Groups with Spanier-Whitehead duality
aa r X i v : . [ m a t h . K T ] F e b GROUPS WITH SPANIER–WHITEHEAD DUALITY
SHINTARO NISHIKAWA AND VALERIO PROIETTI
Abstract.
Building on work by Kasparov, we study the notion of Spanier–Whitehead K -duality for a discrete group. It is defined as duality in theKK-category between two C ∗ -algebras which are naturally attached to thegroup, namely the reduced group C ∗ -algebra and the crossed product for thegroup action on the universal example for proper actions. We compare thisnotion to the Baum–Connes conjecture by constructing duality classes basedon two methods: the standard “gamma element” technique, and the morerecent approach via cycles with property gamma. As a result of our analysis,we prove Spanier–Whitehead duality for a large class of groups, includingBieberbach’s space groups, groups acting on trees, and lattices in Lorentzgroups. Contents γ -element and the ( γ )-element 50.4. Equivariant Kasparov duality 60.5. Main results 71. General framework 81.1. Argument based on the ( γ )-element 101.2. Argument based on the γ -element 131.3. The torsion-free case 172. Examples 192.1. Groups with Spanier–Whitehead K -duality 192.2. Groups with weak Spanier–Whitehead K -duality 243. Some applications 24References 27 Introduction
Alexander duality applies to the homology theory properties of the complementof a subspace inside a sphere in Euclidean space. More precisely, for a finite complex X contained in S n +1 , if ˜ H denotes reduced homology or cohomology with coeffi-cients in a given abelian group, there is an isomorphism ˜ H i ( X ) ∼ = ˜ H n − i ( S n +1 \ X ),induced by slant product with the pullback of the generator µ ∗ ([ S n ]), via the dualitymap µ : X × ( S n +1 r X ) → S n , µ ( x, y ) = ( x − y ) / k x − y k .Ed Spanier and J. H. C. Whitehead generalized this statement and adapted itto the context of stable homotopy theory. Their basic intuition was that sphere Mathematics Subject Classification.
Key words and phrases.
Spanier–Whitehead duality, Poincaré duality, Baum–Connes conjec-ture, direct splitting method, noncommutative topology. complements determine the homology, but not the homotopy type, in general. How-ever, the stable homotopy type can be deduced and provides a first approximationto homotopy type [SW58]. Thus, the modern statement is phrased in terms ofdual objects
X, DX in the category of pointed spectra with the smash product asa monoidal structure, and by taking maps to an Eilenberg-MacLane spectrum onerecovers Alexander duality formally.The modern version of the duality implies Poincaré duality for compact manifoldsand extends in a natural way to generalized cohomology theories such as K -theory.In this setting, a compact spin c -manifold exhibits Poincaré duality in the sense thatthe K -homology class of the Dirac operator induces by cap product an isomorphism K ∗ ( M ) → K ∗ + n ( M ), where the shift is given by the dimension [Kas88].More generally, the bivariant version of K -theory introduced by Kasparov, whichwe shall use extensively in this paper, showcases a close relationship to Alexander-Spanier duality; by this we mean that for X, Y finite complexes one has a chain ofisomorphisms ([KS19])KK ∗ ( C ( X ) , C ( Y )) ∼ = KK ∗ ( C , C ( DX ∧ Y )) ∼ = K ∗ ( C ( DX ∧ Y )) ∼ = K ∗ ( DX ∧ Y ) . Having introduced C ∗ -algebras in this way, as they arise naturally in applicationsto topology, dynamics, and index theory, and are generally noncommutative, it isnatural to seek for generalizations of Spanier–Whitehead duality in the frameworkof noncommutative geometry.For a separable, nuclear C ∗ -algebra A represented on a Hilbert space, the com-mutant of its projection into the Calkin algebra has some of the properties reminis-cent of a Spanier–Whitehead K -dual. This is the Paschke dual of A , and satisfies K ∗ ( P ( A )) ∼ = K ∗ ( A ). However, in general P ( A ) is neither separable nor nuclear,the Kasparov product is not defined, so that it seems desirable to explore differentroutes for the definition of a K -dual.In [Con94] A. Connes introduced the appropriate formalism for this question,which shall be described shortly, and in [Con96] he showed the first nontrivialexample of a noncommutative Poincaré duality algebra, in the form of the irrationalrotation algebra. In [Eme03] H. Emerson proves the same result for the crossedproduct of a hyperbolic group acting on its Gromov boundary. Examples of pairs ofalgebras with Spanier–Whitehead duality were also given by Kaminker and Putnam[KP97] in the case of Cuntz–Krieger algebras associated respectively to M and itstranspose, where M is a square { , } -valued matrix. Their result is a special caseof a more general one, in which the stable and unstable Ruelle algebras of a Smalespace are shown to be in duality [KPW17]. Duality in K -theory also appears inconnection with string theory on noncommutative spacetimes [BMRS08, BMRS09].In this paper, G is a discrete group which admits a G -compact model EG of theclassifying space for proper actions [BCH94]. We study the question of Spanier–Whitehead duality for the pair of C ∗ -algebras C ∗ r ( G ) and C ( EG ) ⋊ G , where thelatter is the crossed product for the group action on EG .This problem is tightly related to the Baum–Connes conjecture and in particularto the so-called Dirac dual-Dirac method. This goes back to the seminal work ofKasparov [Kas88, Sections 4 and 6] and is further explored in [KS91, Section 6]. Ina different direction, the relationship between the assembly map and Fourier-Mukaiduality is discussed in [Blo10].The idea of an underlying noncommutative duality whenever Dirac and dual-Dirac classes are available is well-known to experts (see for example [BMRS08,Example 2.14] and [EEK08, Theorems 2.9 and 3.1]). In particular the work [EM10]by Emerson and Meyer shares many ideas with the present paper, while workingin the context of equivariant KK-theory and groupoids. See Subsection 0.4 andRemark 25 for more details. ROUPS WITH SPANIER–WHITEHEAD DUALITY 3
Below are two main results of this paper. More details on statements and termi-nology are given in the sequel.
Theorem.
Suppose the γ -element exists. Then C ( EG ) ⋊ G is a Spanier–Whitehead K -dual of C ∗ r ( G ) (in a canonical way) if and only if G satisfies the strong Baum–Connes conjecture. Corollary.
For all a-T-menable groups G which admit a G -compact model of EG , C ( EG ) ⋊ G is a Spanier–Whitehead K -dual of C ∗ r ( G ) . Acknowledgements
The first author would like to thank H. Emerson for his helpful comments andsuggestions on this topic. He was supported by Prof. N. Higson during the summer2019 at Penn State University. He would like to thank him for the support.The second author would like to thank B. Mesland and R. Nest for stimulat-ing conversations during the early stages of this paper. This research was partlysupported through V. Proietti’s “Oberwolfach Leibniz Fellowship” by the
Mathe-matisches Forschungsinstitut Oberwolfach in 2019. In addition, the second authorwas partly supported by the Science and Technology Commission of Shanghai Mu-nicipality (STCSM), grant no. 13dz2260400.We would like to thank K. Li for his helpful comments and suggestions on anearlier version of this paper.0.1.
Noncommutative Spanier–Whitehead duality.
Let us see the main no-tions we will be working with. In what follows the C ∗ -tensor product is understoodto be spatial. Definition 1. (cf. [BMRS08, Section 2.7]) Let
A, B be separable C ∗ -algebras. Wesay that B is a weak Spanier–Whitehead K -dual of A if there are elements d ∈ KK i ( A ⊗ B, C ) δ ∈ KK − i ( C , A ⊗ B )such that the induced maps d j : K j ( A ) → K j + i ( B ) d j ( x ) = x b ⊗ A dδ j : K j ( B ) → K j − i ( A ) δ j ( x ) = δ b ⊗ B x are isomorphisms and inverses to each other.Note that, unlike the case of topological spaces, in the noncommutative contextthe existence of d , given δ , is an additional requirement.Some notation: 1 A ∈ KK ( A, A ) stands for the ring unit, σ : A ⊗ B ∼ = B ⊗ A denotes the flip isomorphism. Recall as well the homomorphism τ B : KK ∗ ( A, A ) → KK ∗ ( A ⊗ B, A ⊗ B ), given on cycles as( φ, H, T ) ( φ b ⊗ , H b ⊗ B, T b ⊗ , and equally defined via Kasparov product (over the complex numbers) by τ B ( x ) = x b ⊗ B = 1 B b ⊗ x . Lemma 2 ([Eme03, Lemma 9]) . In the setting of Definition 1, we have the follow-ing identities: ( δ j + i ◦ d j )( y ) = ( − ij y b ⊗ A Λ A ( d j − i ◦ δ j )( y ) = ( − ij Λ B b ⊗ B y, ROUPS WITH SPANIER–WHITEHEAD DUALITY 4 where the elements Λ A ∈ KK ( A, A ) and Λ B ∈ KK ( B, B ) are defined as Λ A = δ b ⊗ B d = ( δ b ⊗ A ) b ⊗ A ⊗ B ⊗ A (1 A b ⊗ σ ∗ ( d ))Λ B = δ b ⊗ A d = ( σ ∗ ( δ ) b ⊗ B ) b ⊗ B ⊗ A ⊗ B (1 B b ⊗ d ) . Definition 3.
Let
A, B denote C ∗ -algebras in weak Spanier–Whitehead duality.With notation from Lemma 2, if we have Λ A = 1 A and Λ B = ( − i B , we say that A and B satisfy Spanier–Whitehead K -duality .Note that this definition is symmetric, so that it can equivalently be phrased bysaying that B is a Spanier–Whitehead K -dual of A , in alignment with the weakform introduced earlier. Remark 4.
In the tensor category (KK , ⊗ ) , where objects are C ∗ -algebras and Hom(
A, B ) = KK ( A, B ) , the previous definition (for i = j = 0 ) can be reinter-preted as the statement that A is a dualizable object and B is its dual. In otherwords the following triangle identity (and its analogue swapping A and B ) holds A ⊗ B ⊗ A d b ⊗ A % % ❏❏❏❏❏❏❏❏❏❏ A A b ⊗ δ tttttttttt A / / A up to the unique isomorphisms coming from braiding and A ⊗ C ∼ = A . The Spanier–Whitehead K -dual respects tensor products in the following sense:if the dual of A is B and the dual of A ′ is B ′ , then the dual of A ⊗ B is KK-equivalentto A ′ ⊗ B ′ , provided it exists (see [KS19]).Throughout this paper G denotes a countable discrete group admitting a G -compact model for its universal example for proper actions. Definition 5.
We say that G has (weak) Spanier–Whitehead K -duality if C ( EG ) ⋊ G is a (weak) dual of C ∗ r ( G ). Remark 6.
It follows from [AD02, Proposition 2.2] that the action of G on EG is amenable. Then by [AD02, Theorem 5.3] the associated full and reduced crossedproducts are isomorphic. In particular, any covariant pair of representations for C ( EG ) and G gives rise to a representation of the reduced crossed product C ( EG ) ⋊ G , namely the integrated form. In short, the aim of this paper is identifying an element x belonging to the“representation ring” KK G ( C , C ), and constructing classes d and δ as above in sucha way that Λ C ∗ r ( G ) and Λ C ( EG ) ⋊ G are both expressible in terms of x . Then thesought duality is reduced to studying the homotopy class of such element.0.2. Baum–Connes conjecture: the duality perspective.
The Baum–Connesconjecture [BCH94] states that the Baum–Connes assembly map µ G : KK G ∗ ( C ( EG ) , C ) → KK ∗ ( C , C ∗ r ( G )) (1)is an isomorphism of abelian groups. A generalization “with coefficients” can beintroduced by inserting a G -algebra A in the right “slot” of the left-hand side of (1),and by considering the corresponding reduced crossed product in the target group: µ GA : KK G ∗ ( C ( EG ) , A ) −→ KK( C , A ⋊ r G ) . (2)Going back to the case with trivial coefficients (i.e., A = C ), since G is a discretegroup, the (dual) Green–Julg isomorphism ([Bla98, KP18, Lan15])KK G ∗ ( C ( EG ) , C ) ∼ = KK ∗ ( C ( EG ) ⋊ r G, C ) ROUPS WITH SPANIER–WHITEHEAD DUALITY 5 allows us to view the assembly map as a morphismKK ∗ ( C ( EG ) ⋊ G, C ) −→ KK ∗ ( C , C ∗ r ( G )) . (3)We shall see that this map is given by Kasparov product with a certain element δ ∈ KK( C , C ∗ r ( G ) ⊗ C ( EG ) ⋊ G )(see Definition 11). Thus, the Baum–Connes conjecture for a discrete group G admitting a G -compact model EG is equivalent to the assertion that the element δ induces the isomorphism δ ∗ : K ∗ ( C ( EG ) ⋊ G ) ∼ = −→ K ∗ ( C ∗ r ( G )) . A priori, this isomorphism itself is not enough to conclude that G has weak Spanier–Whitehead K -duality. In this paper, under an assumption (see below), we identifyan element d ∈ KK( C ∗ r ( G ) ⊗ C ( EG ) ⋊ G, C )which induces a map d ∗ : K ∗ ( C ∗ r ( G )) −→ K ∗ ( C ( EG ) ⋊ G )which is the inverse of δ ∗ in favorable circumstances, namely if the Baum–Connesconjecture holds (it is a left inverse in general). Our assumption for constructingsuch an element d is that the existence of the so-called gamma element, or alterna-tively the ( γ )-element for G . Let us briefly review these notions.0.3. The γ -element and the ( γ ) -element. The following notion of the gammaelement originates in Kasparov’s work [Kas88].
Definition 7. (See [Tu00]) An element x in KK G ( C , C ) is called a gamma element for G if it satisfies the following:(1) for any finite subgroup F ⊆ G , we haveRes FG ( x ) = 1 C ∈ KK F ( C , C ) . (2) for some separable, proper G - C ∗ -algebra P , we have x = β b ⊗ P α where α ∈ KK G ( P, C ) , β ∈ KK G ( C , P ) . A gamma element for G , if it exists, is a unique idempotent in KK G ( C , C ) whichis characterized by the listed properties. Thus, we call it the gamma element for G and denote it by γ . The existence of the gamma element for G implies that theBaum–Connes assembly map is split-injective for all coefficients A (cf. [Tu00]), andfurthermore that the assembly map µ GA is surjective if and only if γ acts as theidentity on K ∗ ( A ⋊ r G ) via ring homomorphismsKK G ( C , C ) → KK G ( A, A ) → KK( A ⋊ r G, A ⋊ r G ) → End( K ∗ ( A ⋊ r G )) . (4)The other composition y = α b ⊗ β is an idempotent in KK( P, P ) which may notbe the identity on P in general. Upon replacing P with its “summand” P C = yP ,which can be defined as a limit of P y −→ P y −→ · · · in the category KK G (cf. [Nee01,Proposition 1.6.8]), we can arrange α (and β ) above to be a weak-equivalence, mean-ing that Res FG ( α ) is an isomorphism for any finite subgroup F of G . In this case,the element α in KK G ( P C , C ) is called the Dirac element and can be characterizedup to equivalence by the fact that α is a weak-equivalence from a “proper object” P C to C . Meyer and Nest [MN06] show that the Dirac element always exists forany group G but, in general, it is not known whether P C can be taken to be aproper C ∗ -algebra. For most of the known examples, P C can indeed be assumedto be proper, meaning that we may think P = P C . However, we emphasize that ROUPS WITH SPANIER–WHITEHEAD DUALITY 6 the algebra P appearing in the definition can be quite arbitrary whereas P C is auniquely characterized object.In [Nis19], the first author introduced a notion called the ( γ )-element, which canbe thought of as an alternative description of the gamma element, bypassing thenecessity of a proper algebra P for its definition.Recall that we assume that G admits a G -compact model for EG . We use [ · , · ]to denote the commutator. Definition 8. ([Nis19, Definition 2.2]) A cycle (
H, T ) representing an element[
H, T ] in KK G ( C , C ) is said to have property ( γ ) if it satisfies the following:(1) for any finite subgroup F ⊆ G we haveRes FG ([ H, T ]) = 1 C ∈ KK F ( C , C ) . (2) there is a non-degenerate G -equivariant representation of C ( EG ) on H such that(2.1) the function g [ g · f, T ]belongs to C ( G, K ( H )), i.e., it vanishes at infinity and is compact-operator-valued for any f ∈ C ( EG ).(2.2) for some cutoff function c ∈ C c ( EG ) (i.e., c is non-negative and satifies P g ∈ G g ( c ) = 1), we have T − X g ∈ G ( g · c ) T ( g · c ) ∈ K ( H ) . An element x in KK G ( C , C ) is called a ( γ )-element for G if it is represented bysome cycle with property ( γ ).A ( γ )-element for G , if it exists, is a unique idempotent in KK G ( C , C ) which ischaracterized by the listed properties. Thus, we call it the ( γ )-element for G . Ifthere is a gamma element γ for G , there is a cycle with property ( γ ) representing γ . Thus the two notions, the γ -element and the ( γ )-element for G , coincide when γ exists. The existence of the ( γ )-element x for G implies that the Baum–Connesassembly map is split-injective for all coefficients A , and furthermore that the as-sembly map µ GA is surjective if and only if x acts as the identity on K ∗ ( A ⋊ r G ) viaring homomorphisms (4).Given the existence of the ( γ )-element, [Nis19] introduced the so-called ( γ )-morphism as a candidate for inverting the assembly map µ G . This is given byKasparov product with a certain element˜ x ∈ KK G ( C ∗ r ( G ) ⊗ C ( EG ) , C ) . The Green–Julg isomorphism allows us to get the corresponding element d ∈ KK( C ∗ r ( G ) ⊗ C ( EG ) ⋊ G, C ).Our proposed strategy aims at realizing weak Spanier–Whitehead duality throughelements δ and d respectively corresponding to the assembly map and the ( γ )-morphism, which seems to be a natural situation. Furthermore, as a result ofLemma 2, the surjectivity and injectivity of the assembly map are controlled re-spectively by Λ C ∗ r ( G ) and Λ C ( EG ) ⋊ G . This gives yet another interpretation ofthese two classes.0.4. Equivariant Kasparov duality.
In [EM10] the authors study several du-ality isomorphisms between equivariant bivariant K -theory groups, generalizingKasparov’s first and second Poincaré duality isomorphisms. For many groupoids,both dualities apply to a universal proper G -space, which is the basis for the Dirac ROUPS WITH SPANIER–WHITEHEAD DUALITY 7 dual-Dirac method. In this setting they explain how to describe the Baum–Connesassembly map via localization of categories as in [MN06].The main notion in [EM10] is that of equivariant Kasparov dual for a G -space X . It involves an X ⋊ G - C ∗ -algebra P , an element α ∈ KK G ( P, C ), and an ad-ditional class Θ ∈ RKK G ( X ; C , P ) (see [EM10, Definition 4.1] for more details).Recall that the category RKK G ( X ) coincides with the range of the pullback func-tor p ∗ X : KK G → KK X ⋊ G via the collapsing map p : X → ∗ .The case X = EG is particularly relevant for our purposes. The class Θ may bethought as the “inverse” of α up to restriction to finite subgroups. More precisely,if a lifting β ∈ KK G ( C , P ) of Θ exists, then the axioms of equivariant Kasparovduality guarantee that β b ⊗ P α is the γ -element and α b ⊗ C β = 1 P . In particular, wehave P = P C and α is a weak equivalence, and hence a Dirac morphism.Let Z denote the unit space of G and suppose the moment map from EG → Z is proper. Then [EM10, Theorem 5.7] establishes a connection to what we mightcall “equivariant” Spanier-Whitehead duality. We summarize it below for the con-venience of the reader (see also Remark 25). Theorem 9.
The triple ( P, α, Θ) is a Kasparov dual for X if and only if C ( X ) and P are dual objects in KK G (cf. Remark 4) with duality unit and counit respectivelyinduced by Θ and α . Concerning the connection with the Baum–Connes assembly map, we have:
Theorem 10 ([EM10, Theorem 6.14]) . Suppose EG admits a local symmetric Kas-parov dual. Then the assembly map µ GA is an isomorphism for all proper coefficientalgebras A . Assuming EG to be G -compact, the proof of the previous theorem roughly goesas follows: the second Poincaré duality isomorphism [EM10, Section 6] combinedwith the Green-Julg isomorphism for proper groupoids [EM09, Theorem 4.2] trans-late the assembly map µ GA into the map K ∗ (( P ⊗ A ) ⋊ G ) → K ∗ ( A ⋊ G ) inducedby α . Now it is easy to see from the definition of equivariant Kasparov dual thatthe element τ A ( α ) ∈ KK G ( P ⊗ A, A ) is invertible when A is a proper C ∗ -algebra.0.5. Main results.
Let us summarize our main results. Recall that G is a count-able discrete group with a G -compact model for EG .As we have explained in the previous sections, our main strategy for obtainingduality relies on(1) the γ element, or(2) the ( γ )-element.The choice of one over the other does not affect the expression for the unit ofSpanier–Whitehead duality, nevertheless the descriptions of the counit and theelements Λ C ∗ r ( G ) and Λ C ( EG ) ⋊ G depend on the method that we are employing. Inpractice, the latter elements will be expressible in terms of the γ -element in thefirst case, and in the terms of the ( γ )-element in the second case.Along this categorization, Theorem A and Corollary B below fall in the firstscenario, while Theorem D is an instance of the second. Section 3 contains simpleexamples of possible applications of duality in K -theory. Theorem A.
Suppose that the γ -element γ ∈ KK G ( C , C ) exists and let P C be thesource of the Dirac morphism α ∈ KK G ( P C , C ) . Then the C ∗ -algebra P C ⋊ G isSpanier–Whitehead K -dual to C ( EG ) ⋊ G . A few more comments about this theorem. The source of the Dirac morphism(the “simplicial approximation” in [MN06]) can be obtained in a variety of ways:
ROUPS WITH SPANIER–WHITEHEAD DUALITY 8 by appealing to the Brown representability theorem, by considering the left adjointto the embedding functor of projective objects, or by constructing the appropriatehomotopy colimit from a projective resolution of C (here, “projective” is to beunderstood in a relative sense, i.e., with respect to the homological ideal of weaklycontractible objects). Even though P C may not be a proper algebra in general,its reduced and maximal crossed products are KK-equivalent. This is because P C belongs to the localizing subcategory of KK G generated by proper algebras andthe reduced and maximal crossed product functors are triangulated functors andcommute with countable direct sums (see [MN06]).Theorem A provides a fourth characterization of P C . Namely as the Spanier–Whitehead K -dual of the classifying space for proper actions. Note that eventhough our statement is only available after descent, that is we can only get P C ⋊ G and not P C via duality, this is only a minor drawback in the case of discrete groups,for the the left-hand side of (3) retains the full information of the “topological” K -theory group through the dual Green–Julg isomorphismKK G ( C ( EG ) , C ) ∼ = KK( C ( EG ) ⋊ G, C ) . In the situation where, at the KK-theory level, the simplicial approximation isequivalent to the data of G acting on the point, we can replace P C ⋊ G with C ∗ r ( G )and obtain Spanier–Whitehead duality for the group as in the next Corollary. If the γ -element exists, we define the strong Baum–Connes conjecture to be the statementthat Gr ( γ ) = 1 C ∗ r ( G ) in KK( C ∗ r ( G ) , C ∗ r ( G )). Corollary B.
Suppose the γ -element exists. Then G has Spanier–Whitehead du-ality if and only if it satisfies the strong Baum–Connes conjecture. In light of the result above, we can view the notion of Spanier–Whitehead K -duality for G as a homotopy-theoretic characterization of the strong Baum–Connesconjecture (cf. Remark 43).The main application of the previous corollary is summarized in the result below. Corollary C.
All a-T-menable groups which admit a G -compact model of EG haveSpanier–Whitehead K -duality. Examples of a-T-menable groups are the following: • All groups which act properly, affine-isometrically and co-compactly on afinite-dimensional Euclidean space. • All co-compact lattices of simple Lie groups
SO( n, or SU( n, . • All groups which act co-compactly on a tree.
Having such an explicit duality should be useful. For example, in principle, itallows us to compute the Lefschetz number of an automorphism of C ∗ r ( G ) or moregenerally of a morphism f in KK( C ∗ r ( G ) , C ∗ r ( G )) (see [DEM14, Eme11]).If a cycle with property ( γ ) is found, then we can deduce the duality in completeanalogy with the case of the γ -element (this is how the definition of property ( γ )was designed). However in this case we do not have information on the localizationat the weakly contractible objects [MN10]. So we get the corresponding statementfor Corollary B, but not for Theorem A. Theorem D.
Suppose there is a ( γ ) -element x ∈ KK G ( C , C ) for G . If Gr ( x ) =1 C ∗ r ( G ) ∈ KK G ( C ∗ r ( G ) , C ∗ r ( G )) , then G has Spanier–Whitehead duality. General framework
Let G be a countable discrete group, and EG be a G -compact model of theuniversal proper G -space. Let A and B be C ∗ -algebras equipped with a G -action. ROUPS WITH SPANIER–WHITEHEAD DUALITY 9
If the G -action on B is trivial, we recall the dual Green–Julg isomorphism ([Bla98,KP18, Lan15]) GJ : KK G ( A, B ) ∼ = KK( A ⋊ G, B ) . Given a ∈ A , define δ ag ∈ C c ( G, A ) ⊆ A ⋊ G to be the function δ ag ( t ) = ( a if t = g t = g .The dual coaction is defined as∆ : A ⋊ G → C ∗ r ( G ) ⊗ A ⋊ Gδ ag g ⊗ δ ag . Let c ∈ C c ( EG ) be a cutoff function, and consider the associated projection p c ∈ C c ( G, C ( EG )) ⊆ C ( EG ) ⋊ G defined by p c ( g ) = cg ( c ). This projection doesnot depend on c up to homotopy, hence we will denote it p G in the sequel. Definition 11.
We define the canonical duality unit to be the class δ = δ G = [∆( p G )] ∈ KK( C , C ∗ r ( G ) ⊗ C ( EG ) ⋊ G ) . The notational dependence on G shall be dropped when clear from the context.In this paper, whenever we say that G has Spanier–Whitehead duality, we implicitlyassume that the duality unit is given as above.Let us recall the definition of Kasparov’s descent homomorphism [Kas88], whichplays an important role in this paper. It will be denoted G below. Suppose ( φ, H, T )is a Kasparov cycle defining an element of KK G ( A, B ). The G -action on H willbe denoted U : G → End C ( H ). The element G ([ φ, H, T ]) ∈ KK( A ⋊ G, B ⋊ G ) isdefined by the cycle ( ˜ φ, H ⋊ G, ˜ T ) given as follows.The Hilbert C ∗ -module H ⋊ G is the completion of C c ( G, H ) with respect to thenorm induced by the following B ⋊ G -valued inner product: h ξ | ζ i ( t ) = X g ∈ G β g − ( h ξ ( g ) | ζ ( gt ) i ) , where ξ, ζ ∈ C c ( G, H ), t ∈ G , and β denotes the given G -action on B . The rightaction of B ⋊ G is uniquely determined by the formula( ξ · f )( t ) = X g ∈ G ξ ( g ) β g ( f ( g − t )) , where ξ ∈ C c ( G, H ) , f ∈ C c ( G, B ) and t ∈ G . The representation of A ⋊ G on H ⋊ G is determined by( ˜ φ ( f )( ξ ))( t ) = X g ∈ G φ ( f ( g ))[ U ( g )( ξ ( g − t ))] , where f ∈ C c ( G, A ) , ξ ∈ C c ( G, H ) and t ∈ G . Finally the operator ˜ T is definedby ( ˜ T ξ )( t ) = T ( ξ ( t )) for ξ ∈ C c ( G, H ) and t ∈ G . By using reduced crossedproducts everywhere, we can similarly defined a “reduced version” of the descenthomomorphism, denoted Gr in the sequel. Lemma 12 ([Lan15, Proposition 4.7]) . Kasparov’s descent homomorphism can befactorized as follows: KK G ( A, C ) GJ (cid:15) (cid:15) G / / KK( A ⋊ G, C ∗ ( G ))KK( A ⋊ G, C ) τ C ∗ ( G ) / / KK( C ∗ ( G ) ⊗ A ⋊ G, C ∗ ( G )) . ∆ ∗ O O ROUPS WITH SPANIER–WHITEHEAD DUALITY 10
When the canonical map A ⋊ G → A ⋊ r G is an isomorphism (e.g., if G actsproperly on A ), the version of the previous lemma with reduced crossed productsalso holds. See Remark 6. Lemma 13 ([KP18, Section 2]) . Let A and B be G - C ∗ -algebras and suppose the G -action on B is trivial. Consider an element x ∈ KK G ( A, A ) . The followingdiagram commutes. KK G ( A, B ) GJ / / x b ⊗ − (cid:15) (cid:15) KK( A ⋊ G, B ) G ( x ) b ⊗ − (cid:15) (cid:15) KK G ( A, B ) GJ / / KK( A ⋊ G, B ) . It follows from Lemma 12 that we have the following commutative diagramKK G ( C ( EG ) , B ) GJ ∼ = (cid:15) (cid:15) µ GB / / KK( C , B ⊗ C ∗ r ( G )) = (cid:15) (cid:15) KK( C ( EG ) ⋊ G, B ) δ b ⊗ C EG ) ⋊ G − / / KK( C , B ⊗ C ∗ r ( G ))Since the definition of the duality counit requires additional information, andwill depend on the choice of “ γ -like” element, the rest of this section gets split intwo parts. The torsion-free case is treated in detail in Subsection 1.31.1. Argument based on the ( γ ) -element. Let (
H, T ) be a G -equivariant Kas-parov cycle with property ( γ ). Let x = [ H, T ] be the corresponding element inKK G ( C , C ). Let˜ x = [ H ⊗ ℓ ( G ) , ρ ⊗ π, ( g ( T )) g ∈ G ] ∈ KK G ( C ∗ r ( G ) ⊗ C ( EG ) , C ) . (5)Here, π : C ( EG ) → B ( H ) is the representation witnessing the conditions for prop-erty ( γ ) of ( H, T ), ρ stands for the right regular representation, and C ∗ r ( G ) hastrivial G -action. By means of the Green-Julg isomorphism, we set d = GJ(˜ x ) ∈ KK( C ∗ r ( G ) ⊗ C ( EG ) ⋊ G, C ) . We set Λ C ∗ r ( G ) = δ b ⊗ C ( EG ) ⋊ G d and Λ C ( EG ) ⋊ G = δ b ⊗ C ∗ r ( G ) d . We shall prove(1) Λ C ∗ r ( G ) = Gr ( x ) in KK( C ∗ r ( G ) , C ∗ r ( G ));(2) Λ C ( EG ) ⋊ G = 1 C ( EG ) ⋊ G in KK( C ( EG ) ⋊ G, C ( EG ) ⋊ G ). Proposition 14.
We have the equality Λ C ∗ r ( G ) = Gr ( x ) .Proof. We claim the Kasparov module[ p G ] b ⊗ C ( EG ) ⋊ G Gr (˜ x ) (6)is equivalent to Gr ( x ), i.e., there is an isomorphism of Hilbert C ∗ -modules inter-twining the representations and the operators (up to a compact perturbation).The class in (6) is represented by( H ⊗ ℓ ( G ) ⋊ r G, ( ρ ⊗ π ⋊ r p G ⊗ − ) , ( g ( T )) g ∈ G ⋊ r . We have an isomorphism of C ∗ r ( G )-modules H ⋊ r G ∼ = ( ρ ⊗ π ⋊ r p G ⊗ H ⊗ ℓ ( G ) ⋊ r G ) (7)given by the assignment ξ ⋊ r u g X h ∈ G π ( c )( h · ξ ) ⊗ δ h ⋊ r u hg , ROUPS WITH SPANIER–WHITEHEAD DUALITY 11 where ξ ∈ H , δ h ∈ ℓ ( G ), and c is a cutoff function defining p G . The inverse of themap above is given by the restriction of( ξ ) h ∈ G ⋊ r u g X h ∈ G h − · ( π ( c ) ξ h ) ⊗ ⋊ r u h − g , where ( ξ ) h ∈ G ∈ H ⊗ ℓ ( G ). Under the isomorphism in (7), the representation( ρ ⊗ π ⋊ r p G ⊗ − ) is identified with the left action of C ∗ r ( G ) on H ⋊ r G , and thecompressed operator( ρ ⊗ π ⋊ r p G ⊗ g ( T )) g ∈ G ⋊ r ρ ⊗ π ⋊ r p G ⊗ T ′ ⋊ r H ⋊ r G , where we defined T ′ = X g ∈ G ( g · c ) T ( g · c ) . Hence the claim follows by definition of property ( γ ).By Lemma 12, we have Gr (˜ x ) = ∆ ⊗ C ( EG ) ⋊ G GJ(˜ x ) . Thus, we have Gr ( x ) = [ p G ] b ⊗ C ( EG ) ⋊ G Gr (˜ x )= [ p G ] b ⊗ C ( EG ) ⋊ G ∆ ⊗ C ( EG ) ⋊ G GJ(˜ x )= δ b ⊗ C ( EG ) ⋊ G d. (cid:3) Proposition 15.
We have the equality Λ C ( EG ) ⋊ G = 1 C ( EG ) ⋊ G . In order to prove the proposition, a few preliminaries are in order. First wegeneralize the construction in (5) to include a coefficient algebra. This is easilydone: simply replace ℓ ( G ) with the right Hilbert A -module ℓ ( G, A ) and definethe right regular representation ρ GA of A ⋊ r G (equipped with trivial G -action) a ( g ( a )) g ∈ G , h ρ h : ( a g ) g ∈ G ( a gh ) g ∈ G for a ∈ A, h ∈ G . Thus we get a class ˜ x A in KK G ( A ⋊ r G ⊗ C ( EG ) , A ). We definea group homomorphism ν GA : KK( C , A ⋊ r G ) → KK G ( C ( EG ) , A )as the one induced by the class ˜ x A via the index pairingKK( C , A ⋊ r G ) × KK G ( A ⋊ r G ⊗ C ( EG ) , A ) → KK G ( C ( EG ) , A ) . This map is referred to as the ( γ )-morphism in [Nis19]. Note also that GJ ◦ ν G C equalsthe map d j from Definition 1 (choosing B = C ( EG ) ⋊ G as usual). The lemmabelow is about the naturality property of the assembly map and the ( γ )-morphism. ROUPS WITH SPANIER–WHITEHEAD DUALITY 12
Lemma 16.
The following diagrams commute for any f ∈ KK G ( A, B ) . KK G ( C ( EG ) , A ) µ GA / / − b ⊗ f (cid:15) (cid:15) KK( C , A ⋊ r G ) − b ⊗ Gr ( f ) (cid:15) (cid:15) KK G ( C ( EG ) , B ) µ GA / / KK( C , B ⋊ r G ) , KK( C , A ⋊ r G ) ν GA / / − b ⊗ Gr ( f ) (cid:15) (cid:15) KK G ( C ( EG ) , A ) − b ⊗ f (cid:15) (cid:15) KK( C , B ⋊ r G ) ν GA / / KK G ( C ( EG ) , B ) Proof.
The first diagram commutes by functoriality of descent and associativity ofthe Kasparov product. By results from [Mey00] any morphism f in KK G ( A, B ) canbe written as a composition of ∗ -homomorphisms and their inverses in KK. Thismeans it suffices to check the commutativity of the second diagram with respect to ∗ -homomorphisms f : A → B . We omit this simple verification. (cid:3) Proof of Proposition 15.
Let B = C ( EG ) ⋊ G and regard it as a G - C ∗ -algebrawith the trivial G -action. We have the following diagramKK G ( C ( EG ) , B ) GJ ∼ = (cid:15) (cid:15) µ GB / / KK( C , C ∗ r ( G ) ⊗ B ) = (cid:15) (cid:15) ν GB / / KK G ( C ( EG ) , B ) GJ ∼ = (cid:15) (cid:15) KK(
B, B ) δ b ⊗ B − / / KK( C , C ∗ r ( G ) ⊗ B ) − b ⊗ C ∗ r ( G ) d / / KK(
B, B )If we prove the composition on the top is the identity, then it follows Λ C ( EG ) ⋊ G =1 C ( EG ) ⋊ G . Let D B : P B → B be a weak equivalence as in [MN06]. Because thefollowing diagram commutes,KK G ( C ( EG ) , P B ) D B ∗ (cid:15) (cid:15) µ GPB / / KK( C , P B ⋊ G ) Gr ( D B ∗ ) (cid:15) (cid:15) ν GPB / / KK G ( C ( EG ) , P B ) D B ∗ (cid:15) (cid:15) KK G ( C ( EG ) , B ) µ GB / / KK( C , B ⋊ r G ) ν GB / / KK G ( C ( EG ) , B ) , it suffices to show that ν GP B is a left inverse of the assembly map µ GP B . Now, µ GP B is invertible, hence it suffices to show that ν GP B yields a right inverse. A minorgeneralization of the proof of Proposition 14 shows that µ GP B ◦ ν GP B coincides withthe induced action of x ∈ KK G ( C , C ) on K ∗ ( P B ⋊ G ). Recall that x equals theidentity when restricted to each finite subgroup H ⊆ G , and P B ⋊ G belongs tothe localizing subcategory of KK generated by the B ⋊ H ’s. Therefore the map x b ⊗ − : K ∗ ( P B ⋊ G ) → K ∗ ( P B ⋊ G ) is the identity by [MN06, Theorem 9.3]. (cid:3) Remark 17.
In parallel with Proposition 14, one can prove that Λ C ( EG ) ⋊ G = Gr ( x b ⊗ C ( EG ) ) . Again, we set B = C ( EG ) ⋊ G and first notice that ν GB ◦ µ GB = x b ⊗ C ( EG ) − . Itis enough to show this when B is replaced by P B , in which case we can invert the ROUPS WITH SPANIER–WHITEHEAD DUALITY 13 assembly map and write ( Gr ( x ) b ⊗ − ) = ( Gr ( x ) b ⊗ − ) ◦ µ GB ◦ ( µ GB ) − µ GB ◦ ν GB = µ GB ◦ ( x b ⊗ C ( EG ) − ) ◦ ( µ GB ) − ν GB ◦ µ GB = x b ⊗ C ( EG ) − . To complete the proof one must show that
GJ( x b ⊗ C GJ − (1 B )) = Gr ( x b ⊗ C ( EG ) ) b ⊗ B B , but this follows from Lemma 13. We now come to the main result of this subsection.
Theorem 18.
Suppose there is a ( γ ) -element x ∈ KK G ( C , C ) for G . If Gr ( x ) =1 ∈ KK G ( C ∗ r ( G ) , C ∗ r ( G )) , then G has Spanier–Whitehead duality. Argument based on the γ -element. Suppose there is a gamma element γ as in Definition 7. Following [GHT00, Chapter 15], we define a map s A for anyproper algebra A . This is the G -equivariant ∗ -homomorphism s A : A ⋊ r G ⊗ C ( EG ) → K ( A ⊗ ℓ ( G ))where A ⋊ r G is equipped with the trivial G -action, defined as the tensor productof the following representation of C ( EG ) on A ⊗ ℓ ( G ): C ( EG ) ∋ φ ( φ ) g ∈ G ∈ L ( A ⊗ ℓ ( G ))and the right regular representation of A ⋊ r G on A ⊗ ℓ ( G ): A ∋ a ( g ( a )) g ∈ G ∈ L ( A ⊗ ℓ ( G )) , G ∋ g ⊗ ρ g where ρ g is the right translation by g . Here, the G -action on the Hilbert module A ⊗ ℓ ( G ) is given by the tensor product of the action on A and the left-regularrepresentation. The ∗ -homomorphism s A defines an element s A ∈ KK G ( A ⋊ r G ⊗ C ( EG ) , A ) . Proposition 19 (cf. [GHT00, Chapter 15]) . For any proper G - C ∗ -algebra A , the ∗ -homomorphism s A defines the inverse s A : KK( C , A ⋊ r G ) → KK G ( C ( EG ) , A ) of the assembly map µ GA : KK G ( C ( EG ) , A ) → KK( C , A ⋊ r G ) . Proof.
The assembly map is an isomorphism for any proper algebra. Hence, we justshow that the composition µ GA ◦ s A is the identity. Take a Kasparov cycle ( E, F ) forKK( C , A ⋊ r G ) where E is a graded A ⋊ r G -module and F is an odd, self-adjointoperator on E satisfying 1 − F ≡ E is A ⊗ H ⋊ r G for somegraded Hilbert space H with the trivial G -action. The map s A sends this cycle ( A ⊗ H ⋊ r G, F ) to the G -equivariant cycle ( A ⊗ H ⊗ ℓ ( G ) , π, ρ ( F )) for KK G ( C ( EG ) , A )where π is a representation of C ( EG ) on A ⊗ H ⊗ ℓ ( G ) defined as: for φ in C ( EG ) π ( φ )( a g ⊗ v g ⊗ δ g ) = φa g ⊗ v g ⊗ δ g and ρ ( F ) is an operator in L ( A ⊗ H ⊗ ℓ ( G )) determined by the map L ( A ⊗ H ⋊ r G ) = M ( A ⊗ K ( H ) ⋊ r G ) ρ −→ M ( A ⊗ K ( H ⊗ ℓ ( G ))) = L ( A ⊗ H ⊗ ℓ ( G )) , which is a natural extension of the right regular representation ρ GA of A ⋊ r G on A ⊗ ℓ ( G ) described before. Hence, the composition µ GA ◦ s A sends the cycle ( A ⊗ H ⋊ r G, F ) to the cycle ( p c ( A ⊗ H ⊗ ℓ ( G ) ⋊ r G ) , p c ρ ( F ) ⋊ r p c ) where we simply ROUPS WITH SPANIER–WHITEHEAD DUALITY 14 denote by p c the image of a cutoff projection p c in C ( EG ) ⋊ G by the representation π ⋊ r A ⋊ r G -modules A ⊗ H ⋊ r G −→ p c ( A ⊗ H ⊗ ℓ ( G ) ⋊ r G )given by for ξ in A ⊗ H , ξ ⋊ r u g X h ∈ G c ( h ( ξ )) ⊗ δ h ⋊ r u hg . The inverse map is given by for ( ξ h ) h ∈ G in A ⊗ H ⊗ ℓ ( G ),( ξ h ) h ∈ G ⋊ r u g X h ∈ G h − ( cξ h ) ⋊ r u h − g . Under this isomorphism, the restriction p c ρ ( F ) ⋊ r p c of ρ ( F ) ⋊ r p c ( A ⊗ H ⊗ ℓ ( G ) ⋊ r G ) of A ⊗ H ⊗ ℓ ( G ) ⋊ r G corresponds to the operator F on A ⊗ H ⋊ r G . In summary, the composition µ GA ◦ s A sends the cycle ( A ⊗ H ⋊ r G, F )to itself up to the isomorphism described above. (cid:3)
For any separable G - C ∗ -algebra B , we have the following commutative diagram µ GB : KK G ( C ( EG ) , B ) − b ⊗ C β (cid:15) (cid:15) / / KK( C , B ⋊ r G ) − b ⊗ B ⋊ rG Gr (id B b ⊗ β ) (cid:15) (cid:15) µ GB ⊗ P : KK G ( C ( EG ) , B ⊗ P ) − b ⊗ P α (cid:15) (cid:15) ∼ = / / KK( C , ( B ⊗ P ) ⋊ r G ) − b ⊗ ( B ⊗ P ) ⋊ rG Gr (id B b ⊗ α ) (cid:15) (cid:15) µ GB : KK G ( C ( EG ) , B ) / / KK( C , B ⋊ r G )where the vertical composition on the left is the identity. With this and by Propo-sition 19, we see that the element( Gr (1 B b ⊗ β )) b ⊗ ( B b ⊗ P ) ⋊ r G s B ⊗ P b ⊗ P α ∈ KK G (( B ⋊ r G ) ⊗ C ( EG ) , B )induces the left-inverse of the assembly map µ GB via Kasparov product. We remarkthat this is the standard technique for proving the split injectivity of the assemblymap in the presence of a γ -element.Now, we set d ′ to be the element in KK( C ∗ r ( G ) ⊗ C ( EG ) ⋊ G, C ) which corre-sponds to d ′′ = ( Gr ( β )) b ⊗ P ⋊ r G s P b ⊗ P α ∈ KK G ( C ∗ r ( G ) ⊗ C ( EG ) , C ) . Let δ = δ G ∈ KK( C , C ( EG ) ⋊ G ⊗ C ∗ r ( G ))as before. We set Λ ′ C ∗ r ( G ) = δ b ⊗ C ( EG ) ⋊ G d ′ , Λ ′ C ( EG ) ⋊ G = δ b ⊗ C ∗ r ( G ) d ′ . Proposition 20.
We have Λ ′ C ∗ r ( G ) = Gr ( γ ) ∈ KK( C ∗ r ( G ) , C ∗ r ( G )) . and Λ ′ C ( EG ) ⋊ G = 1 C ( EG ) ⋊ G ∈ KK( C ( EG ) ⋊ G, C ( EG ) ⋊ G )Before giving a proof of Proposition 20, let us obtain our main results as itsdirect consequences: Theorem 21. If Gr ( γ ) = 1 C ∗ r ( G ) , then G has Spanier–Whitehead duality. The previous result has a converse, see Theorem 39 for further details.
ROUPS WITH SPANIER–WHITEHEAD DUALITY 15
Theorem 22. If µ G C is an isomorphism, G has weak Spanier–Whitehead duality. Theorem 23.
In general, if γ ∈ KK G ( C , C ) exists, then C ( EG ) ⋊ G is a Spanier–Whitehead K -dual of P C ⋊ G .Proof. Note that P C ⋊ G is a direct summand (in the categoy KK) of C ∗ r ( G ) corre-sponding to the idempotent Gr ( γ ) ∈ KK( C ∗ r ( G ) , C ∗ r ( G )) (see [Nee01, Proposition1.6.8])). Namely, we have i P C ⋊ G ∈ KK( P C ⋊ G, C ∗ r ( G )) , q P C ⋊ G ∈ KK( C ∗ r ( G ) , P C ⋊ G ) , so that q P C ⋊ G ◦ i P C ⋊ G = 1 P C ⋊ G and i P C ⋊ G ◦ q P C ⋊ G = Gr ( γ ). We set d P C ⋊ G = i P C ⋊ G b ⊗ C ∗ r ( G ) d ′ ∈ KK( C ( EG ) ⋊ G ⊗ P C ⋊ G, C ) ,δ P C ⋊ G = δ b ⊗ C ∗ r ( G ) q P C ⋊ G ∈ KK( C , C ( EG ) ⋊ G ⊗ P C ⋊ G ) . Then, we have δ P C ⋊ G b ⊗ C ( EG ) ⋊ G d P C ⋊ G = 1 P C ⋊ G , δ P C ⋊ G b ⊗ P C ⋊ G d P C ⋊ G = 1 C ( EG ) ⋊ G . This proves the statement. We only prove the first identity, the other one is similarlyproved. For any C ∗ -algebra D , we have the following commutative diagram:KK( P C ⋊ G, D ) δ P C⋊ G b ⊗ P C⋊ G − (cid:15) (cid:15) / / KK( C ∗ r ( G ) , D ) δ b ⊗ C ∗ r ( G ) − (cid:15) (cid:15) KK( C , C ( EG ) ⋊ G ⊗ D ) − b ⊗ C EG ) ⋊ G d P C⋊ G (cid:15) (cid:15) = / / KK( C , C ( EG ) ⋊ r G ⊗ D ) − b ⊗ C EG ) ⋊ rG d ′ (cid:15) (cid:15) KK( P C ⋊ G, D ) / / KK( C ∗ r ( G ) , D ) . Here, the top and the bottom horizontal arrows are induced by i P C ⋊ G and q P C ⋊ G .The right vertical composition is induced by Gr ( γ ). It follows, the left verticalcomposition is the identity. Taking D = P C ⋊ G , we get δ P C ⋊ G b ⊗ C ( EG ) ⋊ G d P C ⋊ G = 1 P C ⋊ G . (cid:3) Proof of Proposition 20.
We directly compute and prove δ b ⊗ C ( EG ) ⋊ G d ′ = Gr ( γ ) ∈ KK( C ∗ r ( G ) , C ∗ r ( G )) . For simplicity, we prove this for the case when β is represented by a cycle ( P, b )where b is an essential unitary in M ( P ) and if α is represented by a cycle ( H, F )where P is represented on H non-degenerately and F is a G -equivariant essentialunitary modulo P . Then, d ′′ is represented by the cycle of the form( H ⊗ ℓ ( G ) , ρ ⊗ π, N ( g ( b )) g ∈ G + M ( g ( F )) g ∈ G )where the G -action on H ⊗ ℓ ( G ) is the tensor product of the G -action on H and theleft regular representation on ℓ ( G ), π is a representation of C ( EG ) on H ⊗ ℓ ( G )given by φ ( φ ) g ∈ G and where ρ is a representation of C ∗ r ( G ) on H ⊗ ℓ ( G ) bythe right regular representation g ⊗ ρ g . Here, M and N are given by KasparovTechnical Theorem as usual [Hig87, Kas95, Kas80]. If we compute δ ⊗ C ( EG ) ⋊ G d ′ ,we get the cycle isomorphic to( H ⋊ r G, π G , T ⋊ r
1) = Gr (( H, T ))where ( H, T ) is a cycle for KK G ( C , C ), π G is the natural left multiplication by C ∗ r ( G ) and T = N b + M F . Here F is the average of F : F = R G g ( c ) F g ( c ) dµ G ROUPS WITH SPANIER–WHITEHEAD DUALITY 16 and so are N and M . The cycle ( H, T ) is (homotopic to) a Kasparov product of α and β . In other words, the element [ H, T ] is the gamma element γ . It follows δ b ⊗ C ( EG ) ⋊ G d ′ = Gr ( γ ) . Now, we can prove δ b ⊗ C ∗ r ( G ) d ′ = 1 C ( EG ) ⋊ G ∈ KK( C ( EG ) ⋊ G, C ( EG ) ⋊ G )using a simple trick. We have the following diagram for B = C ( EG ) ⋊ G with thetrivial G action:KK G ( C ( EG ) , B ) ∼ = (cid:15) (cid:15) µ GB / / KK( C , C ∗ r ( G ) ⊗ B ) = (cid:15) (cid:15) ( µ GB ) − / / KK G ( C ( EG ) , B ) ∼ = (cid:15) (cid:15) KK(
B, B ) δ b ⊗ B − / / KK( C , C ∗ r ( G ) ⊗ B ) − b ⊗ C ∗ r ( G ) d / / KK(
B, B ) . Here, by ( µ GB ) − we simply mean the left inverse of µ GB . This shows δ ⊗ C ∗ r ( G ) d ′ acts as the identity on KK( B, B ), proving the claim. (cid:3)
Remark 24.
The previous proof also shows that d = d ′ , as it is intuitive from thefact that the γ -element can be represented by a cycle satisfying property ( γ ) [Nis19] . Remark 25.
It is natural to use the duality class Θ from Subsection 0.4 to proveTheorem 23. The argument is based on the following diagram, where we have set d ′ = GJ( s P b ⊗ P α ) , and µ GP ⋊ G,P is a bivariant assembly map (cf. Section 3).
KK( C , P ⋊ G ⊗ C ( EG ) ⋊ G ) − b ⊗ C EG ) ⋊ G d ′ / / KK( P ⋊ G, P ⋊ G )KK G ( C ( EG ) , P ⊗ C ( EG ) ⋊ G ) p ∗ EG ∼ = (cid:15) (cid:15) µ GP ⊗ C EG ) ⋊ G ∼ = O O − b ⊗ C EG ) ⋊ G d ′ / / KK G ( C ( EG ) ⊗ P ⋊ G, P ) µ GP ⋊ G,P O O p ∗ EG ∼ = (cid:15) (cid:15) RKK G ( EG ; C ( EG ) , P ⊗ C ( EG ) ⋊ G ) − b ⊗ C EG ) ⋊ G d ′ / / RKK G ( EG ; C ( EG ) ⊗ P ⋊ G, P ) Set e = GJ − (1 C ( EG ) ⋊ G ) and consider the element δ = Θ b ⊗ C ( EG ) e in the bottomleft group. Suppressing p ∗ EG from the notation, we compute δ b ⊗ C ( EG ) ⋊ G d ′ = Θ b ⊗ C ( EG ) e b ⊗ C ( EG ) ⋊ G GJ( s P b ⊗ P α )= (Θ b ⊗ P α ) b ⊗ C ( e b ⊗ C ( EG ) ⋊ G GJ( s P )) = s P . Now it is routine to check that µ GP ⋊ G,P ( s P ) = 1 P ⋊ G . Hence if we define δ P ⋊ G ∈ KK( C , P ⋊ G ⊗ C ( EG ) ⋊ G ) by sending δ through the left vertical isomorphism in the diagram above, we have δ P ⋊ G b ⊗ C ( EG ) ⋊ G d ′ = 1 P ⋊ G . The other identity is similarly proved, we skip the details.Note that this is an improvement over Theorem 23, because the existence of Θ is strictly weaker than having a gamma element. A similar argument shows thatin general, if P C is a (categorical) direct summand of some proper algebra, theconclusion of Theorem 23 holds, namely C ( EG ) ⋊ G is a Spanier–Whitehead K -dual of P C ⋊ G . ROUPS WITH SPANIER–WHITEHEAD DUALITY 17
The torsion-free case.
We treat the torsion-free case separately, partly be-cause it is particularly simple (e.g., condition (1) of Definition 7 reduces to a state-ment in non-equivariant K -theory), partly because it is among the first cases wherethe duality classes (i.e., unit and counit) have been identified (albeit in a slightlydifferent language, cf. [Kas88, Theorems 6.6 and 6.7]).We assume that G is a countable, discrete, torsion-free group. In this case,because proper actions are automatically free, the space EG is identified as thetotal space EG of the classifying space for principal G -bundles, and our assumptionthat G admits a G -compact model of EG translates into the assumption that G admits a compact model of BG . Denote by [MF], the class[MF] ∈ KK( C , C ∗ r ( G ) ⊗ C ( BG ))associated to the module of sections of the Miščenko bundle. This is the Hermitianbundle of C ∗ -algebras obtained from the associated bundle construction EG × G C ∗ r ( G ) → BG, where G acts diagonally, acting on the reduced group C ∗ -algebra via the left regularrepresentation [MF79]. Proposition 26 ([Con94], for a proof see [KP18]) . The Miščenko module MF isthe finitely generated projective Hilbert C ∗ -module, described as the completion of C c ( EG ) with respect to the norm induced by the following C ∗ r ( G ) b ⊗ C ( BG ) -valuedinner product: h ξ | ζ i ( t )( x ) = X p ( y )= x ¯ ξ ( y ) ζ ( y · t ) , (8) where ξ, ζ ∈ C c ( EG ) , t ∈ G , x ∈ BG and p : EG → BG is the quotient map. Theright action of C ∗ r ( G ) b ⊗ C ( BG ) on M is defined by ( ξ · f )( y ) = X g ∈ G f ( g )( p ( y )) · ξ ( y · g − ) , (9) where ξ ∈ C c ( EG ) , f ∈ C c ( G, C ( BG )) and y ∈ EG . We have for any separable C ∗ -algebra B with trivial G -action [Lan15, KP18],KK( C ( BG ) , B ) ∼ = (cid:15) (cid:15) [MF] b ⊗ C ( BG ) − / / KK( C , C ∗ r ( G ) ⊗ B ) = (cid:15) (cid:15) KK G ( C ( EG ) , B ) µ GB / / KK( C , C ∗ r ( G ) ⊗ B )The vertical isomorphism above is implemented by the strong Morita equivalencebetween C ( BG ) and C ( EG ) ⋊ G [Rie76], whose associated KK-class is denoted[ Y ∗ ] below (we use [ Y ] for the opposite module).If G admits a compact non-positively curved manifold as a model for BG , thenthe element d was introduced by Kasparov [Kas88] as a “dual-Dirac” class d ∈ KK( C ∗ r ( G ) b ⊗ C ( BG ) , C ) . To be more consistent with the terminology of this paper, d should be called theduality counit induced by the γ -element (which exists in this situation). Kasparovwent on to show that d defines a left inverse for the assembly map.Hence we see that we are in a situation where Spanier–Whitehead duality comesinto play very naturally, with the choice MF = unit and d = counit. Note that,while the class d requires structural information on the group, the class of theMiščenko bundle relies on very little structure. This is in complete analogy withthe canonical unit defined previously. ROUPS WITH SPANIER–WHITEHEAD DUALITY 18
Proposition 27.
The class MF coincides with δ G from Definition 11 up to KK -equivalence. More precisely, we have δ G = MF b ⊗ C ∗ r ( G ) b ⊗ C ( BG ) τ C ∗ r ( G ) ([ Y ∗ ]) . Proof.
Let us set Z = GJ − ([ Y ]) ∈ KK G ( C ( EG ) , C ( BG )). It is shown in [KP18]that Z is represented by a G - C ∗ -correspondence satisfying the following isomor-phism of Hilbert modulesMF ∼ = i ∗ ( Y ∗ ) b ⊗ C ( EG ) ⋊ G ( Z ⋊ r G )(we are denoting by i the inclusion C ֒ → C ( BG ) as constant functions). We wantto prove[ p G ] b ⊗ C ( EG ) ⋊ G [∆] = i ∗ ([ Y ∗ ]) b ⊗ C ( EG ) ⋊ G Gr ( Z ) b ⊗ C ∗ r ( G ) b ⊗ C ( BG ) τ C ∗ r ( G ) ([ Y ∗ ]) , or equivalently, by Lemma 12,[ p G ] b ⊗ C ( EG ) ⋊ G [∆] = i ∗ ([ Y ∗ ]) b ⊗ C ( EG ) ⋊ G ([∆] b ⊗ τ C ∗ r ( G ) (GJ( Z ))) b ⊗ C ∗ r ( G ) b ⊗ C ( BG ) τ C ∗ r ( G ) ([ Y ∗ ]) . It is well-known that [ p G ] = i ∗ ([ Y ∗ ]) (see for example [Lan15]), so that by associa-tivity of the Kasparov product we have reduced the problem to showing τ C ∗ r ( G ) (GJ( Z ))) b ⊗ τ C ∗ r ( G ) ([ Y ∗ ]) = τ C ∗ r ( G ) (GJ( Z ) b ⊗ C ( BG ) [ Y ∗ ]) = 1 C ∗ r ( G ) b ⊗ C ( EG ) ⋊ G . Now GJ( Z ) = [ Y ] by construction, hence the proof is complete. (cid:3) Now suppose that G is a general torsion-free group, and that a ( γ )-element x = [ H, T ] exists. Inspired by Kasparov’s construction, we define the class d inKK( C ∗ r ( G ) ⊗ C ( BG ) , C ) by setting d = [ Y ] b ⊗ C ( EG ) ⋊ G d. The element d admits a simple description in terms of the cycle ( H, T ) with prop-erty ( γ ) as follows. The G -equivariant non-degenerate representation π of C ( EG )on H extends to the one of the multiplier algebra C b ( EG ). Together with the rep-resentation π G of G on H , it induces the representation π G ⊗ π of C ∗ r ( G ) ⊗ C ( BG )on H . Here, C ( BG ) is naturally identified as the subalgebra C b ( EG ) consisting of G -invariant functions. The representation π G extends to the one for C ∗ r ( G ) since π G is weakly contained in the left regular representation. Indeed, π G is containedin the (amplified) left regular representation as we have a G -equivariant embeddingfrom H to H ⊗ ℓ ( G ) given by v X h π ( h ( c )) v ⊗ δ h . Proposition 28.
The triple ( H, π G ⊗ π, T ) defines a Kasparov cycle [ π G ⊗ π, H, T ] for KK( C ∗ r ( G ) ⊗ C ( BG ) , C ) . We have [ π G ⊗ π, H, T ] = d .Proof. We need to show that for any G -invariant continuous function φ on EG , thecommutator [ T, φ ] is compact. By the condition (2.2) for property ( γ ), we just needshow that [ T ′ , φ ] is compact where T ′ = P g ∈ G g ( c ) T g ( c ); c is a cutoff function on EG . Take any compactly supported function χ on EG so that cχ = c .We have [ T ′ , φ ] = X g ∈ G g ( c )[ T, g ( χφ )] g ( c ) = X g ∈ G g ( c ) T g g ( c )where T g = [ T, g ( χφ )] are compact operators whose norm vanish as g goes to infinityby the condition (2.1) for property ( γ ). It follows that [ T ′ , φ ] = P g ∈ G g ( c ) T g g ( c )is compact (see Lemma 2.5, 2.6 of [Nis19]). ROUPS WITH SPANIER–WHITEHEAD DUALITY 19
We leave to the reader the straightforward check that the element [
H, π G ⊗ π, T ]in KK( C ∗ r ( G ) ⊗ C ( BG ) , C ) corresponds to d in KK( C ∗ r ( G ) ⊗ C ( EG ) ⋊ G, C ) bythe Morita equivalence between C ( BG ) and C ( EG ) ⋊ G . (cid:3) We set Λ C ∗ r ( G ) = [MF] b ⊗ C ( BG ) d, Λ C ( BG ) = [MF] b ⊗ C ∗ r ( G ) d. The following conclusions are immediate from the discussion above.
Theorem 29.
Let G be a torsion-free group and suppose a ( γ ) -element x ∈ KK G ( C , C ) exists. We have Λ C ∗ r ( G ) = Gr ( x ) , Λ C ( BG ) = 1 C ( BG ) . For example, this is the case when BG is a compact smooth manifold of non-postivesectional curvature. Examples
In this section we give a few examples and computations to put into context theabstract duality results that have been explained previously. We primarily treatthe case of strong Spanier–Whitehead duality, and briefly discuss the weak case asit is mostly covered by other results in the literature (see for example [BMRS08,Example 2.14 & 2.17]).2.1.
Groups with Spanier–Whitehead K -duality. Let G be a countable dis-crete group which satisfies the following two conditions (1), (2) or (1), (3):(1) G admits a G -compact model of EG ;(2) G admits a γ -element γ such that Gr ( γ ) = 1 C ∗ r ( G ) , or(3) G admits a ( γ )-element x such that Gr ( x ) = 1 C ∗ r ( G ) .We recall that the gamma element γ , if exists, is represented by a cycle with property( γ ). Therefore, the condition (2) implies (3). Our previous argument shows thatsuch a group G has Spanier–Whitehead K -duality. Thanks to the Higson–KasparovTheorem [HK01], we obtain the following: Theorem 30.
All a-T-menable groups which admit a G -compact model of EG haveSpanier–Whitehead K -duality. Examples of such a-T-menable groups are the following: • All groups which act properly, affine-isometrically and co-compactly on afinite-dimensional Euclidean space. • All co-compact lattices of simple Lie groups SO( n,
1) or SU( n, • All groups which act co-compactly on a tree (or more generally on a CAT(0)-cube complex).For any a-T-menable group G listed above, the gamma element γ can be representedby an explicit cycle with property ( γ ). Below, we describe an explicit cycle withproperty ( γ ) for these groups. As a consequence, we can obtain an explicit cycle d in KK( C ∗ r ( G ) ⊗ C ( EG ) ⋊ G, C ) which together with δ , induces the duality between C ∗ r ( G ) and C ( EG ) ⋊ G .To begin with, we recall from [Kas88, Val02] that the gamma element exists forany group G which acts properly, isometrically on a simply connected, completeRiemannian manifold M of non-positive sectional curvature which is bounded frombelow. In this case, the gamma element for G is represented by an unbounded G -equivariant Kasparov cycle ( H M , D M ) ROUPS WITH SPANIER–WHITEHEAD DUALITY 20 where H M is the Hilbert space L ( M, Λ ∗ T ∗ C M ) of L -sections of the complexifiedexterior algebra bundles on M and where D M is the self-adjoint operator D M = d f + d ∗ f on M given by the following Witten type perturbation d f = d + df ∧ of the exterior derivative d ; the function f is the squared distance d M ( x , x ) on M for some fixed point x of M . Let F M = D M (1 + D M ) be the bounded transform of D M . The element [ H M , F M ] in KK G ( C , C ) is thegamma element for G . We now suppose furthermore that G action on M is co-compact. In this case, G admits a G -compact model of EG , namely the manifold M . Proposition 31.
The cycle ( H M , F M ) has property ( γ ) .Proof. Since [ H M , F M ] is the gamma element for G , it satisfies the condition (1)of property ( γ ). To show that the condition (2) holds for [ H M , F M ], we shallapply Theorem 6.1 of [Nis19]. We use the natural non-degenerate representationof C ( M ) on H M by pointwise multiplication. We take the dense subalgebra B of C ( M ) consisting of compactly supported smooth functions. Note that B containsa cutoff function of M . For any function h in B , we have[ D M , g ( h )] = [ d + d ∗ , g ( h )] = g ( c ( h ))where c ( h ) is the Clifford multiplication by the gradient of h which is bounded andcompactly supported. We can now use Theorem 6.1 of [Nis19] to conclude that thebounded transform ( H M , F M ) satisfies the condition (2) of property ( γ ). (cid:3) Corollary 32.
For all groups G which act properly, affine-isometrically, and co-compactly on a finite-dimensional Euclidean space R n , the G -equivariant cycle ( H R n , F R n ) has property ( γ ) . Corollary 33.
For all co-compact closed subgroups G of a semi-simple Lie group L , the G -equivariant cycle ( H L/K , F
L/K ) has property ( γ ) , where K is a maximalcompact subgroup of L . Let us look at a few examples.
Poincaré–Langlands duality:
In [NPW16] the authors examine the Baum–Connes correspondence for the (extended) affine Weyl group W a associatedto a compact connected semisimple Lie group G . This group can be realizedas the group of affine isometries of the Lie algebra t of a maximal torus T ⊆ G . The structure of W a is that of a semidirect product Γ ⋊ W , whereΓ is the lattice of translations in t , and W is the Weyl group of the rootsystem of G .Ultimately, it is shown that the Baum–Connes conjecture (which holdsin this case) is equivalent to T -duality for the aforementioned torus T andthe Pontryagin dual ˆΓ of the lattice Γ. From the viewpoint of Lie groups,ˆΓ equivariantly coincides with the maximal torus T ∨ of the Langlandsdual G ∨ of G . In K -theory this is expressed by W -equivariant Spanier–Whitehead duality between the dual tori T and T ∨ , which is referred to as“Poincaré–Langlands duality” in [NPW16]. ROUPS WITH SPANIER–WHITEHEAD DUALITY 21
The results from Subsection 1.3 in this paper can be equivalently appliedto get these results, with C ( B Γ) playing the role of C ( T ) and C ∗ r (Γ) playingthe role of C ( T ∨ ) through the Gelfand transform.The ( γ )-element, which belongs to KK W a ( C , C ), in this case can be con-structed as explained above with M = t and distance function inducedby a W -equivariant metric. Equivalently, the bounded transform of theBott–Dirac operator B t = X i (ext( e i ) + int( e )) x i + (ext( e i ) − int( e i )) ddx i yields a W -equivarant cycle with property ( γ ), provided that interior mul-tiplication is defined through a W -equivariant metric. The cycle obtainedthis way is indeed isomorphic to the one obtained through the Witten typeperturbation of the de Rham operator, and its KK-class coincides with theclassical γ -element which is homotopic to the unit [HK01].In summary, we obtain equivariant duality classes δ W ∈ KK W ( C , C ( t / Γ) ⊗ C ∗ r (Γ)), derived from the Miščenko W -bundle associated to the principalΓ-bundle t → T , and d W ∈ KK W ( C ( t / Γ) ⊗ C ∗ r (Γ) , C ), derived from the( γ )-element described above. We can prove δ W b ⊗ C ( T ) d W = Γ r ( γ ) , where on the right-hand side we mean “partial” descent with respect to thenormal subgroup Γ ⊆ W a . As we know γ = 1 C in KK Γ × W ( C , C ), so thatwe get respectively δ W b ⊗ C ( T ) d W = 1 , δ W b ⊗ C ( T ∨ ) d W = 1in the equivariant groups KK W ( C ( T ∨ ) , C ( T ∨ )) , KK W ( C ( T ) , C ( T )). Lattices in
SO( n, and SU( n, : Let G be a co-compact lattice of a simpleLie group L = SO( n, L = SU( n, K be a maximal compact sub-group of L . Corollary 33 shows that the G -equivariant cycle ( H L/K , F
L/K )has property ( γ ). The corresponding element x = [ H L/K , F
L/K ] is noth-ing but the gamma element γ for G which is shown to be equal to 1 G [HK01, JK95]. Groups acting on trees:
Let G be a countable discrete group which actsproperly and co-compactly on a locally finite tree Y . The tree Y is the unionof the sets Y , Y of the vertices and edges of the tree. Without loss ofgenerality, we assume a G -invariant typing on the tree. Namely, we assumea G -invariant decomposition Y = Y ⊔ Y so that any two adjacent verticeshave distinct types. This can be achieved by the barycentric subdivisionof the tree. We take E as the geometric realization of the tree. This is a G -compact model of the universal proper G -space. We denote by d , theedge path metric on E and hence on Y so that each edge has length 1.The ℓ space ℓ ( Y ) is naturally a graded G -Hilbert space with the evenand odd spaces being ℓ ( Y ), ℓ ( Y ) respectively. Let H R be the gradedHilbert space L ( R , Λ ∗ C ( R )) as before, but now with the trivial G -action.We construct a Kasparov cycle with the property ( γ ) on the graded tensorproduct H Y = H R b ⊗ ℓ ( Y ) . Following [KS91], we define a non-degenerate representation π of C ( E ) on H Y , which is diagonal with respect to Y . This is given by a family ( π y ) y ∈ Y of representations of C ( E ) on H R indexed by y in Y . If y is a vertex, wedefine π y by sending φ in C ( E ) to the multiplication on H R by constant ROUPS WITH SPANIER–WHITEHEAD DUALITY 22 φ ( y ). If y is an edge with vertices y , y of corresponding types, we identify y with the interval [ − , ] via the unique isometry sending y j to ( − j .We define π y by sending φ in C ( E ) to the multiplication on H R by therestriction of φ to the edge y extended to left and right constantly.Now, like the operator D M , we shall define an unbounded, odd, self-adjoint operator D Y with compact resolvent of index 1, which is almost G -equivariant and has nice compatibility with functions in C ( E ). Thebounded transform F Y of D Y will give us a desired Kasparov cycle ( H Y , F Y )with property ( γ ). For this, we fix a base point y from Y . The followingconstruction depends on the choice of y . We have the following decompo-sition of H Y : H Y = H R b ⊗ C δ y ⊕ M y ∈ Y \{ y } (cid:0) H R b ⊗ ( C δ y ⊕ C δ e y ) (cid:1) where for each vertex y = y , e y is the last edge appearing in the geodesicfrom y to y and where the symbol δ ∗ denotes a delta-function in ℓ ( Y ).Our operator D Y is block-diagonal with respect to this decomposition. Itis given by a family ( D y ) y ∈ Y of an unbounded, odd, self-adjoint operatorswith compact resolvent.For a vertex y ∈ Y j of type j , let B R ,y be the Bott–Dirac operator on H R with “origin shifted”: B R ,y = (ext( e ) + int( e ))( x − n y ) + (ext( e ) − int( e )) ddx where n y = ( − j ( + d ( y, y )). For y = y , we simply set D y = B R ,y b ⊗ H R b ⊗ C δ y . For y = y , we set D y = B R ,y b ⊗ M χ y b ⊗ (cid:18) (cid:19) on H R b ⊗ ( C δ y ⊕ C δ e y )where M χ y is the multiplication on H R by the function χ y on R defined as:for y ∈ Y , χ y ( x ) = x < x −
12 ) ≤ x < x −
34 1 ≤ x < d ( y, y ) − ( x − n y ) + d ( y, y ) − d ( y, y ) ≤ x < n y d ( y, y ) − n y ≤ x, for y ∈ Y , χ y ( x ) = d ( y, y ) − x < n y − ( x − n y ) + d ( y, y ) − n y ≤ x < − d ( y, y ) − x − − d ( y, y ) ≤ y < − x + 12 ) − ≤ x < − − ≤ x. ROUPS WITH SPANIER–WHITEHEAD DUALITY 23
Note that for each y = y , D y is a bounded perturbation of a self-adjointoperator B R ,y b ⊗ D y . All D y are hence diagonalizable. Therefore, D Y = ( D y ) y ∈ Y is self-adjoint. Inorder to see that D Y has compact resolvent, we compute D y = B R ,y b ⊗ M χ y b ⊗ (cid:18) − M χ ′ y M χ ′ y (cid:19) b ⊗ (cid:18) (cid:19) where χ ′ y is the derivative of χ y . We see that D y has spectrum far away from0 as y goes to infinity essentially because the derivatives χ ′ y are uniformlybounded in y and because we have( x − n y ) + χ y ≥ (cid:16) d ( y, y )2 − (cid:17) everywhere. It follows D Y has indeed, compact resolvent. Let F Y be thebounded transform F Y = D Y (1 + D Y ) . Proposition 34.
A pair ( H Y , F Y ) is a G -equivariant Kasparov cycle withproperty ( γ ) .Proof. Almost G -equivariance follows from D Y − g ( D Y ) = bounded for g ∈ G which we leave to the reader. To see that [ H Y , F Y ] = 1 F in R ( F ) for anyfinite subgroup F of G , we note that the class [ H Y , F Y ] does not dependon the choice of the base point y . Hence, we may assume that y is avertex fixed by the group F . In this case, it is not hard to see that F Y is odd, F -equivariant, self-adjoint operator whose graded index is the one-dimensional trivial representation of F spanned by ξ b ⊗ δ y in H R b ⊗ C δ y where ξ = e − x . This shows [ H Y , F Y ] = 1 F . To show that is has thecondition (2) of property ( γ ) with respect to the representation π of C ( E ),we shall apply Theorem 6.1 of [Nis19] for the dense subalgebra B of C ( E )consisting of compactly supported functions which are smooth inside eachedge and constant near the vertices. Note that B contains a cutoff functionof E . First, we can see that for each y = y , the operator M χ y b ⊗ (cid:18) (cid:19) commutes with the representation π . This is due to the vanishing of χ y for y ∈ Y (resp. Y ) over x ≤ (resp. over − ≤ x ). For φ in B , wecompute the commutator [ D Y , π ( φ )] as[ D Y , π ( φ )] = [ B R ,y b ⊗ , π ( φ )]+ X y ∈ Y \{ y } [ B R ,y b ⊗ M χ y b ⊗ (cid:18) (cid:19) , π ( φ )]= [ B R ,y b ⊗ , π ( φ )] + X y ∈ Y \{ y } [ B R ,y b ⊗ , π ( φ )]= X y ∈ Y \{ y } [ (cid:18) − ddxddx (cid:19) , π e y ( φ )] b ⊗ π ( φ ′ ) X y ∈ Y \{ y } (cid:18) −
11 0 (cid:19) b ⊗ H R b ⊗ C δ e y andwhere φ ′ is the derivative of φ . Note that each summation is finite sum ROUPS WITH SPANIER–WHITEHEAD DUALITY 24 since φ is compactly supported. We can now use Theorem 6.1 of [Nis19]to conclude that the bounded transform ( H Y , F Y ) satisfies condition (2) inthe definition of property ( γ ). (cid:3) Remark 35.
The construction can be generalized to define a cycle withproperty ( γ ) for a group which acts properly and co-compactly on a Eu-clidean building in a sense of [KS91] . In [BGHN19] , a different constructionis given which provides us a cycle with property ( γ ) for a group which actsproperly and co-compactly on a finite-dimensional CAT(0) cube complex. Groups with weak Spanier–Whitehead K -duality. Let G be a countablediscrete group satisfying the two conditions (1), (2)’ or (1), (3)’ below:(1) G admits a G -compact model of EG ;(2)’ G admits a γ -element γ with Gr ( γ ) acting as the identity on K ∗ ( C ∗ r ( G )),or(3)’ G admits a ( γ )-element x with Gr ( x ) acting as the identity on K ∗ ( C ∗ r ( G )).Our previous argument shows that such a group G has weak Spanier–Whitehead K -duality. For any word-hyperbolic group, the gamma element is shown to existand the Baum–Connes conjecture has been verified [Laf12, KS03, MY02]. Moreover,any hyperbolic group is known to admit a G -compact model of EG [MS02]. Hence,we have: Theorem 36.
All word-hyperbolic groups G have weak Spanier–Whitehead K -duality. As an example of hyperbolic groups, we can take G to be a co-compact latticesof the simple Lie group L = Sp( n, γ -element for G has an explicitrepresentative ( H L/K , F
L/K ) with property ( γ ). We remark that the gamma ele-ment γ = [ H L/K , F
L/K ] is well-known to be not homotopic to 1 G due to Kazhdan’sproperty (T). Furthermore, Skandalis [Ska88] showed that Gr ( γ ) is not equal to1 C ∗ r ( G ) . More precisely, what he showed is that C ∗ r ( G ) is not K -nuclear, which inparticular implies that it cannot be KK-equivalent to any nuclear C ∗ -algebra. Thesame remark that Gr ( γ ) = 1 C ∗ r ( G ) applies to any infinite hyperbolic property (T)group [HG04, Theorem 5.2]. In general, when the gamma element γ exists, theequality Gr ( γ ) = 1 C ∗ r ( G ) implies that C ∗ r ( G ) is KK-equivalent to P C ⋊ G , which sat-isfies the UCT ([MN06, Proposition 9.5]), in particular it is K -nuclear. Therefore,if C ∗ r ( G ) is not K -nuclear, we have Gr ( γ ) = 1 C ∗ r ( G ) .3. Some applications
In this section we prove a few results by applying the theory of K -duality devel-oped in the previous pages. Some of the material presented here has been previouslytreated in the literature via possibly different methods [Dad09, Section 3] [EM10,Section 5], [KPW17, Section 4.4], [RS87, Section 7], nevertheless we provide a briefaccount for completeness, to give a better idea of some applications of our maintheorems.We say a C ∗ -algebra A is KK -compact if the functor sending D to KK ∗ ( A, D )commutes with filtered colimits. If A is a C ∗ -algebra with a Spanier–Whitehead K -dual B , then A is KK-compact because KK ∗ ( A, D ) is naturally isomorphic toKK ∗ ( C , D ⊗ B ) and the K -theory functor is continuous.As explained after Theorem 6.6 of [MN06], a C ∗ -algebra satisfies the UniversalCoefficient Theorem (UCT) ([Bla98, Section 23]) if and only if it belongs to thelocalizing triangulated subcategory of the KK-category generated by the complexnumbers (this category is denoted as h∗i in [MN06]). As in [DEKM11], let us
ROUPS WITH SPANIER–WHITEHEAD DUALITY 25 denote this subcategory by T . It is known that within this subcategory, an objectis dualizable if and only if it is compact: Proposition 37. ( [DEKM11, Proposition 4.1] ) In the subcategory T ⊆ KK , thefull triangulated subcategory T c of compact objects coincides with the (closed) sym-metric monoidal category T d of dualizable objects. Furthermore, both these two sub-categories are equal to the thick triangulated subcategory generated by the complexnumbers. Corollary 38. If G has Spanier–Whitehead duality then C ∗ r ( G ) satisfies the UCT.Proof. We know that C ( EG ) ⋊ G satisfies the UCT [MN06, Proposition 9.5]. Byassumption, C ( EG ) ⋊ G has a Spanier–Whitehead K -dual C ∗ r ( G ). Thus, C ( EG ) ⋊ G is KK-compact. By Proposition 37, it is dualizable in T . Namely, it has aSpanier–Whitehead K -dual, say A , which satisfies the UCT. On the other hand, itis fairly easy to see that a dual object is unique up to equivalence. Hence, C ∗ r ( G )is KK-equivalent to A . The claim follows from this. (cid:3) The strong Baum–Connes conjecture was introduced in [MN06] as the assertionthat the canonical Dirac morphism α in KK G ( P C , C ) induces a KK-equivalence Gr ( α ) from P C ⋊ G to C ∗ r ( G ). In the presence of the gamma element γ for G , thisis equivalent to the assertion that Gr ( γ ) = 1 C ∗ r ( G ) . Theorem 39. If G has Spanier–Whitehead duality then it satisfies the strongBaum–Connes conjecture. Moreover, if the γ -element exists and G satisfies thestrong Baum–Connes conjecture, than G has Spanier–Whitehead duality.Proof. Suppose G has Spanier–Whitehead duality. Then, we know that the Baum–Connes conjecture holds for G , and so the Dirac morphism α induces an isomor-phism Gr ( α ) ∗ on K -groups from P C ⋊ G to C ∗ r ( G ). Furthermore, both P C ⋊ G and C ∗ r ( G ) satisfy the UCT by [MN06, Proposition 9.5] and by Corollary 38 respectively.It follows that Gr ( α ) is a KK-equivalence [Bla98, Theorem 23.10.1]. Conversely, ifthe strong Baum–Connes conjecture holds, we have Gr ( γ ) = 1 C ∗ r ( G ) . Hence, G hasSpanier–Whitehead duality by Theorem 21. (cid:3) As in [Bla98, Theorem 23.10.5], a C ∗ -algebra A satisfies the UCT if and only ifit is KK-equivalent to a commutative C ∗ -algebra C ( X ). Furthermore, this X canbe taken to be a 3-dimensional cell complex (see [Bla98, Corollary 23.10.3], [RS87,Proposition 7.4]). This is because the range of K -theory on such spaces exhaustsall countable Z / (2)-graded abelian groups. If K ∗ ( A ) is finitely generated, then X can be chosen finite, and a Spanier-Whitehead K -dual exists for such spaces [EM10,Proposition 5.9]. Lemma 40.
Suppose A has a Spanier–Whitehead K -dual and satisfies the UCT.Then it has finitely generated K -theory groups.Proof. As in the proof of [RS87, Proposition 7.4], [Bla98, Corollary 23.10.3], let C = C ⊕ C be a commutative C ∗ -algebra KK-equivalent to A , where C isthe mapping cone of a ∗ -homomorphism on direct sums of C ( R ), and C is thesuspension of such a mapping cone. It is easy to see that C is the inductive limitof subalgebras C n where C n has finitely generated K -theory. Since KK ∗ ( A, − ) iscontinuous (since A is KK-compact), the equivalence A → C factors through C n forsome n ∈ N . Then K ∗ ( A ) is finitely generated because it is a quotient of K ∗ ( C n ),which enjoys this property. (cid:3) Proposition 41.
Suppose G satisfies the Baum–Connes conjecture and the γ -element exists. Then C ∗ r ( G ) has finitely generated K -theory groups. ROUPS WITH SPANIER–WHITEHEAD DUALITY 26
Proof. If γ ∈ KK G ( C , C ) exists, then P C ⋊ G is dualizable by Theorem 23. It isknown that P C ⋊ G satisfies the UCT (see [MN06, Proposition 9.5]). Thus, P C ⋊ G has finitely generated K -groups by Lemma 40. Recall that in the localizationpicture the assembly map appears as K ∗ ( P C ⋊ G ) −→ K ∗ ( C ∗ r ( G )) . (10)Therefore if (10) is an isomorphism the right-hand side is finitely generated. (cid:3) Remark 42.
More generally, C ∗ r ( G ) has finitely generated K -theory groups if G satisfies the Baum–Connes conjecture and the source P C of the Dirac morphism isa (categorical) direct summand of a proper algebra. This is because by Remark 25 P C ⋊ G has a Spanier–Whitehead K -dual. Remark 43.
By the results in [DEKM11] , there exists a functor K from the KK -category to the stable homotopy category, satisfying π n ( K ( A )) ∼ = K n ( A ) . This func-tor specializes to a full and faithful functor on the subcategory of dualizable objectssatisfying the UCT, realizing C ∗ -algebras as perfect KU -modules (in particular, fi-nite spectra). Hence the previous results can be also obtained from the well-knownfact that homotopy groups are finitely generated in this context. Define the n -th dimension-drop algebra as I n = { f ∈ C ([0 , , M n ( C )) | f (0) = 0 , f (1) ∈ C n } . We can use this to introduce the mod- n K -theory groups as follows: K ∗ ( B ; Z / ( n )) = KK ∗ ( I n , D ) . It is apparent from this definition that a Baum–Connes conjecture in mod- n K -theory for B would have to introduce coefficients on the left, and we can take thisas motivation to find a satisfactory formulation for the full bivariant version of theBaum–Connes conjecture. The approach via localization immediately generalizesto this context, giving us a mapKK ∗ ( A, ( P C ⊗ B ) ⋊ G ) −→ KK ∗ ( A, B ⋊ r G ) (11)defined as y y ⊗ Gr (1 B b ⊗ α ), where α ∈ KK( P C , C ) is the Dirac morphism, forany (separable) C ∗ -algebra A and G - C ∗ -algebra B .The original definition of the left-hand side (following [BMP03] and [Uuy11]),what is called the “naive” topological K -group in [Uuy11], is given aslim −→ Y ⊆ EG KK G ∗ ( C ( Y, A ) , B ) , where the limit ranges as usual over G -invariant G -compact subspaces of EG . Un-like the simpler case of the conjecture, the definition making use of the naive topo-logical group is not equivalent to the definition in (11). However [Uuy11] showsthat there are natural maps ν Y : KK G ∗ ( C ( Y, A ) , B ) −→ KK ∗ ( A, ( P C ⊗ B ) ⋊ G ) , (12)which make the obvious diagram commute. In addition, if A admits a Spanier–Whitehead K -dual, then (12) induces an isomorphism. Theorem 44 ([Uuy11]) . Suppose A has a Spanier–Whithead K -dual. Then thecomparison map induced by the ν Y ’s is an isomorphism. ROUPS WITH SPANIER–WHITEHEAD DUALITY 27
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