studied in [1]. In our case of foliations, we need to assume that the principal symbol of such G -invariantlongitudinal operator be invertible away from the “zero section” of F G . So by homogeneity, this means thatwe assume that the restriction of the principal symbol of our operator to the subspace S ∗ G F of covectors in F ∗ G of length 1, is pointwise invertible. Following classical normalizations (see [26, 41]), we introduce thefollowing simpler definition: Definition 2.2.
A zero-th order G -invariant longitudinal pseudodifferential operator P acting from the sec-tions of E + to the sections E − is a longitudinal G -transversally elliptic operator or a leafwise G -transversallyelliptic operator, if the symbol of the associated self-adjoint operator P = (cid:18) P ∗ P (cid:19) on E = E + ⊕ E − satisfies the following condition σ ( P ) = id in restriction to S ∗ G F . (3)The principal symbol of such leafwise G -transversally elliptic operator P then represents a class in the G -equivariant Kasparov bivariant group KK G ( C , C ( F G )) and is represented by the Kasparov even cycle( C ( π ∗ E ) , σ ( P )), where C ( π ∗ E ) is the space of continuous sections of the continuous bundle π ∗ E → F G G ( C , C ( F G )) ≃ K G ( F G ), any G -invariant leafwise G -transversally elliptic operator P has a symbol class[ σ ( P )] ∈ K G ( F G ) . In the ungraded case, Definition 2.2 applies to the self-adjoint operator P = P and we get an odd Kasparovcycle and hence a symbol class [ σ ( P )] ∈ KK ( C , C ( F G )) ≃ K ( F G ) . We end this subsection with the following lemma which will be needed in the sequel. Item (2) was usedin [41] to define the notion of G -transversally elliptic symbols for non classical symbols, and in the moregeneral setting of proper actions. Lemma 2.3.
Let ( W, F W ) be a smooth (not necessary compact) foliated manifold, and denote as usual by F W the longitudinal (co)tangent bundle of ( W, F W ) . Let A be an order longitudinal operator on W . Thenthe following are equivalent(1) σ ( A )( x, ξ ) = 0 , ∀ ( x, ξ ) ∈ F WG r T G W ∩ F W r .(2) ∀ ε > , ∀ compact K ⊂ W, ∃ c > , k σ ( A )( x, ξ ) k ≤ c q x ( ξ ) k ξ k + ε , ∀ x ∈ K and ∀ ξ ∈ F Wx r .Proof. (2) implies (1) because q x ( ξ ) = 0 ⇐⇒ ( x, ξ ) ∈ F WG . Let us show that (1) implies (2). Since σ ( A )( x, • )and q x are homogeneous, we only need to prove the relation (2) on the sphere bundle S ∗ F W of F W . Let ǫ > A ǫ,K the subspace of S ∗ F W of those ( x, ξ ) such that k σ ( A )( x, ξ ) k ≥ ǫ and x ∈ K . Then by (1), the continuous positive function q is bounded below on A ǫ,K , by compacity of A ǫ,K .This shows that k σ ( A )( • ) k q is bounded on A ǫ,K . Since k σ ( A )( x, ξ ) k < ǫ on S ∗ F W r A ǫ,K , the proof of (2) iscomplete. We fix a G -invariant selfadjoint longitudinal zero-th order pseudodifferential operator Q acting on thesections of the vector bundle E and with principal symbol given for non-zero ξ by σ ( Q )( x, ξ ) = q x ( ξ ) | ξ | × id E .This can be achieved by using for instance the usual quantization map, see [26]. Proposition 2.4.
Let A be a G -invariant selfadjoint leafwise pseudodifferential operators of order actingon the sections of the hermitian bundle E over M . Assume that the principal symbol σ ( A ) vanishes inrestriction to the subspace F G r of G -transverse leafwise covectors. Then there exist two G -invariantselfadjoint compact operators R and R on the Hilbert module E such that: − ( c Q + ε + R ) ≤ A ≤ c Q + ε + R as self-adjoint operators on E . Proof.
It is a classical result for a single operator even on non compact manifolds but with the proper supportthat such operators R and R exist as smoothing properly supported operators, see [35, 56, 41]. Since weshall only need the condition of compactness of the operators and since our ambiant manifold is compacthere, the proof is immediate. Indeed, by Lemma 2.3, for any ǫ >
0, there exists c > cQ + ǫ id E ± A are non-negative as elements of the C ∗ -algebra C ( S ∗ F , END( E ))of continuous sections of the algebra bundle END( E ) = π ∗ End( E ) over the cosphere bundle S ∗ F of thelongitudinal bundle F . Now, a classical result of Connes [23, 26] gives us a C ∗ -algebra short exact sequenceobtained out of the closure of the zero-th-order pseudodifferential operators along the leaves of F :0 → K C ∗ ( M, F ) ( E ) ֒ → Ψ ( M, F ; E ) σ −→ C ( S ∗ F , END( E )) → , where Ψ ( M, F ; E ) is the closure in L C ∗ ( M, F ) ( E ) of the ∗ -algebra of zero-th order pseudodifferential operatorsalong the leaves (acting on the sections of E ) and σ is the principal symbol map. Hence, we deduce that theoperators cQ + ǫ id E ± A are non-negative up to compact operators and hence the conclusion.13 emark 2.5. Proposition 2.4 can be stated for a non compact foliated manifold and for operators withcompactly supported symbols in S ∗ F . In this case one needs to work with locally compact operators. SeeProposition B.1 in Appendix B where the corresponding generalization is stated and proved using the exactsequence of locally compact pseudodifferential operators as obtained in [26].We are now in position to state the following important result. Theorem 2.6.
The triple (cid:0) E , π, P (cid:1) = (cid:20) E + ⊕ E − , π, (cid:18) P ∗ P (cid:19)(cid:21) is an even Kasparov cycle for the C ∗ -algebras C ∗ G and C ∗ ( M, F ) . In the ungraded case, we similarly have an odd Kasparov cycle which isrepresented by ( E , π, P ) .Proof. By Lemma 1.7, we know that [ π ( ϕ ) , P ] = 0 for any ϕ ∈ C ∗ G . Moreover, P is selfadjoint and odd forthe Z -grading while π obviously respects the Z -grading. It thus remains to check that (id − P ) ◦ π ( ϕ ) ∈K C ∗ (M , F ) ( E ), where K C ∗ ( M, F ) ( E ) stands as usual for the C ∗ -algebra of compact operators in the Hilbertmodule E . As P is a leafwise G -transversally elliptic operator, the principal symbol of id − P whichcoincides with id − σ (P) , satisfies the assumption of Proposition 2.4. Therefore ∀ ε >
0, there exist c , c > R and R on the Hilbert module (in fact leafwise smoothing operators) suchthat − ( c Q + ε + R ) ≤ − P ≤ c Q + ε + R . Let us take for ∆ G the operator d ∗ G d G , see Remark 1.8. Denote also by ∆ a G -invariant longitudinalsecond order differential operator with principal symbol k ξ k × id E for ( x, ξ ) ∈ F x . Modulo longitudinalpseudodifferential operators of negative order, the longitudinal pseudodifferential operator Q then coincideswith the operator d ∗ G (1 + ∆) − d G .By Proposition 1.9, we know that d G ◦ π ( ϕ ) = π ( dϕ ) and hence this latter is a bounded operator on E .Moreover, d ∗ G (1 + ∆) − has negative order, so by Corollary 3 of [46], it is a compact operator of the Hilbertmodule E . It follow that d ∗ G (id + ∆) − d G π ( ϕ ) is compact as well. Again since longitudinal pseudodifferentialoperators of negative order extend to compact operators on the Hilbert module E , we deduce that Qπ ( ϕ ) iscompact. In order to show that the operator (id − P ) ◦ π ( ϕ ) is compact, we first notice that for ψ = ϕ ∗ ϕ ,we have: − π ( ϕ ) ∗ ◦ ( c Q + ε + R ) ◦ π ( ϕ ) ≤ π ( ϕ ) ∗ ◦ (id − P ) ◦ π ( ϕ ) ≤ π ( ϕ ) ∗ ◦ ( c Q + ε + R ) ◦ π ( ϕ ) , i.e. − ( c Q + ε + R ) ◦ π ( ψ ) ≤ (id − P ) ◦ π ( ψ ) ≤ ( c Q + ε + R ) ◦ π ( ψ ) , since all the operators are G -invariant. Therefore, projecting in the Calkin algebra and letting ǫ →
0, wededuce that (id − P ) ◦ π ( ψ ) is compact for any non-negative ψ ∈ C ∗ G . Now, using the spectral theorem,we may write any ϕ ∈ C ∗ G as a linear combination of non-negative elements and conclude. Definition 2.7.
The index class Ind F ( P ) of the G -invariant leafwise G -transversally elliptic operator P isdefined as: Ind F ( P ) := [ E , π, P ] ∈ KK i ( C ∗ G, C ∗ ( M, F )) , with i ∈ Z determined according to Convention (K).We end this subsection by a short explanation for this choice of the non-equivariant index class Ind F ( P ),in opposition to the apparently more interesting equivariant one in KK G ( C ∗ G, C ∗ ( M, F )). This latterobviously exists and covers Ind F ( P ) under the G -forgetful map. Indeed, the triple ( E , π, P ) is G -equivariant,the representation π of C ∗ ( G ) in the adjointable operators of E is for instance G -equivariant when C ∗ G isendowed with the inner adjoint action. Now, recall that for inner actions the equivariant KK groups areisomorphic to the ones corresponding to trivial G -actions. Notice in addition that π allows to recover theaction by the unitaries U g , hence there is no essential lost in concentrating on the non-equivariant classInd F ( P ) ∈ KK( C ∗ G, C ∗ ( M, F )). We are grateful to the referee for pointing this out to us.14 .3 The index map Proposition 2.8.
The index class
Ind F ( P ) only depends on the K -theory class [ σ ( P )] of the principalsymbol σ ( P ) , and this induces for i = 0 , , a group homomorphism: Ind F : K i G ( F G ) −→ KK i ( C ∗ G, C ∗ ( M, F )) . More precisely, the map [ σ ] Ind F ( P ( σ )) is well defined by using any quantization P of σ .Proof. This is classical and we follow [3] and [1]. We only give the graded case, the ungraded being similar andeasier. Let C ( F G ) be the semigroup of 0-homogeneous homotopy classes of transversally elliptic symbols oforder 0 and let C φ ( F G ) ⊂ C ( F G ) be the classes of such symbols whose restriction to the sphere bundle of F G , isinduced by a bundle isomorphism over M . By a standard argument, we know that K G ( F G ) := C ( F G ) / C φ ( F G ).Let now σ t be a homotopy of leafwise 0-th order G -transversally elliptic symbols, then the quantizationof this homotopy gives an operator homotopy and hence by the very definition of the Kasparov groupKK( C ∗ G, C ∗ ( M, F )), the index classes of σ and σ coincide in KK( C ∗ G, C ∗ ( M, F )). On the other hand,given two 0-th order G -invariant leafwise G -transversally elliptic operators P : C ∞ , ( G , r ∗ E ) → C ∞ , ( G , r ∗ E )and P ′ : C ∞ , ( G , r ∗ E ′ ) → C ∞ , ( G , r ∗ E ′ ), we obviously haveInd F ( P ⊕ P ′ ) = [( E ⊕ E ′ , π E ⊕ π E ′ , P ⊕ P ′ )] = [( E , π E , P )] + [( E ′ , π E ′ , P ′ )] = Ind F ( P ) + Ind F ( P ′ ) . Finally, it is clear that any zero-th order G -invariant longitudinal pseudodifferential operator whose symbolis induced by a bundle isomorphism over M has Ind F ( P ) = 0, for more details see for instance [4].We shall also use in the sequel the classical isomorphism K G ( F G ) ≃ k C ( F G ) / k C φ ( F G ) for any k ∈ Z . Here k C ( F G ) and k C φ ( F G ) are the same semi-groups introduced in the previous proof but replacing 0-homogeneousby k -homogeneous. See for instance [3]. Remark 2.9. • Since the commutator [
P, f ] is compact for f ∈ C ( M ), the triple ( E , π, P ) extendsto an element in KK ( C ( M ) ⋊ G, C ∗ ( M, F )). Therefore, the index morphism Ind F factors throughInd M, F : K i G ( F G ) → KK i ( C ( M ) ⋊ G, C ∗ ( M, F )).• When the G -action is locally free so as to generate a smooth regular subfoliation F ′ of the foliation F ,the index morphism Ind M, F can be recast as valued in KK( C ∗ ( M, F ′ ) , C ∗ ( M, F )) and we recover inthis case the index construction given for more general double foliations in [33].We may state the similar proposition when an extra compact group acts on the whole data. Moreprecisely, we have: Proposition 2.10.
Assume that the compact group G acts as before by holonomy diffeomorphisms, and thatan extra compact group G acts also on M by F -preserving isometries (not necessarily preserving the leaves)such that this G -action commutes with the action of G , then the previous construction yields, for G × G -invariant G -transversally elliptic operators along the leaves of ( M, F ) , to a well defined G -equivariantindex map Ind F ,G : K i G × G ( F G ) −→ KK i G ( C ∗ G , C ∗ ( M, F )) , i ∈ Z . The G -equivariant index class of the G × G -invariant leafwise pseudodifferential operator P on ( M, F )which is G -transversally elliptic is represented again by the cycle ( E , π, P ) which is now in addition G -equivariant. Indeed, the Hilbert module E is automatically endowed with the extra G -action so that E is a G -equivariant Hilbert module over the G -algebra C ∗ ( M, F ). Here of course the actions are the usual onesinduced from the action on the holonomy groupoid G and on the bundle and no need to assume that theaction of G preserves the leaves. When the group G is for instance the trivial group, then we recover the G -equivariant index class for G -invariant leafwise elliptic operators as considered for instance in [10].15 emark 2.11. If we assume in the previous proposition that the extra group G also acts by holonomydiffeomorphisms on ( M, F ), then exactly as for the G -action, we can arrange the G -action on E so thatit becomes a G -equivariant Hilbert module over the G -trivial C ∗ -algebra C ∗ ( M, F ). Hence in this case,there are again two ways to define the equivariant index class but they are isomorphic. Remark 2.12.
We shall see in Subsection 3.2 that the index morphism is also well defined when M is notnecessarily compact as a morphism on the (compactly supported) equivariant K -theory of the space F G . K -theory multiplicity of a representation For any irreducible unitary representation of G , we now proceed to define a class in K i ( C ∗ ( M, F )) playingthe role of its multiplicity in the index class Ind F ( P ), and which coincides with the usual multiplicity asdefined by Atiyah in [1] in the case of a single operator, corresponding for us to the maximal foliation witha single leaf.So let ρ : G → U ( X ) be an (irreductible) unitary representation of G in the finite dimensional space X . For simplicity, we shall refer to such representation by X when no confusion can occur. Recall that thespace of isomorphism classes of irreducible unitary representations of G is the discrete dual b G of G , hencewe have fixed X ∈ b G .Recall that the representation X corresponds to a projection which represents a K-theory class in K ( C ∗ G ) (see [36]), equivalently X corresponds to a class [ X ] ∈ KK( C , C ∗ G ) given by [( X, C v,w the coefficient of the representation X corresponding to the vectors v, w ∈ X ,i.e. the map g < v, ρ ( g ) w > . Then the Schur Lemma gives the relations C v ,w ∗ C v ,w = 1dim X h w , v i C v ,w and C ∗ v ,v = C v ,v , v , v , w , w ∈ X, which in turn imply that for any v ∈ X such that k v k = 1, the element p v = (dim X ) C v,v is a minimalprojection in C ∗ G , and the map φ v : X → p v C ∗ G given by φ v ( w ) = √ dim XC v,w is an isomorphism ofHilbert C ∗ G -modules. Therefore, [ X ] = [ p v C ∗ G,
0] independently of the unital vector v .Using our representation π , we thus have a well defined Kasparov cycle ( p v E , P p v ) with P p v being therestriction of P to p v E = π ( p v ) E , which represents by definition the class of the Kasparov product [ X ] ⊗ C ∗ G Ind F ( P ). Definition 2.13.
The K-multiplicity m P ( X ) of the irreducible unitary representation ρ : G → U ( X ) in theindex class Ind F ( P ) is the image of the above class [ p v E , P p v )] ∈ KK i ( C , C ∗ ( M, F )) under the isomorphismKK i ( C , C ∗ ( M, F )) ∼ = −→ K i ( C ∗ ( M, F )). Hence we end up with the well defined K -multiplicity map: m P : b G −→ K i ( C ∗ ( M, F )) which assigns to X the multiplicity m P ( X ) ∈ K i ( C ∗ ( M, F )) . Let us compare our definition of the K -multiplicity with Atiyah’s definition. Recall the Hilbert modulestructure of X ⊗ E over C ∗ ( M, F ):( v ⊗ ξ ) a := v ⊗ ξa and h v ⊗ ξ, v ′ ⊗ ξ ′ i := < v, v ′ > h ξ, ξ ′ i , ξ, ξ ′ ∈ E , v, v ′ ∈ X and a ∈ C ∗ ( M, F ) . Denote by E GX the Hilbert C ∗ ( M, F )-submodule of X ⊗ E composed of the G -invariant elements for theaction of G given by ρ ⊗ U where U has been introduced in Section 1.1 using that the action is by holonomydiffeomorphims, i.e. with the previous notations, E GX = ( X ⊗ E ) G := { ξ ∈ X ⊗ E such that ( ρ ( g ) ⊗ U g ) ξ = ξ, ∀ g ∈ G } . Lemma 2.14.
The following standard formula holds: DX v i ⊗ η i , X w j ⊗ ξ j E = X h η i , π ( C v i ,w j ) ξ j i , for X v i ⊗ η i ∈ X ⊗ E and X w j ⊗ ξ j ∈ E GX . roof. By linearity, we forget the sums. We have for any γ ∈ G , η, ξ ∈ C ∞ c ( G , r ∗ E ) and since ´ G ρ ( g )( w ) ⊗ U g ( ξ ) dg = w ⊗ ξ : h v ⊗ η, w ⊗ ξ i ( γ ) = ˆ G r ( γ ) h ( v ⊗ η )( γ ) , ( w ⊗ ξ )( γ γ ) i dν r ( γ ) ( γ )= ˆ G r ( γ ) (cid:28) ( v ⊗ η )( γ ) , ˆ G ( ρ ( g )( w ) ⊗ U g ξ )( γ γ ) dg (cid:29) dν r ( γ ) ( γ )= ˆ G r ( γ ) ˆ G h v, ρ ( g ) w i h η ( γ ) , ( U g ξ )( γ γ ) i dgdν r ( γ ) ( γ )= ˆ G r ( γ ) (cid:28) η ( γ ) , ˆ G C v,w ( g )( U g ξ )( γ γ ) dg (cid:29) dν r ( γ ) ( γ )= h η, π ( C v,w ) ξ i ( γ ) . By restricting the operator id X ⊗ P ∈ L C ∗ ( M, F ) ( X ⊗E ) to the G -invariant elements, we get the adjointableoperator P GX ∈ L C ∗ ( M, F ) ( E GX ), i.e. P GX := (id X ⊗ P ) | E GX . Proposition 2.15.
For any (irreducible) unitary representation ρ : G → U ( X ) , (cid:0) E GX , P GX (cid:1) is a Kasparovcycle whose class in KK i ( C , C ∗ ( M, F )) coincides with the class of the cycle ( p v E , P p v ) for any unital vector v ∈ X . In particular, the image of (cid:2) E GX , P GX (cid:3) under the isomorphism KK i ( C , C ∗ ( M, F )) ∼ = −→ K i ( C ∗ ( M, F )) coincides with the K -multiplicity m P ( X ) . Remark 2.16.
Our definition of the K -multiplicity is therefore the exact generalization of Atiyah’s defini-tion, and the multiplicity map m P defined in 2.13 is just the corresponding inteprepretation of the indexclass Ind F ( P ), in the spirit of the series decomposition used for the distributional index in [1]. Proof.
We have for any unital vector v ∈ X , a unitary equivalence between ( p v C ∗ G ⊗ C ∗ G E , Id ⊗ C ∗ G P ) and( X ⊗ C ∗ G E , Id ⊗ C ∗ G P ) given by φ v ⊗ Id : X ⊗ C ∗ G E → p v C ∗ G ⊗ C ∗ G E . On the other hand, we have an isomorphismAv : X ⊗ C ∗ G E −→ E GX = ( X ⊗ E ) G given by ˆ G ( ρ ( g ) ⊗ U g )( • ) dg. We compute for w ∈ X and η ∈ E : ˆ G ( ρ ( g ) ⊗ U g )( w · ϕ ⊗ η ) dg = ˆ G ( ρ ( g ) ⊗ U g )( w ⊗ π ( ϕ ) η ) dg and ˆ G ( ρ ( g ) ⊗ U g )( w ⊗ η ) dg = X k e k ⊗ π ( C e k ,w ) η where ( e k ) k is a given orthonormal basis of X . The first relation shows that the map Av is well definedand we observe that the range of Av is exactly equal to E GX . Using the second relation, Lemma 2.14, andthe standard fact that for any ( w , w ) ∈ X we have P k C w ,e k C e k ,w = C w ,w , in C ∗ G , we see that Avis indeed a unitary between the Hilbert C ∗ ( M, F )-modules X ⊗ C ∗ G E and E GX . That (cid:0) E GX , P GX (cid:1) is a Kasparovcycle is now a consequence of the above unitary equivalence. Remark 2.17.
If ( M, F ) admits a holonomy invariant Borel transverse measure Λ, then applying theassociated additive map K ( C ∗ ( M, F )) → R , we get a well defined Λ-multiplicity morphism, in the gradedcase, for the G -transversely elliptic operator P : m Λ P : b G −→ R , in the spirit of the Murray-von Neumann dimension theory.17 The Atiyah axioms for our index morphism
As before, we denote by F G the closed subspace of F defined by F ∩ T G M . In this subsection, we let G and H be both compact Lie groups. Let M be a smooth compact manifold andlet F be a given smooth foliation of M . We suppose that the compact group G × H acts on M by leaf-preserving diffeomorphisms that we may assume to be isometries of the ambiant manifold M , by averagingthe metric. We further assume that H acts freely on M so that the projection q : M → M/H correspondsto a G -equivariant principal H -fibration which sends leaves to leaves. So, we insist that we assume here that H preserves the leaves upstairs and induces the corresponding leaves downstairs, this is automatic when H is connected. Notice that the leaf of ( M, F ) through a given point m ∈ M coincides here with the inverseimage of the leaf through q ( m ) in the quotient manifold M/H . The induced foliation downstairs in
M/H will be denoted F /H in the sequel. We denote again by π : F → M the vector bundle projection and by π : F/H → M/H the induced vector bundle projection downstairs. The foliations ( M, F ) and ( M/H, F /H )then have the same codimension and under our assumptions do actually have the same space of leaves as weexplain below. The action of H on F then preserves the subspace F G and we have an isomorphism q ∗ : K i G (( F/H ) G ) → K i G × H ( F G × H ) . To be specific, this isomorphism identifies the classes of the G -invariant G -transversally elliptic F /H -leafwisesymbols over M/H , with those of the symbols of G × H -invariant G × H -transversally elliptic F -leafwisesymbols over M . At the level of cycles, q ∗ associates with ( E, a ) the cycle ( q ∗ E, q ∗ a ) with q ∗ a ( m, ξ ) = a ( q ( m ) , q ∗ ξ ), identifying again covectors with vectors. Said differently, the space F G × H is just the fiberedproduct ( F/H ) G × M/H M and using the proper G -equivariant map ˜ q : F G × H → ( F/H ) G , we can see thatthe map q ∗ is the functoriality map ˜ q ∗ .Let ˆ H be the set of isomorphism classes of irreducible unitary representations of the compact group H . We shall sometimes refer to an element α : H → End( W α ) of ˆ H simply as α , and the correspondingcharacter on H will be denoted χ α . Associated with such representation we have the homogeneous bundle W α → M/H associated with the principal H -bundle q : M → M/H . We thus have the classical map: R ( H × G ) −→ K G ( M/H ) V V ∗ where V ∗ is the dual representation.Using a distinguished open cover for the foliated manifold ( M/H, F /H ) which trivializes the principalfibration q : M → M/H as well, it is easy to see that the foliations ( M, F ) and ( M/H, F /H ) have Moritaequivalent C ∗ -algebras. If we denote by G ( M/H, F /H ) the holonomy groupoid of the foliation ( M/H, F /H ),then this Morita equivalence is implemented by the Hilbert module associated with the graph space G q := { ( m, α ) ∈ M × G ( M/H, F /H ) | q ( m ) = r ( α ) } . This is the graph of the morphism of groupoids induced by the projection q : M → M/H . The action of G on G q is given by γ · ( m, α ) = ( γm, q ( γ ) α ) and we leave it as an exercise for the interested reader to show thatwe get in this way a principal G -bundle in the sense of [54] and that this bundle indeed embodies the Moritaequivalence. As a consequence, we can define the imprimitivity Hilbert bimodule which realizes the Moritaequivalence between the corresponding C ∗ -algebras as the completion of the pre-Hilbert module C c ( G q ).There is a left prehilbert C c ( G )-module structure on C c ( G q ) given by f · ϕ ( m, α ) = ˆ G m f ( γ ) ϕ ( s ( γ ) , q ( γ ) − α ) dν m ( γ )18nd G h ϕ, ψ i ( γ ) = ˆ G ( M/H, F /H ) qr ( γ ) ϕ ( s ( γ ) , q ( γ − ) β ) ψ ( r ( γ ) , β ) dν qr ( γ ) ( β ) . There is similarly a right prehilbert C c ( G ( M/H, F /H ))-module structure on C c ( G q ) given by ϕ · ξ ( m, α ) = ˆ β ∈G ( M/H, F /H ) s ( α ) ϕ ( m, αβ ) ξ ( β − ) dν s ( α ) ( β )and h ϕ, ψ i G ( M/H, F /H ) ( β ) = ˆ γ ∈G m ϕ ( s ( γ ) , q ( γ ) − ) ψ ( s ( γ ) , q ( γ ) − β ) dλ m ( γ ) , for a chosen m ∈ q − { r ( β ) } . Notice that the last integral does not depend on the choice of m due to the H -invariance of our Haar system. We can now state our theorem. Theorem 3.1.
Denote by χ the class of the trivial representation in KK( C , C ∗ H ) . Then for i ∈ Z , thefollowing diagram commutes: K i G (( F/H ) G ) q ∗ / / Ind F /H (cid:15) (cid:15) K i G × H ( F G × H ) χ ⊗ C ∗ H Ind F ( • ) (cid:15) (cid:15) KK i ( C ∗ G, C ∗ ( M/H, F /H )) KK i ( C ∗ G, C ∗ ( M, F )) . ⊗ C ∗ ( M, F ) E q o o So if a ∈ K i G (( F/H ) G ) then ignoring the quasi-trivial Morita isomorphism, we may write: Ind F /H ( a ) ∼ = χ ⊗ C ∗ H Ind F ( q ∗ a ) . Proof.
Recall that H acts freely on M and preserves the leaves of F . The holonomy groupoid upstairs is aprincipal H -fibration over G q , this latter is an H -fibration over the holonomy groupoid downstairs. Moreprecisely, G can be identified with the smooth pull-back groupoid ˆ G given byˆ G := { ( m, α, m ′ ) ∈ M × G ( M/H, F /H ) × M | q ( m ) = r ( α ) and q ( m ′ ) = s ( α ) } . The Haar system on G is supposed to be H -invariant and normalized. More precisely, we assume that theHaar system ( ν m ) m ∈ M on ˆ G combines the normalized Haar measure on H with a Haar system downstairs(¯ ν m ) m ∈ M/H on G ( M/H, F /H ). Let A be a G -invariant leafwise G -transversally elliptic pseudodifferentialoperator representing a , of order 1 and supported as close as we please to the units M/H . So the operator A can be seen as a G ( M/H, F /H )-invariant operator along the leaves of the groupoid G ( M/H, F /H ), i.e. A = ( A x ) x ∈ M/H with A x : C ∞ c ( G ( M/H, F /H ) x , r ∗ E ) −→ C ∞ c ( G ( M/H, F /H ) x , r ∗ E ) and with the usual equivariance . Using a partition of unity argument, we may lift A to a G × H -invariant leafwise G × H -transversallyelliptic pseudodifferential operator ˆ A on ( M, F ), which represents q ∗ a . Roughly speaking, the operator ˆ A corresponds in the identification G ≃ ˆ G to tensoring locally by the identity on both sides and can be denotedabusively by id ⊗ A ⊗ id. The index class of q ∗ a can then be represented in KK i ( C ∗ ( G × H ) , C ∗ ( M, F ))by the unbounded Kasparov cycle with the closure of ˆ A as an operator acting on the Hilbert module E corresponding to the pull-back bundle q ∗ E over M , and with the usual representation π G × H of C ∗ ( G × H ).Let us first compute the Kasparov product Ind F ( q ∗ a ) ⊗ C ∗ ( M, F ) E q . This is by definition the class of thetriple (cid:18) E ⊗ C ∗ ( M, F ) C ∗ ( G q ) , π G × H ⊗ C ∗ ( M, F ) id , ˆ A ⊗ C ∗ ( M, F ) id (cid:19) .
19f we denote as well by r : G q → M/H the map r ( m, α ) := r ( α ) = q ( m ) then it is easy to check that theHilbert module E ⊗ C ∗ ( M, F ) C ∗ ( G q ) is isomorphic to C ∗ ( G q , r ∗ E ), i.e. the completion of C c ( G q , r ∗ E ) withrespect to the prehilbertian structure given for e , e ∈ C c ( G q , r ∗ E ) by h e , e i ( β ) = ˆ G m h e ( s ( γ ) , q ( γ ) − ) , e ( s ( γ ) , q ( γ ) − β ) i dν m ( γ ) . (4)To be specific, this identification can be described by a unitary V which is given for u ∈ C c ( G , r ∗ E ) and f ∈ C c ( G q ) by the formula V ( u ⊗ f )( m, β ) = ˆ G m u ( γ ) f ( s ( γ ) , q ( γ ) − β ) dν m ( γ ) . One then checks immediately that for any ϕ ∈ C ( G × H ), we have V ◦ ( π G × H ( ϕ ) ⊗
1) = ˜ π G × H ( ϕ ) ◦ V , wherefor any u ∈ C c ( G q , r ∗ E ),[˜ π G × H ( ϕ )( u )] ( m, η ) = ˆ H × G ϕ ( g, h ) ( g, h ) (cid:16) u (cid:0) ( g, h ) − ( m, η ) θ ( g,h ) − G × H ( s ( η )) (cid:1)(cid:17) dg dh. Similarly, we have that
V ◦ ( ˆ A ⊗ C ∗ ( M, F ) id) = (id ⊗ A ) ◦ V , where id ⊗ A stands for a first order lift of theoperator A to G q using the H -fibration G q → G ( M/H, F /H ). To sum up, we see thatInd F ( q ∗ a ) ⊗ C ∗ ( M, F ) E q = (cid:2) C ∗ ( G q , r ∗ E ) , ˜ π G × H , id ⊗ A (cid:3) . It now remains to compute the Kasparov product of this latter class with the trivial representation of H .We shall use the identification C ∗ ( G q , r ∗ E ) θ H ≃ C ∗ ( G ( M/H, F /H ) , r ∗ E ) . Notice indeed that for θ H -invariant sections e and e , we have: h e , e i ( β ) = ´ G m h e ( s ( γ ) , q ( γ − )) , e ( s ( γ ) , q ( γ − ) β ) i dν m ( γ )= ´ G ( M/H, F /H ) r ( β ) h e ( β − ) , e ( β − β ) i dν r ( β ) ( β ) . In the first expression m ∈ M is any chosen element of the fiber over r ( β ), and the last equality is a conse-quence of the θ H -invariance together with our choice of Haar system upstairs which uses the normalized Haarmeasure on H . Now id ⊗ C ∗ H ˜ π G × H and id ⊗ C ∗ H (id ⊗ A ) both make sense and by using the previous isomorphismwe can see that the first coincides with the representation π G of C ∗ G on C ∗ ( G ( M/H, F /H ) , r ∗ E ) while thesecond is just the operator A . The verification is an exact rephrasing of the same proof for a single operatorand is therefore omitted here. Whence we eventually get the allowed equalityInd F /H ( a ) = χ ⊗ C ∗ ( H ) (cid:20) Ind F ( q ∗ a ) ⊗ C ∗ ( M, F ) E q (cid:21) . Associativity of the Kasprov product allows to conclude.If we replace a by the symbol corresponding to the twist of a by a given unitary representation ( α, W α ),then the same proof yields to the following result: Theorem 3.2. [ ] Let ( W α , α ) be a given finite dimensional unitary representation of H and denote asbefore by χ α the corresponding class in KK( C , C ∗ H ) . Then the following diagram commutes: K i G (( F/H ) G ) q ∗ / / W ∗ α ⊗ • (cid:15) (cid:15) K i G × H ( F G × H ) Ind F / / KK i ( C ∗ ( G × H ) , C ∗ ( M, F )) χ α ⊗ C ∗ H • (cid:15) (cid:15) K i G (( F/H ) G ) Ind F /H / / KK i ( C ∗ G, C ∗ ( M/H, F /H )) KK i ( C ∗ G, C ∗ ( M, F )) . ⊗ C ∗ ( M, F ) E q o o n other words, if a ∈ K iG (( F/H ) G ) and ignoring the Morita isomorphism ⊗ C ∗ ( M, F ) E q , we have Ind F /H ( W ∗ α ⊗ a ) ∼ = χ α ⊗ C ∗ H Ind F ( q ∗ a ) . Remark 3.3.
In particular, as an element of Hom (cid:0) R ( H ) , KK i ( C ∗ G, C ∗ ( M, F )) (cid:1) we haveInd F ( q ∗ a ) = X α ∈ ˆ H ˆ χ α ⊗ Ind F /H ( W ∗ α ⊗ a ) , where ˆ χ α is the element of Hom( R ( H ) , Z ) given by the usual multiplicity.When the group G is the trivial group, we obtain the following expected relation between the Connes-Skandalis index of leafwise elliptic operators downstairs and the index of leafwise H -transversally ellipticoperators upstairs. Corollary 3.4.
Let q : ( M, F ) → ( M/H, F /H ) be as above a principal H -fibration of smooth foliations,recall that H preserves the leaves of ( M, F ) . Then for any leafwise elliptic pseudodifferential symbol σ on ( M/H, F /H ) so that a = [ σ ] ∈ K i ( F/H ) , we have the following equality: χ α ⊗ C ∗ H Ind F /H ( q ∗ a ) ⊗ C ∗ ( M, F ) E q = Ind F ( W ∗ α ⊗ a ) , where Ind F ( W ∗ α ⊗ a ) ∈ KK i ( C , C ∗ ( M, F )) ≃ K i ( C ∗ ( M, F )) is the Connes-Skandalis index, as defined in[26], for the leafwise elliptic symbol W ∗ α ⊗ σ on the compact foliated manifold ( M, F ) . Remark 3.5.
The previous corollary can as well be stated with the extra action of the compact Lie group G and gives a relation between the corresponding G -indices [8]. In this subsection, we show an excision property for the index class of G -invariant leafwise G -transversallyelliptic operators. More precisely, we shall first extend our definition of the index morphism to the case ofany smooth foliated (open) manifold ( U, F U ) which is again endowed with a leaf-preserving action of G byholonomy diffeomorphisms, and obtain an index morphismInd F U : K i G ( F UG ) −→ KK i ( C ∗ G, C ∗ ( U, F U )) . Then we shall show the compatibility of this morphism with foliated open embeddings, in particular inclosed foliated manifolds, this is the expected excision result. Again C ∗ ( U, F U ) is the Connes algebra ofthe foliation ( U, F U ), i.e. the C ∗ -completion of the convolution algebra of compactly supported continuousfunctions on the holonomy groupid G ( U, F U ) of ( U, F U ). As usual, we have fixed a G -invariant metric on U and used it to identify for instance the colongitudinal bundle ( F U ) ∗ with the longitudinal bundle F U .We shall use the following classical lemma which is shown for instance in [1, lemma 3.6] in the non-foliatedcase, see also [3] for the original proof in the elliptic case and [26] for the leafwise elliptic case. The proof forthe foliated G -transversely elliptic case is similar with the same standard techniques and hence is omitted.Let π U : F U → U be the projection of the tangent space to the foliation F U . We denote as before by F UG = F U ∩ T G U . Lemma 3.6.
Each element a ∈ K G ( F UG ) can be represented by a G -equivariant morphism π ∗ U E + σ −→ π ∗ U E − over the whole of F U , which is zero-homogeneous for large ξ with E ± being G -equivariant vector bundlesover U , and such that: • Outside some compact G -subspace L in U , the bundles E ± are trivialized and the restriction of σ to π − U ( U r L ) is the identity morphism modulo the trivializations of E ± . The morphism σ ( x, ξ ) : E + x → E − x is an isomorphism for ( x, ξ ) ∈ F UG r U . So, the first item means that there exist bundle G -equivariant isomorphisms over U r L (or rather overeach of its connected components) ψ ± : E ± | U r L −→ ( U r L ) × C dim( E ± ) such that ∀ ( x, ξ ) ∈ π − U ( U r L ) : σ ( x, ξ ) = ( ψ − x ) − ◦ ψ + x : E + x → E − x . We endow the vector bundles E ± with G -invariant hermitian structures and consider the Hilbert modules E ± over C ∗ ( U, F U ) which, as in the previous sections, are the completions of the prehilbertian C c ( G ( U, F U ))-modules C c ( G ( U, F U ) , r ∗ E ± ). Moreover, using the equivalence relation of stable homotopies with compactsupport as in [3], the bundle trivialization ψ ± can be assumed to be bounded and in fact even fiberwiseunitaries for the hermitian structures. We thus assume as well that σ ∗ σ and σσ ∗ are the identity bundleisomorphisms of E + and E − respectively, over U r L . By using the holonomy action as in Section 2, wecan endow the Hilbert modules E ± with the structure of Hilbert G -modules when C ∗ ( U, F U ) is triviallyacted on by the compact Lie group G . We can now quantize any such zero-degree homogeneous symbol σ and choose a uniformly supported zero-th order G -invariant pseudodifferential G ( U, F U )-operator P : C ∞ c ( G ( U, F U ) , r ∗ E + ) → C ∞ c ( G ( U, F U ) , r ∗ E − ) with the principal symbol equal to σ . More precisely, uniformsupport is taken in the sense of [55], see also [26, Proposition 4.6]. Here, we can in fact ensure that P is theidentity operator outside some compact G -subspace L ′ whose interior contains L , i.e. that we have P ( η )( γ ) = ( ψ − x ) − ◦ ( ψ + x )( η ( γ )) , for any η ∈ C ∞ c ( G ( U, F U ) U \ L ′ x , r ∗ E + ) . Hence P ∗ P and P P ∗ reduce to the identity operators on the sections which are supported above U r L ′ ,say in r − ( U r L ′ ).The operator P hence reduces to multiplication by the unitary bundle morphism ( ψ − ) − ◦ ( ψ + ) over U r L ′ , and it is easy to deduce that it extends to an adjointable G -equivariant operator from E + to E − [51]that we still denote by P , see also [59, 14]. We denote as usual by P the self-adjoint G -invariant operator P := (cid:18) P ∗ P (cid:19) (5)acting on the Hilbert G -module E = E + ⊕ E − . Recall that we are considering the trivial G -action on C ∗ ( U, F U ) and that E is endowed with the structure of a Z -graded Hilbert G -module. Proposition 3.7.
The triple ( E , π, P ) is a Z -graded Kasparov cycle over the pair of C ∗ -algebras ( C ∗ G, C ∗ ( U, F U )) and defines an index class in KK( C ∗ G, C ∗ ( U, F U )) which only depends on the G -equivariant stable homotopy class a = [ σ ] ∈ K G ( F UG ) and is denoted Ind F U ( a ) . Moreover, we get inthis way a well defined index morphism for the open foliation ( U, F U ) : Ind F U : K G ( F UG ) −→ KK( C ∗ G, C ∗ ( U, F U )) . We have similarly a well defined (ungraded) index map
Ind F U : K ( F UG ) −→ KK ( C ∗ G, C ∗ ( U, F U )) . Proof.
We freely use notations from Section 2 and only treat the even case. Since σ is bounded on L andassumed to be unitary outside L , we get that σ is bounded. Now notice that k σ ( P )( x, ξ ) − id k 6 = 0 onlyfor x ∈ L which implies, using Lemma 2.3, that ∀ ε > , ∃ c > k σ ( P )( x, ξ ) − id k ≤ cσ ( Q )( x, ξ ) + ε, (6)where σ ( Q )( x, ξ ) = q x ( ξ ) on S ∗ F U . Using Proposition B.1, we get − ( cQ + ε + R ) ≤ P − id ≤ cQ + ε + R as self-adjoint operators on E (7)22here R i ∈ K U ( E ). Denote by θ ∈ C ∞ c ( U, [0 , L ′ . If ζ ∈ C c ( U, [0 , L and 0 outside L ′ , we have σ ( P ) ( r ( γ ) , ξ ) − id = ζ ( r ( γ ))( σ ( P ) ( r ( γ ) , ξ ) − id)and ζθ = ζ . We use as usual an oscillatory integral to define the quantization map, see for instance [26].More precisely, the G -invariant operator P − id is given through its Schwartz kernel, a distribution k P − id on G given by an expression of the following type k P − id ( γ ) = ˆ F U ∗ r ( γ ) χ ( γ − ) e − i h Φ( γ − ) ,ξ i (cid:0) σ ( P ) ( r ( γ ) , ξ ) − id (cid:1) dξ = k ( P − id) r ∗ ζ ( γ ) = k r ∗ ζ ( P − id) ( γ ) , where Φ is a diffeomorphism from a uniform neighbourhood W of U in G ( U, F U ) to a neighbourhood ofthe zero section in F U with d Φ = id and χ is a cut off function with support inside W which is equal to 1in a smaller neighborhood of U whose closure is contained in W , see [26] as well as [51] or [59]. With theappropriate choice of these small neighborhood of the unit manifold U in G ( U, F U ) in coherence with L ′ , weobtain that r ∗ θ ( P − id) = P − id = ( P − id) r ∗ θ. Multiplying on both sides the inequality (7) by r ∗ θ , we get − ( cr ∗ θQr ∗ θ + εr ∗ θ + r ∗ θR r ∗ θ ) ≤ P − id ≤ cr ∗ θQr ∗ θ + εr ∗ θ + r ∗ θR r ∗ θ. Furthermore, modulo K U ( E ), Q can be represented by d ∗ G (1 + ∆) − d G . Notice that σ ( d ∗ G (1 + ∆) − )( x, ξ )is bounded, therefore d ∗ G (1 + ∆) − ∈ Ψ ( U, F U , E ) U and its zero-th order symbol vanishes, so that d ∗ G (1 +∆) − ∈ K U ( E ). In particular, the operator r ∗ θd ∗ G (1 + ∆) − is compact. Now we can conclude exactly as inthe proof of Theorem 2.6. Indeed, recall that we have d G ( r ∗ θπ ( ϕ ) η )( γ ) = X ∂θ∂V k ( r ( γ )) π ( ϕ ) η ⊗ v k + θ ( r ( γ )) π ( dϕ ) η ( γ ) , and since d G r ∗ θπ ( ϕ ) is bounded we deduce that r ∗ θd ∗ G (1 + ∆) − d G r ∗ θπ ( ϕ ) is compact. Moreover, theoperators r ∗ θR i r ∗ θ are also compact since each R i ∈ K U ( E ). Now for any non-negative ψ = ϕ ∗ ϕ ∈ C ∗ G , − π ( ϕ ) ∗ ( cr ∗ θQr ∗ θ + εr ∗ θ + r ∗ θR r ∗ θ ) π ( ϕ ) ≤ π ( ϕ ) ∗ ( P − id) π ( ϕ ) ≤ π ( ϕ ) ∗ ( cr ∗ θQr ∗ θ + εr ∗ θ + r ∗ θR r ∗ θ ) π ( ϕ ) . Since P is G -invariant, this can be rewritten as − π ( ϕ ) ∗ ( cr ∗ θQr ∗ θ + εr ∗ θ + r ∗ θR r ∗ θ ) π ( ϕ ) ≤ ( P − id) π ( ψ ) ≤ π ( ϕ ) ∗ ( cθQr ∗ θ + εr ∗ θ + r ∗ θR r ∗ θ ) π ( ϕ ) . Therefore, projecting in the Calkin algebra and letting ε →
0, we deduce that ( P − id) ◦ π ( ψ ) is compact.This implies, using the spectral theorem, that ( P − id) ◦ π ( ϕ ) is compact for any ϕ ∈ C ∗ G .Assume now that there exists an open foliated G -embedding j : ( U, F U ) ֒ → ( M, F ) of smooth foliatedmanifolds. This means that j is a G -equivariant embedding of U as an open submanifold of the foliated G -manifold M which transports the foliation F U into the restriction of the foliation F to the open submanifold j ( U ). We assume again that G acts on all foliations by holonomy diffeomorphisms. We shall mainly beinterested in the present paper in the case M compact, but this is not needed so far. The embedding j theninduces an open embedding at the level of holonomy groupoids that we still denote by j for simplicity, i.e. j : G ( U, F U ) ֒ → G ( M, F ) = G . The C ∗ -algebra C ∗ ( U, F U ) is hence isomorphic to the C ∗ -algebra of thefoliation of j ( U ) induced by F , but this latter can be seen as a C ∗ -subalgebra of C ∗ ( M, F ) in an obviousway. We therefore end-up with a well defined class j ! ∈ KK( C ∗ ( U, F U ) , C ∗ ( M, F )). In the notations of[26], the map j : U → M induces in particular a submersion from U to M/ F , and we therefore have awell defined Morita extension Kasparov class which is exactly the class j ! , but the construction is simplerin our open embedding case. Finally, the differential dj : F U → F M = F of the G -embedding j restrictsto an open G -map sending the space F UG in the space F MG = F G , therefore and by functoriality, we get an R ( G )-morphism j ∗ : K i G ( F UG ) −→ K i G ( F G ) . We are now in position to state the main theorem of this subsection.23 heorem 3.8.
Under the above assumptions and for any i ∈ Z , the following diagram commutes: K i G ( F UG ) j ∗ / / Ind F U (cid:15) (cid:15) K i G ( F G ) Ind F (cid:15) (cid:15) KK i ( C ∗ G, C ∗ ( U, F U )) j ! / / KK i ( C ∗ G, C ∗ ( M, F )) . Remark 3.9.
If the action of G on ( U, F U ) is not necessarily a holonomy action while it is a holonomyaction on ( M, F ), then the index morphism Ind F U is not well defined anymore, and one can use Theorem3.8 precisely to define it for any given such embedding, as a class in KK i ( C ∗ G, C ∗ ( M, F )) with the usualcompatibility with embeddings.Recall first the notion of support of a Kasparov ( A, B )-cycle ( E , π, F ), for given separable C ∗ -algebras A and B , as introduced in [26, Appendix A]. This is the Hilbert submodule of E which is generated by K E ,with K the C ∗ -subalgebra of the C ∗ -algebra K ( E ) of compact operators on the Hilbert B -module E , whichis generated by the operators [ π ( a ) , F ] , π ( a )( F −
1) and π ( a )( F − F ∗ ) and their multiples by A , F and F ∗ .Here a runs over A of course. Then obviously E is a Hilbert ( A, B )-bimodule and F restricts automaticallyto E to yield the operator F so that ( E , π, F ) is again a Kasparov ( A, B )-cycle. We quote the followinginteresting observation from [26] which will be used in the sequel.
Lemma 3.10. [26] Let ( E , π, F ) be a Kasparov ( A, B ) -cycle where A and B are given separable C ∗ -algebras.Let ( E , π, F ) be the Kasparov ( A, B ) -cycle obtained by restricting to the support E . Then ( E , π, F ) definesthe same KK -class, i.e. [( E , π, F )] = [( E , π, F )] ∈ KK(
A, B ) . Proof of Theorem 3.8.
We concentrate again on the even case i = 0. Let a ∈ K G ( F UG ) be fixed. We denoteas before by π U : F U → U and by π M : F → M the respective bundle projections. We start by representing a by a symbol of order 0 on F U according to Lemma 3.6: π ∗ U E + σ −→ π ∗ U E − , which is thus trivial outside a compact set L of U . By using the trivializations ψ ± , a standard argumentallows to extend the hermitian bundles E ± viewed over j ( U ) to hermitian G -equivariant vector bundles j ∗ E ± over M with the obvious extension j ∗ σ so that ( j ∗ E + , j ∗ E − , j ∗ σ ) represents the push-forward class j ∗ a , seefor instance [3, 1]. The bundle trivilizations ψ ± then give the extended bundle isomorphisms, still denoted ψ ± , over M r j ( L ). Associated with the hermitian G -bundles j ∗ E ± , we then obtain the correspondingHilbert G -modules over the C ∗ -algebra C ∗ ( M, F ) that we denote by j ∗ E ± . Recall that j induces as well a ∗ -homomorphism j ∗ : C ∗ ( U, F U ) −→ C ∗ ( M, F ) , which allows to represent C ∗ ( U, F U ) as adjointable operators on C ∗ ( M, F ) when this latter is viewed asa Hilbert module over itself. We can therefore consider the Z -graded Hilbert G -module over C ∗ ( M, F ),obtained by composition, and denoted as usual E ⊗ C ∗ ( U, F U ) C ∗ ( M, F ). This latter Hilbert G -module can beidentified with a Hilbert G -submodule of j ∗ E , i.e. there is a Hilbert module isometry V : E ⊗ C ∗ ( U, F U ) C ∗ ( M, F ) −→ j ∗ E , which is given for η ∈ C c ( G ( U, F U ) , r ∗ E ) and f ∈ C c ( G ) by V ( η ⊗ f ) = ˜ η · f , that is, the convolution of ˜ η ,the extension by 0 of η outside G ( U, F U ), with f , i.e. V ( η ⊗ f )( γ ) := ˆ G r ( γ ) ˜ η ( γ ) f ( γ − γ ) dν r ( γ ) ( γ ) .
24e identify for simplicity U with j ( U ) for the rest of this proof. The Hilbert submodule V (cid:18) E ⊗ C ∗ ( U, F U ) C ∗ ( M, F ) (cid:19) can be identified with the completion C ∗ ( G U , r ∗ E ) of C c ( G U , r ∗ E ) in j ∗ E , where G U is the space of elementsof G which end inside U . See [26, Proposition 4.3]. To finish the proof, we only need to compare the supportsof the two Kasparov cycles, and to apply Lemma 3.10.We choose a uniformly supported G -invariant leafwise pseudodifferential operator P on ( U, F U ) withsymbol σ as in the above construction of the index class on ( U, F U ). So, P can be seen as a G ( U, F U )-operator in the sense of [23] that we denote again by P : C ∞ c ( G ( U, F U ) , r ∗ E ) −→ C ∞ c ( G ( U, F U ) , r ∗ E ) , which acts along the fibers of the groupoid and is an invariant family ( P ,x ) x ∈ U . Here of course we assume,as we did in the construction of the index class, that P is the identity outside some compact subspace L ′ of U , modulo the trivializations ψ ± . For simplicity of notations, this operator is also the one over j ( U ) with itsfoliation induced from F . In order to quantize the pushforward class j ∗ a , we can then consider the uniformlysupported G -invariant leafwise operator on M defined as follows.Let θ ∈ C ∞ c ( M, [0 , G -invariant bump function which is equal to 1 on L ′ , and whose supportis a compact subspace of j ( U ) outside of which the operator P is trivial. Denote by ψ ± r the isomorphisms ψ ± viewed between the bundles r ∗ E ± and which are only well defined over r − ( M r j ( L )). Then j ∗ P canbe taken as the G -operator on ( M, F ) defined by j ∗ P := ˜ P r ∗ θ + ( ψ − r ) − ◦ ψ + r (1 − r ∗ θ ) . We use here the same cut-off function used to extend σ to F . Hence j ∗ P is obviously a zero-th order leafwise G -operator which is G -invariant and has the principal symbol equal to j ∗ σ = σ θ + ( ψ − ) − ◦ ψ + (1 − θ )and so represents j ∗ a . Recall that the index class Ind F U ( a ) is represented by the adjointable extension ofthe operator P = (cid:18) P ∗ P (cid:19) acting on the Hilbert module E , while the class Ind( j ∗ a ) can obviously berepresented by the adjointable extension of the operator j ∗ P = (cid:18) j ∗ P ∗ j ∗ P (cid:19) acting on the Hilbert module j ∗ E .Notice that Ind F U ( a ) ⊗ j ! = (cid:20) ( E ⊗ C ∗ ( U, F U ) C ∗ ( M, F ) , π ⊗ , P ⊗ (cid:21) and using the isometry V definedabove we deduce that the Kasparov cycle ( E ⊗ C ∗ ( U, F U ) C ∗ ( M, F ) , π ⊗ , P ⊗
1) is unitarily equivalent tothe cycle [ C ∗ ( G U , r ∗ E ) , π, j ∗ P | C ∗ ( G U ,r ∗ E ) ]. Indeed, the representations of the C ∗ -algebra C ∗ G are clearlycompatible, and we have V ( P η ⊗ f ) = f P η · f = ˜ P ˜ η ⊗ f = ˜ P (˜ η · f ) = j ∗ P | C ∗ ( G U ,r ∗ E ) V ( η ⊗ f ) , with ˜ P being as before the G -operator obtain from P by extending trivially its distributional kernel. Tocomplete the proof, thanks to the Connes-Skandalis Lemma 3.10, we only need to show that the supportsof Ind F ( j ∗ a ) and [ C ∗ ( G U , r ∗ E ) , π, j ∗ P | C ∗ ( G U ,r ∗ E ) ] are the same. But using the cut off function θ which issupported in U , we can write (( j ∗ P ) − id) r ∗ θ = ( j ∗ P ) − idand the same equality is true when replacing j ∗ P by j ∗ P | C ∗ ( G U ,r ∗ E ) and θ by θ | U . Therefore the supports docoincide as allowed. 25 .3 Multiplicativity of the index morphism Recall that G is a compact Lie group. Let M and M ′ be two smooth closed manifolds endowed with smoothfoliations that we denote respectively by F and F ′ . We assume that G acts by holonomy diffeomorphisms on( M, F ) and on ( M ′ , F ′ ). We assume in addition that another compact Lie group H acts on the first manifold M also by holonomy diffeomorphisms, and that the actions of G and H commute. So said differently, thecompact Lie group G × H acts by holonomy diffeomorphisms which are isometries (for the ambiant manifoldmetric) on ( M, F ) and ( M ′ , F ′ ) and we assume that the action of H on the second manifold M ′ is trivial.Recall that in this situation the compact Lie groups G and H act by inner automorphisms on the Connes’ C ∗ -algebras of the foliations ( M, F ) and ( M ′ , F ′ ). We thus get for instance the following C ∗ -algebra isomorphismwhich will be used later on (see Corollary 1.3)Ψ : C ∗ ( M, F ) ⋊ G −→ C ∗ ( M, F ) ⊗ C ∗ G, and which is induced by the map C ( G, C ∗ ( M, F )) → C ( G, C ∗ ( M, F )) given for f ∈ C ( G, C ∗ ( M, F )) and g ∈ G by Ψ( f )( g ) := f ( g ) U g . This isomorphism allows indeed to replace, the crossed product C ∗ -algebra C ∗ ( M, F ) ⋊ G by the tensor product C ∗ -algebra C ∗ ( M, F ) ⊗ C ∗ G . We denote by [Ψ] ∈ KK( C ∗ ( M, F ) ⋊ G ) , C ∗ ( M, F ) ⊗ C ∗ G ) the induced KK-equivalence. This element is just the Kasparov descent of the KK G -equivalence between C ∗ ( M, F ) endowed with the inner action and C ∗ ( M, F ) endowed with the trivial action.Recall the Kasparov descent map [40] for given G - C ∗ -algebras A and B . Let E be a Hilbert G -moduleon B . Define a right prehilbertian C ( G, B )-module structure on the space C ( G, E ) of continuous E -valuedfunctions on G , by setting e · d ( s ) = ˆ G e ( t ) td ( t − s ) dt and h e , e i ( s ) = ˆ G t − ( h e ( t ) , e ( ts ) i E ) dt, for e , e , e ∈ C ( G, E ) and d ∈ C ( G, B ). Then the completion of C ( G, E ) with respect to this Hilbertstructure defines by classical arguments, a B ⋊ G -Hilbert module that we denote by E ⋊ G . If π : A → L B ( E )is a G -equivariant ∗ -morphism from A to the C ∗ -algebra of adjointable operators on E then the map π ⋊ G : A ⋊ G → L B ⋊ G ( E ⋊ G ) given by ( π ⋊ G ( a ) e )( t ) = ˆ G π ( a ( s )) s ( e ( s − t )) ds is a ∗ -morphism. Finally, if T ∈ L B ( E ) then T induces an operator T ⋊ G ∈ L B ⋊ G ( E ⋊ G ) defined by( T ⋊ G ) e ( s ) := T ( e ( s )). It was then proved in [40] that if ( E , π, T ) is an ( A, B ) Kasparov cycle, then( E ⋊ G, π ⋊ G, T ⋊ G ) is an ( A ⋊ G, B ⋊ G ) Kasparov cycle and that this induces a well defined groupmorphism at the level of KK-theory. More precisely, Definition 3.11. [40] For i ∈ Z , the Kasparov descent map for the given G -algebras A and B is the welldefined induced map j G : KK i G ( A, B ) −→ KK i ( A ⋊ G, B ⋊ G ) is given by [( E , π, T )] [( E ⋊ G, π ⋊ G, T ⋊ G )] . Back to our foliations, recall from Proposition 2.10 the well defined G -equivariant index map for G × H -invariant leafwise symbols on ( M, F ) which are H -transversally elliptic along the leaves, i.e.Ind F ,G : K iG × H ( F H ) −→ KK i G ( C ∗ H, C ∗ ( M, F )) . (8)If we compose this index map with the Kasparov descent map for the G -algebras C ∗ H (with trivial G -action) and C ∗ ( M, F ) for the standard action induced from the G -action along the leaves, and further usethe isomorphism Ψ, then we end up with an index map d Ind F ,G : K iG × H ( F H ) −→ KK i ( C ∗ ( H × G ) , C ∗ ( M, F ) ⊗ C ∗ G ) . emark 3.12. Since G acts by holonomy diffeomorphisms here, we can recast the representative of theindex class given by Equation (8) so that it rather represents a G -equivariant class for the trivial G -actionon C ∗ ( M, F ). If we denote by KK iG trivial ( C ∗ H, C ∗ ( M, F )) the equivariant Kasparov group for the trivial G -action, then this yields an index morphismInd F ,G trivial : K iG × H ( F H ) −→ KK i G trivial ( C ∗ H, C ∗ ( M, F )) . (9) Lemma 3.13.
Denote by j G trivial : KK iG trivial ( C ∗ H, C ∗ ( M, F )) → KK i ( C ∗ ( H × G ) , C ∗ ( M, F ) ⊗ C ∗ G ) theKasparov descent morphism for the trivial G -action, then the following relation holds: d Ind F ,G = j G trivial ◦ Ind F ,G trivial . Proof.
We denote by C ∗ ( M, F ) t the C ∗ -algebra of the foliation ( M, F ) endowed with the trivial action.Denote by [ ψ ] ∈ KK G ( C ∗ ( M, F ) , C ∗ ( M, F ) t ) the KK G -equivalence class defined using that the action on C ∗ ( M, F ) is inner. We then have Ind F ,G ⊗ C ∗ ( M, F ) [ ψ ] = Ind F ,G trivial . Since j G ([ ψ ]) = [Ψ] and j G (Ind F ,G ⊗ C ∗ ( M, F ) [ ψ ]) = j G (Ind F ,G ) ⊗ C ∗ ( M, F ) ⋊ G j G ([ ψ ]) , we get the result by applying the Kasparov descent on both side.Using the action of G on the second foliation ( M ′ , F ′ ) we also have the index map for leafwise G -transversally elliptic symbols Ind F ′ : K G ( F ′ G ) −→ KK( C ∗ G, C ∗ ( M ′ , F ′ )) . A classical construction then allows to build up from a G × H -invariant leafwise H -transversally ellipticsymbol a on ( M, F ) and a G -invariant leafwise G -transversally elliptic symbol b on ( M ′ , F ′ ) a new symbolwhich is a leafwise symbol on the cartesian product ( M × M ′ , F × F ′ ) of the two foliated manifolds, is G × H -invariant and G × H -transversally elliptic.More precisely, there is a well defined product for all i, j ∈ Z ,K i G × H ( F H ) ⊗ K j G ( F ′ G ) −→ K i + j G × H (( F × F ′ ) G × H ) , (10)which assigns to [ σ ] ⊗ [ σ ′ ] the class of the sharp product σ♯σ ′ that we proceed to recall now. The cartesianproduct F × F ′ fibers over M × M ′ and generates the foliation of M × M ′ whose leaf through any given( m, m ′ ) is just the cartesian product L m × L ′ m ′ of the leaf of ( M, F ) through m by the leaf of ( M ′ , F ′ )through m ′ . The compact group G × H acts obviously by leaf-preserving diffeomorphisms of this productfoliation and the subspace ( F × F ′ ) G × H of vectors transverse to this action, is well defined. Notice as wellthat this product action of G × H is also a holonomy action. For the convenience of the reader, let usdescribe the above product in the case i = j = 0 for simplicity. The other cases can be deduced by replacing M by M × S with the foliation F ×
0. Recall that any class b in K G ( F ′ G ) can be represented by a classical G -invariant pseudodifferential symbol σ ′ along the leaves of the foliation F ′ , that is defined over F ′ , andwhose restriction to F ′ G \ M ′ is pointwise invertible. In the same way, any class a in K G × H ( F H ) can berepresented by a classical G × H -invariant pseudodifferential symbol σ along the leaves of the foliation F ,that is defined over F , and whose restriction to F H \ M is pointwise invertible. We may assume that σ and σ ′ both have positive order, see [3] and also [1]. The product a♯b is then the class in K G × H (( F × F ′ ) G × H )which is represented by the leafwise G × H -invariant symbol on the foliation F × F ′ over M × M ′ definedby: σ♯σ ′ := (cid:18) σ ⊗ − ⊗ σ ′∗ ⊗ σ ′ σ ∗ ⊗ (cid:19) . σ♯σ ′ to ( F × F ′ ) G × H r ( M × M ′ ) is pointwise invertibleas allowed and hence represents our announced sharp product.We are now in position to prove the multiplicativity axiom which computes the index of the sharp product a♯b in terms of the indices of a and b . Notice that C ∗ ( M × M ′ , F × F ′ ) ≃ C ∗ ( M, F ) ⊗ C ∗ ( M ′ , F ′ ) . Theorem 3.14.
For any i, j ∈ Z , the following diagram commutes: K i G × H ( F H ) ⊗ K j G ( F ′ G ) • ♯ • / / c Ind F ,G ⊗ Ind F′ (cid:15) (cid:15) K i + j G × H (( F × F ′ ) G × H ) Ind
F×F′ (cid:15) (cid:15) KK i ( C ∗ ( G × H ) , C ∗ ( M, F ) ⊗ C ∗ G ) ⊗ KK j ( C ∗ G, C ∗ ( M ′ , F ′ )) • ⊗ C ∗ G • / / KK i + j ( C ∗ ( G × H ) , C ∗ ( M × M ′ , F × F ′ )) . In other words, if a ∈ K i G × H ( F H ) and b ∈ K j G ( F ′ G ) and if a♯b ∈ K i + j G × H (( F × F ′ ) G × H ) is their sharpproduct, then we have Ind
F×F ′ ( a♯b ) = d Ind F ,G ( a ) ⊗ C ∗ G Ind F ′ ( b ) . Proof.
We treat the case i = 0 = j , the other cases are similar. If P is a longitudinal pseudodifferentialoperator of positive order then we denote again by P the closure of the formally self-adjoint longitudinaloperator (cid:18) P ∗ P (cid:19) in the corresponding Hilbert module. We also recall that the Woronowicz transform of P is the adjointable operator Q ( P ) = P (1 + P ) − / .Let A : C ∞ c ( G , r ∗ E + ) → C ∞ c ( G , r ∗ E − ) be a G × H -invariant, leafwise H -transversally elliptic operatorof order 1 whose principal symbol represents the class a . Let similarly B : C ∞ c ( G ′ , r ∗ E ′ + ) → C ∞ c ( G ′ , r ∗ E ′ − )be a G -invariant, leafwise G -transversally elliptic operator of order 1 whose principal symbol lies in the class b . The index classes associated respectively are then by definitionInd F ,G ( A ) = [( E , π H , Q ( A ))] and Ind F ′ ( b ) = [( E ′ , π G , Q ( B ))] , where the first class is G -equivariant for the G trivial -action on E , i.e. viewed as a Hilbert G -module for thetrivial G -action on C ∗ ( M, F ) by using the holonomy hypothesis.Hence, the image of [( E , π H , Q ( A ))] under the Kasparov descent is represented, with our previous nota-tions and using Lemma 3.13, by the Kasparov ( C ∗ H ⋊ G, C ∗ ( M, F ) ⊗ C ∗ G ) cycle( E ⋊ G trivial , π H ⋊ G trivial , Q ( A ) ⋊ G trivial ) . Recall that the action of G on the C ∗ -algebra C ∗ ( M ′ , F ′ ) is also inner through unitary multipliers thatwe denote by ( U ′ g ) g ∈ G . Let U : C ( G, E ) ⊗ E ′ → E ⊗ E ′ be the map defined by U ( ρ ⊗ η ′ ) := ˆ G ρ ( g ) ⊗ U ′ g η ′ dg, for ρ ∈ C ( G, E ) and η ′ ∈ E ′ . Here the integral makes sense in the norm topology of the Hilbert module closure, denoted as usual
E ⊗ E ′ ,over the C ∗ -algebra C ∗ ( M, F ) ⊗ C ∗ ( M ′ , F ′ ). From the very definition of the representation π G , we easilydeduce that for ϕ ∈ C ( G ), one has U ( ρ · ϕ ⊗ η ′ ) = U ( ρ ⊗ π G ( ϕ ) η ′ ) ,
28o that U is well defined. Moreover, we can check now that U extends to a unitary isomorphism whichidentifies ( E ⋊ G trivial ) ⊗ π G E ′ with the spatial tensor product Hilbert module E⊗E ′ . Indeed, a direct computationshows that U is isometric, and it is also straightforward to check that U ( C ( G, E ) ⊗E ′ ) is dense in E ⊗ E ′ .Indeed, given η ∈ E and η ′ ∈ E ′ , we may use an approximate unit ( e α ) α of the C ∗ -algebra C ∗ G , composed ofcontinuous functions on G which are supported as close as we please to the neutral element of G , to see that π G ( e α )( η ′ ) converges in E ′ to η ′ . Hence, the net U (( η ⊗ e α ) ⊗ η ′ ) = η ⊗ π G ( e α )( η ′ ) converges to η ⊗ η ′ in thespatial tensor product E ⊗ E ′ . It thus remains to check that U intertwines representations and operators,but this is as well a straightforward verification.The Kasparov product of d Ind F ,G ( A ) and Ind F ′ ( B ) can be represented by the unbounded cycle (cid:18) ( E ⋊ G trivial ) ⊗ π G E ′ , ( π H ⋊ G trivial ) ⊗ π G id , ( A ⋊ G trivial ) ⊗ π G id + id ⊗ π G B (cid:19) where the operator id ⊗ π G B is well defined here since B commutes strictly with the representation π G . Indeed,although this is not an external Kasparov product, this strict commutation allows to apply the argumentgiven in [5][Lemma 3.1 & Theorem 3.2] which adapts mutatis mutandis to our situation. We thank thereferee for pointing out this observation to us. We now use the previous axioms to investigate the induction property of our index morphism with respectto closed subgroups, and then more specifically to a maximal torus.We recall first some standard constructions from [1]. Let G be a compact connected Lie group andlet H be a closed subgroup of G . Denote by i : H ֒ → G the inclusion. Then the functoriality class[ i ] ∈ KK( C ∗ G, C ∗ H ) is defined as follows, see [37]. We fix Haar measures on H and G and consider theright L ( H )-module structure on the space C ( G ), which is induced by the right action of H on G . Moreprecisely, we set for f ∈ C ( G ) and ψ ∈ L ( H ): f · ψ ( g ) = ˆ H f ( gh − ) ψ ( h ) dh, and define the L ( H )-valued hermitian structure by setting for f , f ∈ C ( G ): h f , f i ( h ) = ˆ G f ( g ) f ( gh ) dg. The completion of this prehilbertian L ( H )-module is then a Hilbert C ∗ H -module that we shall denote by J ( G, H ). The left action of G on itself by translation allows to define, after completing, the representation π G : C ∗ G → L C ∗ H ( J ( G, H )). The triple ( J ( G, H ) , π G ,
0) is then a Kasparov cycle over the pair of C ∗ -algebras ( C ∗ H, C ∗ G ), see again [37]. Definition 4.1. [37] The functoriality class [ i ] is the class of the Kasparov cycle ( J ( G, H ) , π G , i ] := [( J ( G, H ) , π G , ∈ KK( C ∗ G, C ∗ H ) . (11) Remark 4.2.
Since the crossed product C ∗ -algebra C ( G/H ) ⋊ G for the induced left action of G on thehomogeneous manifold G/H , is Morita equivalent to C ∗ H , it is easy to reinterpret the class [ i ] as the classinduced through the descent map for the G -action, by the trivial representation of H viewed as a trivial G -equivariant vector bundle over G/H . 29ince the underlying closed manifold G is endowed with the G × H -action given by ( g, h ) · g ′ = gg ′ h − for g, g ′ ∈ G and h ∈ H , we may use the product defined in Equation (10) for any given smooth foliation F on a closed manifold M as soon as this latter is endowed with a smooth leaf-preserving H -action, which isa holonomy action. Indeed, we are considering here the trivial top-dimensional foliation on G and we thusget the following product for j ∈ Z K G × H (cid:0) T G G (cid:1) ⊗ K j H (cid:0) F H (cid:1) −→ K j G × H (cid:0) ( T G × F ) G × H (cid:1) . (12)The space T G G is just G × { } ≃ G , and hence since H acts freely on G :K G × H ( T G G ) ≃ K G ( G/H ) ≃ R ( H ) . Moreover, H also acts freely on the cartesian product G × M preserving the product foliation T G × F andthe quotient manifold Y := G × H M inherits a foliation that we denote by F Y and which is automaticallyendowed with the action of G by holonomy diffeomorphisms, as can be checked easily. The receptacle groupK j G × H (cid:0) ( T G × F ) G × H (cid:1) in (12) is then given byK j G × H (cid:0) ( T G × F ) G × H (cid:1) ≃ K j G ( F YG ) . (13)Notice that the space F YG = G × H F H is G -equivariantly Morita equivalent as a groupoid to ( G × F H ) ⋊ H and we deduce the following list of Morita equivalences F YG ⋊ G ∼ [( G × F H ) ⋊ H ] ⋊ G ≃ [( G × F H ) ⋊ G ] ⋊ H ∼ [( G × F H ) /G ] ⋊ H ≃ F H ⋊ H. In particular, the group K j G ( F YG ) is isomorphic to the group K j H ( F H ), the isomorphism i ∗ : K j H ( F H ) −→ K j G ( F YG ) is given explicitely as follows. There is a privileged element in the group K G × H ( T G G ) whichcorresponds to the class, in R ( H ), of the trivial representation of H . This class is in fact the class of the G × H -equivariant G -transversally elliptic symbol on G , associated with the zero operator 0 : C ∞ ( G ) → i ∗ : K j H ( F H ) −→ K j G ( F YG ) . (14)As we prove below, this isomorphism allows to reduce the index problem for leafwise H -transversally ellipticoperators on foliated H -manifolds to the index problem for leafwise G -transversally elliptic operators onfoliated G -manifolds. Notice that Y is the base of the principal H -fibration G × M → Y and we are exactlyin position to apply the properties of the index morphism with respect to free actions, see Subsection 3.1.Furthermore, since the compact Lie group G is assumed to be connected here, the C ∗ -algebra upstairs, thatis C ∗ ( G × M, G × F ) is Morita equivalent, and in fact isomorphic when F is not the zero foliation [11, 32],to C ∗ ( M, F ). Hence we end up with a KK-equivalence that we denote by ǫ ∈ KK( C ∗ ( M, F ) , C ∗ ( Y, F Y )). Theorem 4.3. [1] For j ∈ Z , the following diagram commutes K j H (cid:0) F H (cid:1) i ∗ / / Ind F (cid:15) (cid:15) K j G ( F YG ) Ind F Y (cid:15) (cid:15) KK j (cid:0) C ∗ H, C ∗ ( M, F ) (cid:1) [ i ] ⊗ C ∗ H • ⊗ C ∗ ( M, F ) ǫ / / KK j (cid:0) C ∗ G, C ∗ ( Y, F Y ) (cid:1) . Proof.
Recall the Kasparov class E q ∈ KK (cid:0) C ∗ ( G × M, G × F ) , C ∗ ( Y, F Y ) (cid:1) ≃ KK (cid:0) K ( L ( G )) ⊗ C ∗ ( M, F ) , C ∗ ( Y, F Y ) (cid:1) H -fibration q : G × M → Y = G × H M . If wedenote by µ ( G ) ∈ KK( C , K ( L ( G ))) the standard KK-equivalence then we have by definition ǫ = µ ( G ) ⊗ K ( L ( G )) E q . Let now a ∈ K j H ( F H ) be fixed. By Theorem 3.1, we know thatInd F Y ( i ∗ a ) = χ H ⊗ C ∗ H Ind G ×F ( q ∗ ( i ∗ a )) ⊗ C ∗ ( G × M,G ×F ) E q , where χ H ∈ KK( C , C ∗ H ) is the class of the trivial representation of H and where in the present case q ∗ ( i ∗ a )is just the isomorphic class to i ∗ a through the identification (13) and thus coincides by definition of i ∗ with[ σ (0)] · a in the product (12). ThusInd G ×F ( q ∗ ( i ∗ a )) = Ind G ×F ([ σ (0)] · a ) ∈ KK j ( C ∗ H ⊗ C ∗ G, K ( L ( G )) ⊗ C ∗ ( M, F )) . We can now apply the multiplicative property of the index from Theorem 3.14 to computeInd G ×F ([ σ (0)] · a ) = d Ind G ,H ([ σ (0)]) ⊗ C ∗ H Ind F ( a ) . For simplicity the KK-equivalence class µ ( G ) is often removed from the formulae, it is only used to naturallyidentify, in K -theory, K ( L ( G )) with C . The index class d Ind G ,H ([ σ (0)]) ∈ KK( C ∗ G ⊗ C ∗ H, C ∗ H ) reduceshere to the image under the Kasparov descent map, for the trivial H -action, of the H -equivariant indexclass in KK H ( C ∗ G, C ), of the G -transversally elliptic operator 0 : C ∞ ( G ) →
0. By gathering the previousequalities, we finally getInd F Y ( i ∗ ( a )) = χ H ⊗ C ∗ H Ind G ×F ( q ∗ ( i ∗ a )) ⊗ C ∗ ( G × M,G ×F ) E q = χ H ⊗ C ∗ H (cid:20)d Ind G ,H ([ σ (0)]) ⊗ C ∗ H Ind F ( a ) (cid:21) ⊗ C ∗ ( G × M,G ×F ) E q = χ H ⊗ C ∗ H (cid:20)d Ind G ,H ([ σ (0)]) ⊗ C ∗ H Ind F ( a ) ⊗ C µ ( G ) (cid:21) ⊗ C ∗ ( G × M,G ×F ) E q = (cid:18) χ H ⊗ C ∗ H d Ind G ,H ([ σ (0)]) (cid:19) ⊗ C ∗ H Ind F ( a ) ⊗ C ∗ ( Y, F Y ) (cid:0) µ ( G ) ⊗ K ( L ( G )) E q (cid:1) where we have used associativity of the Kasparov product. The proof is now complete since we have χ H ⊗ C ∗ H d Ind G ,H ([ σ (0)]) = [ i ] and µ ( G ) ⊗ K ( L ( G )) E q = ǫ. Remark 4.4.
Theorem 4.3 allows to extract information on the index morphism for the action of thecompact Lie group H using all such compact connected Lie groups G and their induced actions on theMorita equivalent foliation ( Y, F Y ). Such G always exists as any compact Lie group is isomorphic to aclosed subgroup of a unitary group.We fix for the rest of this section a compact connected Lie group G and a smooth closed foliated manifoldwhich is endowed with an action of G by leaf-preserving diffeomorphisms. For simplicity, we shall denotethis new G -foliation again by ( M, F ) since we shall again need to build up the new foliation ( Y, F Y ) byusing a particular closed subgroup of G , so no confusion should occur. Since G is connected this action is a31olonomy action and we may apply all the results of the previous sections. In order to compute the indexmorphism for leafwise G -transversally elliptic operators, we shall use a maximal torus T in G and we usethe induced action of T to define the Morita equivalent G -foliation ( Y, F Y ) as explained above. However,since the action of T on ( M, F ) is now the restriction of an action of the whole group G , this foliation iseasier to describe. More precisely, the map ( g, m ) → ( gH, g · m ) is a G -equivariant diffeomorphism whichallows to identify the foliation ( Y, F Y ) with the foliation ( G/ T × M, G/ T × F ). We quote for later use that C ∗ ( Y, F Y ) coincides here with C ∗ ( M, F ) ⊗ K ( L ( G/ T )) which in turn, when F is not the zero foliation, iseven isomorphic to C ∗ ( M, F ). Notice also that there is hence a well defined productK j G ( F G ) ⊗ K G ( T ( G/ T )) −→ K j G ( F YG ) . (15)Recall that G/ T carries a G -invariant complex structure and we may use the Dolbeault operator ∂ . Thisis an elliptic G -invariant operator on the rational variety G/ T whose G -index equals 1 ∈ R ( G ) since onlythe zero-degree Dolbeault cohomology space is non trivial, see [1], i.e.Ind( ∂ ) = 1 ∈ R ( G ) . The product by the symbol class [ σ ( ∂ )] ∈ K G ( T ( G/H )) in (15) allows to define the morphism β : K G ( F G ) −→ K G ( F YG ) . Recall the isomorphism i ∗ defined in Equation (14) as well as the KK-class [ i ] introduced in Definition4.1. We use these notations for the torus closed subgroup H = T to state the following Theorem 4.5.
Let T be a maximal torus of the compact connected Lie group G . Denote by r G T : K j G ( F G ) → K j T ( F T ) the composite map r G T := ( i ∗ ) − ◦ β . Then for j ∈ Z the following diagram commutes: K j G ( F G ) r G T / / Ind F (cid:15) (cid:15) K j T ( F T ) Ind F (cid:15) (cid:15) KK j ( C ∗ G, C ∗ ( M, F )) KK j ( C ∗ T , C ∗ ( M, F )) . [ i ] ⊗ C ∗ T • o o Proof.
We apply the multiplicative property of our index morphism from Theorem 3.14. In the notationsof Theorem 3.14, we take for H the trivial group, for ( M, F ) the G -manifold G/ T with one leaf, and for( M ′ , F ′ ) our G -foliation here, that is the foliation ( M, F ) used in the statement of Theorem 4.5. Then weobtain the commutativity of the following diagram (recall that C ∗ ( Y, F Y ) = K ( L ( G/ T )) ⊗ C ∗ ( M, F ) andhence can be replaced by C ∗ ( M, F )): K G ( T ( G/ T )) ⊗ K j G ( F G ) • ♯ • / / c Ind G ⊗ Ind F (cid:15) (cid:15) K j G ( F YG ) Ind F Y (cid:15) (cid:15) KK( C ∗ G, C ∗ G ) ⊗ KK j ( C ∗ G, C ∗ ( M, F )) • ⊗ C ∗ G • / / KK j ( C ∗ G, C ∗ ( M, F )) . We recall that c Ind G = j G ◦ Ind G where Ind G : K G ( T ( G/ T )) → R ( G ) ≃ KK G ( C , C ) is the usual Atiyah-Singer G -index that we view as valued in the Kasparov group KK G ( C , C ) and j G : KK G ( C , C ) → KK( C ∗ G, C ∗ G ) is theKasparov descent map for the trivial G action on C . In particular, c Ind G ( ∂ ) coincides with the unit of the ringKK( C ∗ G, C ∗ G ). If we thus apply this multiplicativity result to a given a ∈ K j G ( F G ) and to the Dolbeault symbol,then we get Ind F Y ( β ( a )) = c Ind G ( ∂ ) ⊗ C ∗ G Ind F ( a ) = Ind F ( a ) and so Ind F Y ◦ β = Ind F . The proof is now complete since we already proved in Theorem 4.3 the compatibility of the index morphism with themap i ∗ . Naturality of the index morphism
We now apply the previous results to give the allowed topological construction of an index map which willbe compared with our analytical index map from Proposition 2.8.
Let ι : ( M, F ) ֒ → ( M ′ , F ′ ) be a foliated embedding of G -foliations. So we assume that the compact Lie groupacts on M and on M ′ by leaf-preserving holonomy diffeomorphisms and that ι : M ֒ → M ′ is a G -equivariantembedding which sends leaves inside leaves. We assume for simplicity that M is compact, since this is theonly needed situation for the proof of our index theorem. We denote by N := ι ∗ T M ′ /T M the normal bundleto ι . In view of the construction of the topological index in Subsection 5.1, we shall only need the case wherethe transverse bundles τ := T M/F and τ ′ = T M ′ /F ′ do fit under ι , i.e. that ι ∗ τ ′ ≃ τ . In other words,the manifold M embeds transversally in M ′ to the foliation F ′ , i.e. ∀ x ∈ M , F ι ( x ) + dι ( T x M ) = T ι ( x ) M ′ and the foliation F = ι ∗ F ′ is the pull back foliation, see [24, 26] for more details. As a consequence, the G -equivariant embedding dι : F → F ′ , obtained by differentiating ι and restricting to F , is K -oriented by a G -equivariant complex structure. Indeed, under this assumption, the normal bundle N is identified with thenormal bundle to the leaves of F inside the leaves of F ′ , and it is easy then to see that the normal bundle N ′ to dι is isomorphic to the bundle π ∗ F ( N ⊗ C ) with π F : F → M being the bundle projection. Following[1], we deduce for any j ∈ Z , a well defined Thom R ( G )-morphism ι ! : K j G ( F G ) −→ K j G ( F ′ G ) . More precisely, denote by π : N ′ → F the bundle projection of the normal bundle N ′ to F in F ′ , andlet ( π F ◦ π ) ∗ (Λ • ( N ⊗ C )) be the associated exterior algebra over N ′ . Together with exterior multiplicationby the underlying vector, this defines a complex over N ′ which is exact off the zero section F ⊂ N ′ andwhich is denoted λ ( N ⊗ C ). The usual Thom isomorphism K G ( F ) → K G ( N ′ ) is defined by assigning to agiven compactly supported G -complex ( E, σ ) over F the compactly supported G -complex over N ′ given by π ∗ ( E, σ ) · λ ( N ⊗ C ). See [3] for more details. On the other hand, the total space of the fibration π : N ′ → F is G -equivariantly diffeomorphic to a G -stable open tubular neighborhood p : U ′ → F of dι ( F ) in F ′ and thisallows to define classically the Gysin map ι ! : K G ( F ) → K G ( F ′ ). As explained in [1], if we only assume that( E, σ ) represents a class in K G ( F G ), then the complex π ∗ ( E, σ ) · λ ( N ⊗ C ) over N ′ extends to an elementof K G ( F ′ G ). More precisely, if we assume that ( E, σ ) is only compactly supported when restricted to F G ,that is Supp( E, σ ) ∩ F G is compact, then the G -complex π ∗ ( E, σ ) · λ ( N ⊗ C ) yields a compactly supported G -complex over an open subspace U ′ G of F ′ G defined as follows. If we identify similarly the total space N with a G -stable open tubular neighborhood U of ι ( M ) in M ′ , then the foliation F ′ induces by restrictionto the open submanifold U a foliation F U . Then U ′ can be naturally identified with the total space F U of the leafwise tangent bundle of the foliation F U . The subspace U ′ G is then simply F UG = F U ∩ T G U . Tosum up, we deduce in this way a well defined Thom homomorphism of R ( G )-modules K G ( F G ) −→ K G ( U ′ G )(see again [1]). Since U ′ G is an open subspace of the locally compact space F ′ G , the C ∗ -algebra C ( U ) isa G -stable ideal in the G -algebra C ( F G ) and we have the extension R ( G )-morphism K G ( U ′ G ) → K G ( F ′ G ).Composing the Thom homomorphism with this extension map, we end up with our Gysin R ( G )-morphism ι ! : K G ( F G ) −→ K G ( F ′ G ) . Starting with a class in K ( F G ) we get in the same way a class in K ( F ′ G ) and we finally get the morphism ι ! : K j G ( F G ) −→ K j G ( F ′ G ) for j ∈ Z . The G -embedding ι gives a submersion M → M ′ / F ′ in the sense of [26], we hence deduce from [26, Section4] the well defined Connes-Skandalis Morita extension element ǫ ι ∈ KK( C ∗ ( M, F ) , C ∗ ( M ′ , F ′ )). Indeed,33he submanifold ι ( M ) is a transverse G -submanifold in ( M ′ , F ′ ) which inherits a foliation F ι ( M ) which isdiffeomorphic to ( M, F ), hence identifying C ∗ ( M, F ) with C ∗ ( ι ( M ) , F ι ( M ) ) and using the G -equivariantMorita equivalence of ( ι ( M ) , F ι ( M ) ) with a foliation ( U, F U ) obtained as an open tubular neighborhood of ι ( M ) in M ′ , we get the easy definition of the Connes-Skandalis map in our case. Theorem 5.1.
Let ( M ′ , F ′ ) be a smooth G -foliation. Let ι : M ֒ → M ′ be a G -equivariant embedding of aclosed G -manifold M which is transverse to the foliation F ′ and denote by F = ι ∗ F ′ the pull back foliation.We assume that G acts by leaf-preserving holonomy diffeomorphisms on the foliations ( M, F ) and ( M ′ , F ′ ) .Then for any j ∈ Z , the following diagram commutes: K j G ( F G ) ι ! / / Ind F (cid:15) (cid:15) K j G ( F ′ G ) Ind F′ (cid:15) (cid:15) KK j ( C ∗ G, C ∗ ( M, F )) ⊗ C ∗ ( M, F ) ǫ ι / / KK j ( C ∗ G, C ∗ ( M ′ , F ′ )) . Here the index morphism
Ind F ′ is defined according to Proposition 3.7.Proof. For simplicity, we shall identify ( ι ( M ) , F ι ( M ) ) with ( M, F ) and assume that M is a smooth transversesubmanifold to the foliation ( M ′ , F ′ ) whose foliation F coincides with the restricted foliation generated by T F ′ | M ∩ T M . By definition, ι ! is the composite map of a Thom morphism from K G ( F G ) to K G ( F UG ) withthe extension corresponding to the inclusion of the open submanifold U . Notice that since the G -action on( M, F ) is a holonomy action, it is also a holonomy action on ( U, F U ). By the excision theorem, the indexmorphism does automatically respect the latter extension map and the following diagram commutes:K j G ( F UG ) / / Ind F U (cid:15) (cid:15) K j G ( F ′ G ) Ind F′ (cid:15) (cid:15) KK j ( C ∗ G, C ∗ ( U, F U )) / / KK j ( C ∗ G, C ∗ ( M ′ , F ′ )) . Now the open submanifold U can be identified with a vector bundle N → M which is the normal bundle to M in M ′ . Moreover, the foliation F U is then identified with the foliation of N whose leaves are given by the totalspaces of the restrictions of the bundle π N : N → M to the leaves of ( M, F ). It is thus sufficient to show thetheorem in the case of a real G -vector bundle N over M foliated by F N := ker( T N → T M/F ) ≃ π ∗ N ( F ⊕ N )and with ι ! : K G ( F G ) → K G ( F NG ) being the Thom homomorphism associated to the 0-section ζ : M → N .Following [1], we can write N = P × O ( n ) R n , where q : P → M is a G -equivariant O ( n )-principal bundleover M , foliated by F P := ker( T P → T M/F ). Denote by q : P × R n → N the G -equivariant projectioncorresponding to moding out by the action of O ( n ). Let F P × R n be the foliation given by F P × R n on P × R n and by F P × R n the tangent bundle to this foliation.By using the product defined in (10) with G replaced by G × O ( n ) and with trivial H , we obtain the welldefined product: K j G × O(n) ( F PG × O ( n ) ) ⊗ K G × O(n) ( T R n ) −→ K j G × O(n) ( F P × R n G × O ( n ) ) . (16)But since O ( n ) acts freely on P , we also have the following identifications: q ∗ : K j G ( F G ) ≃ −→ K j G × O(n) ( F PG × O ( n ) ) and q ∗ : K j G ( F NG ) ≃ −→ K j G × O(n) ( F P × R n G × O ( n ) ) . Therefore, we end up with the product:K j G ( F G ) ⊗ K G × O(n) ( T R n ) −→ K j G ( F NG ) . (17)34ince G acts trivially on R n , the inclusion i : { } ֒ → R n induces the Bott morphism i ! : R ( G × O ( n )) → K G × O(n) ( T R n ) and we have Ind( i ! (1)) = 1 ∈ KK G × O(n) ( C , C ), see [3]. Now, multiplication by i ! (1) in (17)is exactly the Thom morphism that we denote ζ ! since ζ is the zero section here, i.e. ζ ! : K j G ( F G ) −→ K j G ( F NG ) . Moreover, we may as well consider the multiplication by i ! (1) in the product (16), and we then obviouslyhave the following commutative diagram:K j G ( F G ) ζ ! / / q ∗ (cid:15) (cid:15) K j G ( F NG ) q ∗ (cid:15) (cid:15) K j G × O(n) ( F PG × O ( n ) ) · i ! (1) / / K j G × O(n) ( F P × R n G × O ( n ) )We deduce that for any a ∈ K j G ( F G ):Ind F P × R n ( q ∗ ( ζ ! ( a )) = Ind F P × R n ( q ∗ ( a ) · i ! (1)) = j G × O ( n ) (cid:0) (cid:1) ⊗ C ∗ ( G × O ( n )) Ind F P ( q ∗ ( a )) ⊗ µ ( R n ) , where the last equality is a consequence of the multiplicativity axiom satisfied by our index morphism, asstated in Theorem 3.14, and where µ ( R n ) ∈ KK( C , C ∗ ( R n × R n )) is the Morita equivalence. On the otherhand, by the axiom for free actions stated in Theorem 3.1, and denoting by χ O ( n )1 the trivial representationof O ( n ), we have:Ind F ( a ) = χ O ( n )1 ⊗ C ∗ O ( n ) Ind F P ( q ∗ ( a )) ⊗ C ∗ ( P, F P ) E q , whileInd F N ( ζ ! ( a )) = χ O ( n )1 ⊗ C ∗ O ( n ) Ind F P × R n ( q ∗ ( ζ ! ( a )) ⊗ C ∗ ( P × R n , F P × R n ) E q . We finally conclude by gathering the previous relations as follows:Ind F N ( ζ ! ( a )) = χ O ( n )1 ⊗ C ∗ O ( n ) (cid:18) Ind F P ( q ∗ ( a )) ⊗ µ ( R n ) (cid:19) ⊗ C ∗ ( P × R n , F P × R n ) E q , = χ O ( n )1 ⊗ C ∗ O ( n ) Ind F P ( q ∗ ( a )) ⊗ C ∗ ( P, F P ) (cid:0) µ ( R n ) ⊗ C ∗ ( R n , R n ) E q (cid:1) = χ O ( n )1 ⊗ C ∗ O ( n ) Ind F P ( q ∗ ( a )) ⊗ C ∗ ( P, F P ) (cid:0) E q ⊗ C ∗ ( M, F ) E ζ (cid:1) = (cid:18) χ O ( n )1 ⊗ C ∗ O ( n ) Ind F P ( q ∗ ( a )) ⊗ C ∗ ( P, F P ) E q (cid:19) ⊗ C ∗ ( M, F ) E ζ = Ind F ( a ) ⊗ C ∗ ( M, F ) E ζ . We have used that E q ⊗ C ∗ ( M, F ) E ζ = µ ( R n ) ⊗ C ∗ ( R n , R n ) E q which is a consequence of the equality ζ ◦ q = q ◦ s where s : P ֒ → P × R n is the zero section of this trivial bundle. We prove the following important proposition.
Proposition 5.2.
Let ( M, F ) be a smooth foliated riemannian G -manifold such that G acts by leaf-preserving holonomy diffeomorphisms. Assume that we are given an isometric G -embedding i : M ֒ → E of M in a finite dimensional euclidean G -representation E . Let A := { ( x, ξ, η ) ∈ M × T E, η ∈ di x ( F x ) ⊥ } and ι : F ֒ → A be the G -embedding given by ι ( x, ξ ) = ( x, ξ, . Then . The map ι is K -oriented by a complex G -structure;2. A is diffeomorphic to a smooth G -submanifold A of the cartesian product M × T ( E ) , which is an opentransversal to the smooth foliation F × ;3. Setting A G := A ∩ ( M × T G ( E )) , the usual Thom construction yields, for j ∈ Z , a well defined R ( G ) -homomorphism ι ! : K j G ( F G ) −→ K j G ( A G ) ;4. The space A G is a topological ( G -stable) transversal to the foliated space ( M × T G ( E ) , F × .Proof. A direct inspection shows that the normal bundle to the G -embedding ι is isomorphic to di x ( F x ) ⊥ ⊕ di x ( F x ) ⊥ . This gives the first item. For ε >
0, denote by A ε the set of points ( x, v, w ) ∈ A with k w k < ε .The map A → M × T E given by ( x, v, w ) ( x, v, i ( x ) + w ) then clearly identifies, for ε > A ε with an open transversal to ˜ F = F × { } in M × T E . The transversal condition is given in ( x, ξ, F x + T ( x,ξ, A = F x + ( T x M ) ⊕ T x E ⊕ di x ( F x ) ⊥ = T ( x,ξ, ( M × T E ). The third and fourth items areeventually easily deduced by standard arguments that we already explained in the previous section, see [1]and [26].We point out that, exactly as in the case of smooth foliations, the topological transversal A G to thefoliated space ( M × T G ( E ) , F × ǫ ∈ KK( C ( A G ) , C ∗ ( M × T G ( E ) , F × R ( G )-morphism ǫ : K j G ( A G ) −→ K G j ( C ( T G ( E ) , C ∗ ( M, F ))) . The class ǫ can be described as follows. By using that the normal bundle to A in M × T ( E ) is isomorphicto the vector bundle F × A , we may consider an open tubular neighborhood N of A in M × T ( E ) which is a disc-bundle over A whose fibers are small disc-placques which correspond to therestricted foliation F × N . It is then clear by construction that N G := N ∩ ( M × T G ( E )) is also adisc-fibration by the same placques but now over the space A G , so the base is no more a smooth manifold.The C ∗ -algebra C ∗ ( N G , F N G ) of the lamination F N G of the open subspace N G which is the restriction ofthe foliation F × M × T G ( E ), is then Morita equivalent to C ( A G ). Hence using the trivial extensionmap K G j ( C ∗ ( N G , F N G )) −→ K G j ( C ∗ ( M × T G ( E ) , F × ≃ K G j ( C ( T G ( E ) , C ∗ ( M, F ))) , corresponding to the open subspace N G in the space M × T G ( E ), we finally obtain the allowed quasi-trivial G -equivariant extension map ǫ .Following Kasparov, we define a Dirac element [ D E ] ∈ KK( C ( T G ( E )) ⋊ G, C ) which, according to themain result of [41], computes the index of G -invariant G -transversally elliptic operators on the orthogonal G -representation E , through the descent morphism j G . There are though some technical details which arepassed over here and which would need to be expanded elsewhere. One especially needs to replace C ( T G E )by a better (although non-separable) symbol C ∗ -algebra denoted by S G ( E ) in [41], and therefore one needsas well to use the extended version of Kasparov’s KK-theory, adapted to non-separable algebras. All thesedetails with their generalizations to foliations will be dealt with in a forthcoming paper.We only mention here that since C ∗ ( M, F ) is endowed with the trivial G -action, we have a well definedmorphismK G j ( C ( T G ( E )) ⊗ C ∗ ( M, F )) j G −→ KK j ( C ∗ G, [ C ( T G ( E )) ⋊ G ] ⊗ C ∗ ( M, F )) ⊗ [ D E ] −→ KK j ( C ∗ G, C ∗ ( M, F )) , that we denote by ∂ E ⊗ C ∗ ( M, F ). Roughly speaking and using the main result of [41], the map ∂ E ⊗ C ∗ ( M, F )is the expected index map for G -invariant G -transversally elliptic operators on E with coefficients in the G -trivial C ∗ -algebra C ∗ ( M, F ). Remark 5.3.
The composite morphism Ind F ,top :Ind F ,top : K j G ( F G ) ι ! −→ K j G ( A G ) ǫ −→ K G j ( C ( T G ( E )) ⊗ C ∗ ( M, F )) ∂ E ⊗ C ∗ ( M, F ) −→ KK j ( C ∗ G, C ∗ ( M, F ))is independent of the choice of euclidean G -representation E with the isometric G -embedding i .36 efinition 5.1. The morphismInd F ,top : K j G ( F G ) −→ KK j ( C ∗ G, C ∗ ( M, F )) , will be called the topological index morphism for G -invariant leafwise G -transversally elliptic operators.If G is the trivial group then the topological index morphism reduces to the topological index morphismfor leafwise elliptic operators as defined in [26], and it then coincides with the analytic index morphismInd F , this is precisely the Connes-Skandalis index theorem. For general G and when the foliation is topdimensional, the naturality of the index distribution proved in [1] together with the Kasparov index theoremproved in [41] implies again the equality of the topological index morphism with the analytical one. Remark 5.4.
When G and M are no more compact, but the G -action is supposed to be proper andcocompact as in [41], then the proofs given here allow to still define the index morphismInd M, F : K j G ( F G ) −→ KK j ( C ( M ) ⋊ G, C ∗ ( M, F )) . We finally point out that most of the constructions given in the present paper apply, with minor changes,to the more general category of foliated spaces (e.g. laminations) as studied in [50] using sections and oper-ators which are leafwise smooth and transversally continuous. However, the construction of the topologicalindex for instance is not clear in general since the G -embedding in E is not ensured a priori. A Unbounded version of the index class
We define in this appendix the index class for operators of order 1, using the unbounded version of Kasparov’stheory [5]. The unbounded version simplifies the computation of some Kasparov products and was used inthe present paper. In order, to achieve this construction, we shall need the following independent theoremwhich generalizes results from [58], see also [59]. We assume as in this whole paper that G acts on ( M, F )by holonomy diffeomorphisms. This is true for instance when G is connected. Most of this appendix will bedevoted to the proof of the following Theorem A.1.
Let P be a G -invariant leafwise G -transversally elliptic operator. Then the closure P of P is a regular operator. Moreover, the adjoint operator P ∗ of P coincides with the closure of the leafwiseformal adjoint P ♮ of P , i.e. P ∗ = P ♮ . If P is a leafwise pseudodifferential operator of order m > M acting between the hermitian bundles E and E ′ , then we have h P η, η ′ i = h η, P ♮ η ′ i , η ∈ C ∞ c ( G , r ∗ E ) and η ′ ∈ C ∞ c ( G , r ∗ E ′ ) . The Hilbert module completions of C ∞ c ( G , r ∗ E ) and C ∞ c ( G , r ∗ E ′ ) are respectively denoted E and E ′ . The op-erator P is densely defined with domain C ∞ c ( G , r ∗ E ) and has a well defined closure P . The same observationholds for the leafwise pseudodifferential operator P ♮ . Then P ♮ ⊂ P ∗ .We shall use notations and discussions from [58, 59] and especially the following Lemma A.2. [59] Let A , B be compactly supported leafwise pseudodifferential operators such that ord A +ord B ≤ and ord B ≤ . Then we have AB = A B , an equality of adjointable operators.
The following proposition from [59] will also be needed:
Proposition A.3. [59] Let P be an elliptic, compactly supported leafwise pseudodifferential operator. Thenthe operator P is a regular operator. Recall that if E is a Hilbert G -module over a G -trivial C ∗ -algebra A , and π : C ∗ G → L A ( E ) is the inducedrepresentation, then for any α ∈ ˆ G the α -isotypical component E α of E is the Hilbert A -submodule which isthe image of the projection p α = π ((dim α ) χ α ). Here χ α is the character of α .37 roposition A.4. Let A be a G -trivial C ∗ -algebra and let E , E ′ be Hilbert G -modules on A . Denote as aboveby E α and E ′ α the α -isotypical component of E and E ′ respectively, and by p α the corresponding projections.We also denote by i α the inclusion of E α in E as well as the inclusion of E ′ α in E ′ . Then1. E = L α ∈ ˆ G E α as a countable Hilbert A -module decomposition and the same holds for E ′ . In particular,any G -invariant adjointable operator from E to E ′ is diagonal with respect to this decomposition, i.e. L A ( E , E ′ ) G = L α ∈ ˆ G L A ( E α , E ′ α ) G .2. Let P = ( P, dom( P )) be a closable G -invariant operator from E to E ′ such that ( i α p α ) dom( P ) ⊂ dom( P ) , then P α = ( P | E α , p α dom( P )) is closable and P α = (cid:0) P (cid:1) α , for any α ∈ ˆ G . Remark A.5.
Hence, the α -component of any closed G -invariant operator is closed. Indeed, in this casethe inclusion ( i α p α ) dom( P ) ⊂ dom( P ) is automatically fulfilled. Proof.
1. Notice first that G is a compact Lie group therefore ˆ G is countable. Our hypothesis that the G -action on A is trivial implies that G acts by adjointable unitaries ( U g ) g ∈ G on E . This action extends toan adjointable representation π of C ∗ G given for any ϕ ∈ C ( G ) by π ( ϕ ) η = ´ G ϕ ( g ) U g η dg. Now recall that[(dim α ) χ α ] α ∈ ˆ G is a family of projections in C ∗ G such that ((dim α ) χ α )((dim β ) χ β ) = 0 whenever α = β andId C ∗ G = P α ∈ ˆ G (dim α ) χ α as a multiplier of C ∗ G . Therefore, the family of (adjointable) projections ( p α ) α ∈ ˆ G defined by p α = π ((dim α ) χ α ) satisfies the same properties, in particular one has Id E = P α ∈ ˆ G p α and p α p β =0 if α = β. Notice that under our assumptions, π ( C ∗ G ) E is automatically dense in E as can be seen by usingan approximate identity in C ∗ G composed of continuous non-negative functions and the precise expressionof π . Since a G -invariant operator T ∈ L A ( E , E ′ ) G commutes with the representations both denoted π in E and E ′ , it commutes with each p α and hence the first item is proved.2. We have P α = p α P i α where again we have denoted by the same letters p α and i α the operators for E and E ′ . In particular P α p α = p α P on dom( P ) and i α P α = P i α on dom( P α ) = p α dom( P ). The graph of P α is the image under p α × p α of the graph of P . Since p α and i α are continuous, the conclusion follows byusing that dom( P ) is preseved by i α p α .It is worthpointing out that under the assumptions of Proposition A.4, if the closure P of P is regularthen for any α ∈ b G , P α = P α is regular. The converse is also true and is used below to deduce our theoremA.1. Indeed, notice the following general observation whose proof is a direct inspection of the graphs: Lemma A.6.
Let ( E k ) k ∈ N and ( E ′ k ) k ∈ N be two sequences of Hilbert A -modules and let E = ⊕ k ≥ E k and E ′ = ⊕ k ≥ E ′ k be the Hilbert direct sums. Let ( T k : dom( T k ) ⊂ E k → E ′ k ) k ≥ be a sequence of regularoperators. Let T : dom( T ) ⊂ E → E ′ be the direct sum operator given by dom( T ) := { x = ( x k ) k ≥ ∈ ⊕ k dom( T k ) | ( T k ( x k )) k ≥ ∈ E ′ } and T (( x k ) k ≥ ) = ( T k ( x k )) k ≥ . Then ( T, dom( T )) is a regular operator. We are now in position to prove our Theorem A.1.
Proof of Theorem A.1.
Let ∆ G be the Laplace Casimir operator along the orbits introduced in Subsection1.2, i.e. ∆ G = P L ( V k ) for an orthonormal basis ( V k ) k of g , see again Subsection 1.2. We use again thesame notation for the Casimir operators on both E and E ′ . Set B := P ♮ P + ∆ mG and C = P P ♮ + ∆ mG ,then B and C are G -invariant leafwise elliptic operators of order 2 m . Indeed, the principal symbol of B isa pointwise non-negative linear map and we have for any ( x, ξ ) ∈ F : h σ ( B )( x, ξ ) u, u i = | σ ( P )( x, ξ ) u | + q x ( ξ ) m | u | . q x ( ξ ) = 0 only happens if ξ ∈ ( F G ) x , and since σ ( P )( x, ξ ) is invertible for any ξ ∈ ( F G ) x r G -transverse ellipticity of P , we deduce that σ ( B )( x, ξ ) is invertible for any ξ ∈ F x r
0. The argument for C is completely similar. Denote then by Q a G -invariant leafwise parametrix for B and similarly by ˜ Q a G -invariant leafwise parametrix for C . Set R = id − BQ, S = id − QB, ˜ R = id − C ˜ Q and ˜ S = id − ˜ QC.
Since S and Q are negative order operators they extend to adjointable operators. In particular, for any λ ∈ R , S + λQ = S + λQ = S + λQ . We have ord( P ) + ord( QP ♮ ) = 0 and ord( P ) + ord( S + λQ ) ≤ P QP ♮ = P QP ♮ and P ( S + λQ ) = P ( S + λQ ) . Therefore, im( QP ♮ ) + im( S + λQ ) ⊂ dom( P ) . Denote by b ∆ G the Laplacian on G , we have that ∆ mG π ( ϕ ) = π ( b ∆ mG ϕ ) for any ϕ ∈ C ∞ ( G ). But b ∆ G χ α = λ α χ α with λ α ≥ α and positive if α isdifferent from the trivial representation. Therefore(∆ mG ) α = ∆ mG p α i α = π (cid:16) (dim α ) b ∆ mG χ α (cid:17) i α = λ mα Id E α . On the other hand composition with p α on the left and with i α on the right in the first parametrix relationyields using Proposition A.4 to the following relation on p α C ∞ c ( G , r ∗ E ):( QP ♮ ) α P α = p α − λ mα Q α − S α . This allows to prove the inclusion dom( P α ) ⊂ im( QP ♮α ) + im( S α + λ mα Q α ). Recall that we already provedthe opposite inclusion (before reducing to the α -isotypical component), in particular, we already proved thatim( QP ♮α ) + im( S α + λ mα Q α ) ⊂ dom( P α ) . Let then x ∈ dom( P α ) and x n ∈ dom( P α ) = p α C ∞ c ( G , r ∗ E ) such that x n → x and P α x n → P α x . We canthen write x n = ( S α + λ mα Q α ) x n + ( QP ♮ ) α P α x n . Since ( S + λ mα Q ) and QP ♮ are adjointable, we obtain that ( S + λ mα Q ) α and QP ♮α are adjointable. It followsby continuity that x = ( S + λ mα Q ) α x + ( QP ♮ ) α P α x. Therefore dom( P α ) ⊂ im( QP ♮α ) + im( S α + λ mα Q α ). We thus obtained the equalitydom( P α ) = im( QP ♮α ) + im( S α + λ mα Q α ) . (18)We similarly have by the same method the inclusion im( ˜ Q ♮ P ) + im( ˜ R ♮ + λ mα ˜ Q ♮ ) ⊂ dom( P ♮ ) . From theprevious assertions, we can now deduce that P ∗ α ⊂ P ♮α . Recall that ( P ♮ ˜ Q ) ∗ α P ∗ α ⊂ ( P α ( P ♮ ˜ Q ) α ) ∗ and let x ∈ dom( P ∗ α ) then x = ( P α ( P ♮ ˜ Q ) α ) ∗ x + ( λ mα ˜ Q + ˜ R ) ∗ α x = ( P ♮ ˜ Q ) ∗ α P ∗ α x + ( λ mα ˜ Q ∗ + ˜ R ∗ ) α x. It follows that x ∈ im(( P ♮ ˜ Q ) ∗ α ) + im(( λ mα ˜ Q ∗ + ˜ R ∗ ) α ). Since P ♮ ˜ Q and λ mα ˜ Q + ˜ R are negative order operators,we have ( P ♮ ˜ Q ) ♮ = ( P ♮ ˜ Q ) ∗ and ( λ mα ˜ Q + ˜ R ) ♮ = λ mα ˜ Q ♮ + ˜ R ♮ = ( λ mα ˜ Q + ˜ R ) ∗ = λ mα ˜ Q ∗ + ˜ R ∗ . This gives that x ∈ im(( P ♮ ˜ Q ) ♮α ) + im(( λ mα ˜ Q ♮ + ˜ R ♮ ) α ) = im( ˜ Q ♮ P α ) + im(( λ mα ˜ Q ♮ + ˜ R ♮ ) α ) . But we proved that im( ˜ Q ♮ P ) + im( ˜ R ♮ + λ mα ˜ Q ♮ ) ⊂ dom( P ♮ ) , hence x ∈ dom( P ♮α ) as allowed.39t remains to prove that P α is regular. The equality (18) shows that the graph of P α is given by G ( P α ) = { ( Q α P ♮α x + ( S α + λ mα Q α ) y, P α Q α P ♮α x + P α ( S α + λ mα Q α ) y ) , ( x, y ) ∈ E ′ α × E α } . Hence this graph coincides with the range of the adjointable operator from E ′ ⊕ E to E ⊕ E ′ given by U α = QP ♮α S α + λ mα Q α P α Q α P ♮α P α ( S α + λ mα Q α ) . Hence G ( P α ) = im( U α ) is orthocomplemented in E ⊕ E ′ as the range of an adjointable operator (with closedrange). It follows that P is regular by Lemma A.6. Since P ♮ is also G -transversally elliptic, P ♮ is alsoregular. Now we have seen above that P ♮α = P ∗ α for any α ∈ ˆ G , and this implies that P ♮ = P ∗ .In the sequel we will denote simply by P the regular operator obtained from a formally selfadjoint G -invariant leafwise G -transversally elliptic operator. Definition A.7 (Unbounded Kasparov module [5]) . Let A and B be C ∗ -algebras. An ( A, B )-unboundedKasparov cycle (
E, φ, D ) is a triple where E is a Hilbert B -module, φ : A → L ( E ) is a graded ⋆ -homomorphism and ( D, dom(D)) is an unbounded regular seladjoint operator such that:1. (1 + D ) − φ ( a ) ∈ k ( E ), ∀ a ∈ A ,2. The subspace of A composed of the elements a ∈ A such that φ ( a )(dom(D)) ⊂ dom(D) and [ D, φ ( a )] = Dφ ( a ) − φ ( a ) D is densely defined and extends to an adjointable operator on E , is dense in A .When E is Z -graded with D odd and π ( a ) even for any a , we say that the Kasparov cycle is even. Otherwise,it is odd.In [5], appropriate equivalence relations are introduced on such (even/odd) unbounded Kasparov cycles,which allowed to recover the groups KK ∗ ( A, B ). When the compact group G acts on all the above data,one recovers similarly KK ∗ G ( A, B ) by using the equivariant version of the Baaj-Julg unbounded cycles of theprevious definition. We are now in position to state the second main result of this appendix.
Theorem A.8.
Let P : C ∞ c ( G , r ∗ E + ) → C ∞ c ( G , r ∗ E − ) be a G -invariant leafwise G -transversally ellipticpseudodifferential operators of order , and let P be the associated regular self-adjoint operator defined by (cid:18) P ∗ P (cid:19) . Then the triple ( E , π, P ) is an even ( C ∗ G, C ∗ ( M, F )) -unbounded Kasparov cycle, which de-fines a class in KK( C ∗ G, C ∗ ( M, F )) . The similar statement holds in the ungraded case giving a class in KK ( C ∗ G, C ∗ ( M, F )) .Proof. For any ϕ ∈ C ( G ), it is easy to see that π ( ϕ ) preserves the domain of P and by Remark 1.7, we have[ π ( ϕ ) , P ] = 0. It remains to check that (1 + P ) − ◦ π ( ϕ ) ∈ K ( E ). We may take for our operator ∆ G theCasimir operator, which is a leafwise differential operator of order 2. We already noticed that the operator P + ∆ G is elliptic. We hence deduce that the resolvent (1 + P + ∆ G ) − is a compact operator in E .We now show that for any ϕ ∈ C ∞ ( G ), the operator(1 + P + ∆ G ) − ◦ π ( ϕ ) − (1 + P ) − ◦ π ( ϕ ) , is compact, which will ensure that (1 + P ) ◦ π ( ϕ ) is also compact. Denote by b ∆ G the Laplacian on G viewedas a riemannian G -manifold. Using that ∆ G π ( ϕ ) = π ( b ∆ G φ ) and that [ π ( ϕ ) , P ] = 0, we have:(1 + P + ∆ G ) − ◦ π ( ϕ ) − (1 + P ) − ◦ π ( ϕ ) = − (1 + P + ∆ G ) − ∆ G (1 + P ) − ◦ π ( ϕ )= − (1 + P + ∆ G ) − π ( b ∆ G ϕ )(1 + P ) − . π ( b ∆ G ϕ ) and (1 + P ) − are adjointable operators and since (1 + P + ∆ G ) − is compact, wededuce that (1 + P + ∆ G ) − ◦ π ( ϕ ) − (1 + P ) − ◦ π ( ϕ ) is a compact operator on E . Definition A.9.
The index class Ind F ( P ) of a G -invariant leafwise G -transversally elliptic pseudodifferen-tial operator of positive order m , is the class in KK ∗ ( C ∗ G, C ∗ ( M, F )) of the ( C ∗ G, C ∗ ( M, F ))-unboundedKasparov cycle ( E , π, P ).The relation with the bounded version is obtained by using the Woronowicz transform, see [5]. Moreprecisely, if P is a G -invariant leafwise G -transversally elliptic pseudodifferential operator of order 1 forinstance, then the triple ( E , π, P (1 + P ) − / ) is a (bounded) Kasparov cycle.The following proposition was kindly suggested to us by the referee. It allows to avoid using the regularityof the unbounded operator P and can be exploited by the interested reader to simplify many of the statementsof the present paper. Proposition A.10.
Let P be a formally selfadjoint G -invariant leafwise G -transversally elliptic operator oforder . Let g D G be the order pseudodifferential operator d ∗ G (1 + ∆) − / d G . Then1. T = P g D G g D G − P ! is elliptic and therefore its closure is regular;2. the triple ( E ⊕ E , π ⊕ , T ) defines a ( C ∗ G, C ∗ ( M, F )) -unbounded Kasparov cycle. In the even case withgrading γ of E , we take the grading γ ⊕ − γ .3. The Kasparov cycle ( E ⊕ E , π ⊕ , T √ T ) represents the index class Ind F ( P ) in KK( C ∗ G, C ∗ ( M, F )) .Proof. . We have σ ( T ) = σ ( P ) + σ ( g D G ) σ ( P ) + σ ( g D G ) ! which is invertible on S ∗ F . Indeed, σ ( T ) is a sum of non-negative endomorphisms and σ ( P ) is invertible on F G and σ ( g D G ) isinvertible along the orbits. It follows from Proposition A.3 that T is regular.2 . Since T is elliptic, we get that (id + T ) − is compact. If ϕ ∈ C ∞ ( G ) then g D G π ( ϕ ) = d ∗ G (id +∆) − / π ( dϕ )extends in an adjointable operator since d ∗ G (id +∆) − / has 0 order. Moreover, ( π ( ϕ ) ⊕ ⊂ domTby G -invariance of P and g D G , therefore (cid:2) T , π ( ϕ ) ⊕ (cid:3) is adjointable, and by density this is still true over C ∗ G . The operator T is of course odd for the grading since P is odd and D G is even.3 . Any homotopy of symbols yields an operator homotopy by the standard argument. The cycle( E , π, P √ id + P + D G ) is equivalent to ( E⊕E , π ⊕ , T √ T ) because ( E⊕E , π ⊕ , T t √ T t ) where T t = (cid:18) P tD G tD G − P (cid:19) is a homotopy with ( E ⊕ E , π ⊕ , (cid:18) P − P (cid:19) (1 + P + D G ) − / ) which is a Kasparov cycle because D G π ( ϕ )is adjointable. Indeed, we can assume ϕ non-negative in C ∗ G . Then D G π ( ϕ ∗ ∗ ϕ ) = π ( dϕ ) ∗ (id +∆) − / d G d ∗ G (id +∆) − / π ( dϕ ) , ∀ ϕ ∈ C ∞ ( G ) . Furthermore, (
E ⊕ E , π ⊕ , (cid:18) P − P (cid:19) (1 + T ) − / ) = ( E , π, P √ id + P + D G ) + ( E , , − P √ id + P + D G ), where thelast cycle is degenerate. It eventually follows that ( E , π, P √ id + P + D G ) is homotopy equivalent to ( E , π, P √ P ).41 The technical proposition in the non-compact case
Let us fixe a non compact foliated manifold ( U, F U ) and denote by Ψ ( U, F U , E ) the C ∗ -subalgebra of theadjointable operators L C ∗ ( U, F U ) ( E ), which is generated by the closures of the zero-th order pseudodifferentialoperator with symbols in C ∞ c ( S ∗ F U , End( E )), see [26, 51, 59]. Here E is the Hilbert C ∗ ( U, F U )-moduledefined before for the foliation ( U, F U ). Then setting K U ( E ) := { T ∈ L C ∗ ( U, F U ) ( E ) | T f & f T ∈ K C ∗ ( U, F U ) ( E ) , ∀ f ∈ C ( U ) } andΨ ( U, F U , E ) U := { T ∈ L C ∗ ( U, F U ) ( E ) | T f & f T ∈ Ψ ( U, F U , E ) , ∀ f ∈ C ( U ) } , the following exact sequence holds (see [26, Proposition 4.6]):0 → K U ( E ) / / Ψ ( U, F U , E ) U / / C b ( S ∗ F U , End( E )) → . (19) Proposition B.1.
Let ( U, F U ) be a (non compact) foliated manifold. Let A ∈ Ψ ( U, F U , E ) U be selfadjoint.Suppose that the principal symbol σ ( A ) of A satisfies ∀ ε > , ∃ c > such that ∀ ( x, ξ ) ∈ S ∗ F U : k σ ( A )( x, ξ ) k ≤ c σ ( Q )( x, ξ ) + ε. (20) Then ∀ ε > , there exist two selfadjoint operators R and R ∈ K U ( E ) such that: − ( c Q + ε + R ) ≤ A ≤ c Q + ε + R as self-adjoint operators on E . Proof.
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