Bivariant KK -Theory and the Baum-Connes conjecure
aa r X i v : . [ m a t h . K T ] J un BIVARIANT KK -THEORY AND THE BAUM-CONNESCONJECURE SIEGFRIED ECHTERHOFF
Abstract.
This is a survey on Kasparov’s bivariant KK -theory in connectionwith the Baum-Connes conjecture on the K -theory of crossed products A ⋊ r G by actions of a locally compact group G on a C*-algebra A . In particular weshall discuss Kasparov’s Dirac dual-Dirac method as well as the permanenceproperties of the conjecture and the “Going-Down principle” for the left handside of the conjecture, which often allows to reduce K -theory computations for A ⋊ r G to computations for crossed products by compact subgroups of G . Wegive several applications for this principle including a discussion of a methoddeveloped by Cuntz, Li and the author in [CEL13] for explicit computationsof the K -theory groups of crossed products for certain group actions on totallydisconnected spaces. This provides an important tool for the computation of K -theory groups of semi-group C*-algebras. Introduction The extension of K -theory from topological spaces to operator algebras providesthe most powerful tool for the study of C ∗ -algebras. On one side there now exist farreaching classification results in which certain classes of C ∗ -algebras can be classifiedby their K -theoretic data. This started with the early work of Elliott [Ell76] on theclassification of AF -algebras – inductive limits of finite dimensional C ∗ -algebras.It went on with the classification of simple, separable, nuclear, purely infinite C ∗ -algebras by Kirchberg and Phillips [KP00, Phi00]. In present time, due to the workof many authors (e.g., see [Win16] for a survey on the most recent developments)the classification program covers a very large class of nuclear algebras.On the other hand, the K -theory groups of group algebras C ∗ ( G ) and C ∗ r ( G ) serveas recipients of indices of G -invariant elliptic operators and the study of such indiceshas an important impact in modern topology and geometry. To get a rough idea,the Baum-Connes conjecture implies that every element in the K -theory groupsof the reduced C ∗ -group algebra C ∗ r ( G ) of a locally compact group G appears assuch index of some generalised G -invariant elliptic operator. To be more precise,these generalised elliptic operators form the cycles of the G -equivariant K -homology(with G -compact supports) K G ∗ ( EG ), in which EG is a certain classifying space for The content of this note will appear as Chapter 3 of the book “ K -theory for group C*-algebrasand semigroup C*-algebras” which will appear in the Oberwolfach-seminar series of the Birkh¨auserpublishing company. The research for this paper has been supported by the DFG through CRC878 Groups, Geometry & Actions proper actions of G (often realised as a G -manifold) and the index map µ G : K G ∗ ( EG ) → K ∗ ( C ∗ r ( G ))is then a well defined group homomorphism. It is called the assembly map for G .The Baum-Connes conjecture (with trivial coefficients) asserts that the assemblymap is an isomorphism for all G .The construction of the assembly map naturally extends to crossed-products andprovides a map µ ( G,A ) : K G ∗ ( EG, A ) → K ∗ ( A ⋊ r G ) . The
Baum-Connes conjecture with coefficients asserts that this more general as-sembly map should be an isomorphism as well. Although this general version ofthe conjecture is now known to be false in general (e.g. see [HLS02]), it is known tobe true for a large class of groups, including the class of all amenable groups, andit appears to be an extremely useful tool for the computation of K -theory groupsin several important applications.In this chapter we want to give a concise introduction to the Baum-Connes conjec-ture and to some of the applications which allow the explicit computation of K -theory groups with the help of the conjecture. We start with a very short reminderof the basic properties of C ∗ -algebra K -theory before we give an introduction ofKasparov’s bivariant K -theory functor which assigns to each pair of G - C ∗ -algebras A, B a pair of abelian groups KK G ∗ ( A, B ), ∗ = 0 ,
1. Kasparov’s theory is not onlyfundamental for the definition of the groups K G ∗ ( EG ) and K G ∗ ( EG, A ) and theconstruction of the assembly map, but it also provides the most powerful tools forproving the Baum-Connes conjecture for certain classes of groups. In this chapterwe will restrict ourselves to Kasparov’s picture of KK -theory and we will not touchon other descriptions or variants like the Cuntz picture of KK -theory or E -theoryas introduced by Connes and Higson. We refer to [Bla86] for a treatment of theseand their connections to Kasparov’s theory.As part of our introduction to KK -theory we will give a detailed and a fairly elemen-tary proof of Kasparov’s Bott-periodicity theorem in one dimension by constructingDirac and dual Dirac elements which implement a KK -equivalence between C ( R )and the first complex Clifford algebra Cl . We shall later use these computationsto give a complete proof of the Baum-Connes conjecture for R and Z with the helpof Kasparov’s Dirac-dual Dirac method. This method is the most powerful toolfor proving the conjecture and has been successfully applied to a very large classof groups including all amenable groups. As corollaries of our proof of the con-jecture for R and Z , we shall also present proofs of Connes’s Thom isomorphismfor crossed products by R and the Pimsner-Voiculescu six-term exact sequence forcrossed products by Z .In the last part of this chapter we shall present the “Going-Down” principle whichroughly says the following: Suppose G satisfies the Baum-Connes conjecture withcoefficients. Then any G -equivariant ∗ -homomorphism (or KK -class) between two G -algebras A and B which induces isomorphisms between the K -theory groups of A ⋊ K and B ⋊ K for all compact subgroups K of G also induces an isomorphismbetween the K -groups of A ⋊ r G and B ⋊ r G . We shall give a complete proofof this principle if G is discrete and we present a number of applications of this IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 3 result. In particular, as one application we shall present a theorem about possibleexplicit computations of the K -theory of crossed products C (Ω) ⋊ r G in whicha discrete group G acts on a totally disconnected space Ω with some additional“good” properties which we shall explain in detail. This result is basic for the K -theory computations of the reduced semi-group C ∗ -algebras as presented in [Li17]and [Cun17].There are many other surveys on the Baum-Connes conjecture which look at theconjecture from quite different angles. The reader should definitely have a lookat the paper [BCH94] of Baum, Connes, and Higson, where a broad discussion ofvarious applications of the conjecture is given. The survey [Val03] by Alain Valetterestricts itself to a discussion of the Baum-Connes conjecture for discrete groups,but also provides a good discussion of applications to other important conjectures.The survey [MV03] by Mislin discusses the conjecture from the topologist’s point ofview, where the left hand side (the topological K -theory of G ) is defined in termsof the Bredon cohomology – a picture of the Baum-Connes conjecture first given byDavis and L¨uck [DL98]. We also want to mention the paper [HG04] of Higson andGuentner, which gives an introduction of the Baum-Connes conjecture based on E -theory. Last but not least, we suggest to the interested reader to study the book[HR00] by Higson and Roe, where many of the relevant techniques for producingimportant KK -classes by elliptic operators (as the Dirac-class in K -homology) aretreated in a very nice way.Throughout this chapter we assume that the reader is familiar with the basicson C ∗ -algebras, the basic constructions and properties of full and reduced crossedproducts and the notion of Morita equivalence and Hilbert C ∗ -modules. A detailedintroduction to these topics is given in the first six sections of [Ech17].The author of this chapter likes to thank Heath Emerson and Michael Joachim forhelpful discussions on some of the topics in this chapter.2. Operator K -Theory In this section we want to give a very brief overview of the definition and some basicproperties of the K -theory groups of C ∗ -algebras. We urge the reader to have a lookat one of the standard books on operator K -theory (e.g., [Bla86, RLL00, WO93])for more detailed expositions of this theory.Let us fix some notation: If A is a C ∗ -algebra, we denote by M n ( A ) the C ∗ -algebraof all n × n -matrices over A and by A [0 ,
1] the C ∗ -algebra of continuous functions f : [0 , → A . Moreover, we denote by A = A L C C ∗ -algebra withunderlying vector space A L C a + λ b + µ
1) = ab + λb + µa + λµ a + λ ∗ = a ∗ + ¯ λ , for a + λ , b + µ ∈ A + C
1. Let ǫ : A → C ; ǫ ( a + λ
1) = λ . We call A the unitisation of A (even if A already has a unit). We write M ∞ ( A ) := ∪ n ∈ N M n ( A ) SIEGFRIED ECHTERHOFF where we regard M n ( A ) as a subalgebra of M n +1 ( A ) via T (cid:18) T
00 0 (cid:19) . We denoteby P ( A ) the set of projections p ∈ M ∞ ( A ), i.e., p = p ∗ = p . Definition 2.1.
Let A be a unital C ∗ -algebra and let p, q ∈ P ( A ). Then p, q arecalled • Murray-von Neumann equivalent (denoted p ∼ q ) if there exist x, y ∈ M ∞ ( A ) such that p = xy and q = yx . • unitarily equivalent (denoted p ∼ u q ) if there exists some n ∈ N and aunitary u ∈ U ( M n ( A )) such that p, q ∈ M n ( A ) and q = upu ∗ in M n ( A ). • homotopic (denoted p ∼ h q ), if there exists a projection r ∈ P ( A [0 , p = r (0) and q = r (1).All three equivalence relations coincide on P ( A ) (but not on the level of M n ( A ) forfixed n ∈ N ). If A is unital and if p, q ∈ M n ( A ), then (cid:18) p q (cid:19) and (cid:18) q p (cid:19) areMurray-von Neumann equivalent in M n ( A ) with x = (cid:18) pq (cid:19) and y = (cid:18) qp (cid:19) .This allows us to define an abelian semigroup structure on P ( A ) / ∼ with additiongiven by [ p ] + [ q ] = (cid:20)(cid:18) p q (cid:19)(cid:21) . For every unital C ∗ -algebra A , we define K ( A ) asthe Grothendieck group of the semigroup P / ∼ , that is K ( A ) = (cid:8)(cid:2) [ p ] − [ q ] (cid:3) : [ p ] , [ q ] ∈ P ( A ) / ∼ (cid:9) where we write (cid:2) [ p ] − [ q ] (cid:3) = (cid:2) [ p ′ ] − [ q ′ ] (cid:3) if and only if there exists h ∈ P ( A ) suchthat [ p ] + [ q ′ ] + [ h ] = [ p ′ ] + [ q ] + [ h ] in P ( A ) / ∼ . Example 2.2. If A = C , then two projections p, q ∈ P ( C ) are homotopic, if andonly if they have the same rank. It follows from this that P ( C ) / ∼∼ = N as semigroupand hence we get K ( C ) ∼ = Z .If Φ : A → B is a unital ∗ -homomorphism between the unital C ∗ -algebras A and B ,then there exists a unique group homomorphism Φ : K ( A ) → K ( B ) such thatΦ ([ p ]) = [Φ( p )].We then define K ( A ) := ker (cid:0) K ( A ) ǫ → K ( C ) ∼ = Z (cid:1) for any C ∗ -algebra A . If A is unital, then A ∼ = A L C as a direct sum of the C ∗ -algebras A and C (the isomorphism is given by a + λ ( a − λ A , λ )) and itis not difficult to check that in this case both definitions of K ( A ) coincide. Any ∗ -homomorphism Φ : A → B extends to a unital ∗ -homomorphism Φ : A → B ; Φ ( a + λ
1) = Φ( a ) + λ : K ( A ) → K ( B ) factorsthrough a well defined homomorphism Φ : K ( A ) → K ( B ).For the construction of K ( A ) let U n ( A ) denote the group of unitary elementsof M n ( A ). We embed U n ( A ) into U n +1 ( A ) via U (cid:18) U
00 1 (cid:19) , and we define U ∞ ( A ) = ∪ n ∈ N U n ( A ). Let U ∞ ( A ) denote the path-connected component of IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 5 U ∞ ( A ), where we say that two unitaries can be joined by a path in U ∞ ( A ) if andonly if they can be joined by a continuous path in U n ( A ) for some n ∈ N . For u, v ∈ U n ( A ), one can check that(2.1) (cid:18) uv
00 1 (cid:19) ∼ h (cid:18) u v (cid:19) ∼ h (cid:18) v u (cid:19) in U n ( A ) ⊆ U ∞ ( A ). Therefore, if we define K ( A ) := U ∞ ( A ) /U ∞ ( A ) with addition given by [ u ] + [ v ] = (cid:20)(cid:18) u v (cid:19)(cid:21) , (which by (2.1) is equal to [ uv ]) we see that K ( A ) is an abelian group. As for K ,for unital A we can alternatively construct K ( A ) without passing to the unitization A as K ( A ) = U ∞ ( A ) /U ∞ ( A ) . Example 2.3.
Since U n ( C ) is path connected for all n ∈ N , we have K ( C ) = { } .If Φ : A → B is a ∗ -homomorphism, there is a well defined group homomorphismΦ : K ( A ) → K ( B ); Φ ([ u ]) = (cid:2) Φ ( u ) (cid:3) , where, as before, Φ : A → B denotes the unique unital extension of Φ to A . Proposition 2.4.
The assignments A K ( A ) , K ( A ) are homotopy invariantcovariant functors from the category of C ∗ -algebras to the category of abelian groups. Homotopy invariance means that if Φ , Ψ : A → B are two homotopic ∗ -homomorphisms, then Φ ∗ = Ψ ∗ : K ∗ ( A ) → K ∗ ( B ), ∗ = 0 ,
1. Here a homotopybetween Φ and Ψ is a ∗ -homomorphism Θ : A → B [0 ,
1] such that Φ = ǫ ◦ Θand Ψ = ǫ ◦ Θ, where for all t ∈ [0 , ǫ t : B [0 , → B denotes evaluation at t .Of course, the homotopy invariance is a direct consequence of the fact that “ ∼ ”coincides with “ ∼ h ” on P ( B ).Recall that a C ∗ -algebra is called contractible , if the identity id : A → A ishomotopic to the zero map 0 : A → A . As an example, let A be any C ∗ -algebra, then A (0 ,
1] := { a ∈ A [0 ,
1] : a (0) = 0 } is contractible. A homotopybetween id : A (0 , → A (0 ,
1] and 0 is given by the path of ∗ -homomorphismΦ t : A (0 , → A (0 , (cid:0) Φ t ( a ) (cid:1) ( s ) = a ( ts ). Corollary 2.5.
Suppose that A is a contractible C ∗ -algebra. Then K ( A ) = { } = K ( A ) . If 0 → I ι → A q → B → C ∗ -algebras, then functorialityof K and K gives two sequences(2.2) K ( I ) ι → K ( A ) q → K ( B ) and K ( I ) ι → K ( A ) q → K ( B )which both can be shown to be exact in the middle. If there exists a splittinghomomorphism s : B → A for the quotient map q , it induces a splitting homomor-phism s ∗ : K ∗ ( B ) → K ∗ ( A ), ∗ = 0 ,
1, and in this case the groups K ( A ) and K ( A )decompose as direct sums K ( A ) = K ( I ) M K ( B ) and K ( A ) = K ( I ) M K ( B ) . SIEGFRIED ECHTERHOFF
In particular, in this special case the sequences in (2.2) become short exact se-quences of abelian groups.
Six-term exact sequence:
In general, the sequences in (2.2) can be joined intoa six-term exact sequence K ( I ) ι −−−−→ K ( A ) q −−−−→ K ( B ) ∂ x y exp K ( B ) ←−−−− q K ( A ) ←−−−− ι K ( I )which serves as a very important tool for explicit computations as well as for provingtheorems on K -theory. We refer to [Bla86, RLL00] for a precise description of theboundary maps exp and ∂ . Note that the six-term sequence is natural in the sensethat if we have a morphism between two short exact sequences, i.e., we have acommutative diagram0 −−−−→ I −−−−→ A −−−−→ B −−−−→ ϕ y ψ y y θ −−−−→ J −−−−→ C −−−−→ D −−−−→ C ∗ -algebras, then we havecorresponding commutative diagrams −−−−→ K i ( A ) −−−−→ K i ( B ) −−−−→ K i +1 ( I ) −−−−→ K i +1 ( A ) −−−−→ ψ i y θ i y y ϕ i +1 y ψ i +1 −−−−→ K i ( C ) −−−−→ K i ( D ) −−−−→ K i +1 ( J ) −−−−→ K i +1 ( C ) −−−−→ Aside of theoretical importance, this fact can often be used quite effectively forexplicit computations of the boundary maps in the six-term sequence.If we apply the six-term sequence to the short exact sequence0 → C (0 , ⊗ A ι → A (0 , q → A → q : A (0 , → A is given by evaluation at 1, we get thesix-term sequence K ( C (0 , ⊗ A ) ι −−−−→ q −−−−→ K ( A ) ∂ x y exp K ( A ) ←−−−− q ←−−−− ι K ( C (0 , ⊗ A )which shows that the connecting maps exp : K ( A ) → K ( C (0 , ⊗ A ) and ∂ : K ( A ) → K ( C (0 , ⊗ A ) are isomorphisms which are natural in A . Hence,by identifying (0 ,
1) with R and C ( R ) ⊗ C ( R ) with C ( R ), we can deduce thefollowing important results from the above six-term sequence Theorem 2.6 (Bott-periodicity) . For each C ∗ -algebra A there are natural isomor-phisms K ( A ) ∼ = K ( C ( R ) ⊗ A ) and K ( C ( R ) ⊗ A ) ∼ = K ( A ) . IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 7 Moreover, if we apply the first isomorphism to B = C ( R ) ⊗ A , we obtain isomor-phisms K ( A ) ∼ = K ( C ( R ) ⊗ A ) ∼ = K ( C ( R ) ⊗ A ) . We should note that the proof of the six-term sequence usually uses the Bott-periodicity theorem, so we do present the results in the wrong order. In any case,the proofs of the six-term sequence and of the Bott-periodicity theorem are quitedeep and we refer to the standard literature on K -theory (e.g., [Bla86]) for thedetails. We shall later provide a proof of Bott-periodicity in KK -theory, whichdoes imply Theorem 2.6. We close this short section with two more importantfeatures of K -theory: Continuity: If A = lim i A i is the inductive limit of an inductive system { A i , Φ ij } of C ∗ -algebras, then K ∗ ( A ) = lim i K ∗ ( A i ) , ∗ = 0 , . Morita invariance: If e ∈ M ( n, C ) is any rank-one projection, then Φ e : A → M n ( A ); Φ e ( a ) = e ⊗ a induces an isomorphism between K ∗ ( A ) and K ∗ ( M n ( A )), ∗ = 0 ,
1. More generally, if e is any rank-one projection in K = K ( ℓ ( N )), then thehomomorphism a e ⊗ a induces isomorphisms K ∗ ( A ) ∼ = K ∗ ( K ⊗ A ), ∗ = 0 , σ -unital C*-algebras A and B are stably isomorphic if andonly if they are Morita equivalent, it follows that Morita equivalent σ -unital C*-algebras have isomorphic K -theory groups.3. Kasparov’s equivariant KK -theory We now come to Kasparov’s construction of the G -equivariant bivariant K -theory,which in some sense is built on the correspondence category as described in Sec-tion 1.5. Readers who are not familiar with the notion of Hilbert modules andcorrespondences are advised to read that section before going on here. Since someof the constructions require that the C ∗ -algebras are separable and that Hilbert B -modules E are countably generated (which means that there is a countable sub-set C ⊆ E such that
C · B is dense in E ), we shall from now on assume that theseconditions will hold throughout, except for the multiplier algebras of separable C ∗ -algebras and the algebras of adjointable operators on a countably generated Hilbertmodule. We refer to [Bla86] or Kasparov’s original paper [Kas88] for a more detailedaccount on where these conditions can be relaxed.3.1. Graded C ∗ -algebras and Hilbert modules. We write Z for the groupwith two elements. A Z grading of a G - C ∗ -algebra ( A, α ) is given by an action ǫ A : Z → Aut( A ) which commutes with α . We then might consider A as a G × Z - C ∗ -algebra with action α × ǫ A and a graded G -equivariant correspondence betweenthe graded G - C ∗ -algebras ( A, α × ǫ A ) and ( B, β × ǫ B ) is just a G × Z -equivariantcorrespondence ( E , u × ǫ E , Φ) between these algebras.Moreover, if ǫ A is a grading of A , we write A := { a ∈ A : ǫ A ( a ) = a } and A = { a ∈ A : ǫ A ( a ) = − a } , for the eigenspaces of the eigenvalues 1 and − ǫ A ,and similar for gradings on Hilbert modules. The elements in A and A are calledthe homogeneous elements of A . We write deg( a ) = 0 if a ∈ A and deg( a ) = 1 SIEGFRIED ECHTERHOFF if a ∈ A . deg( a ) is called the degree of the homogeneous element a . Note that A is a C ∗ -subalgebra of A and every element a ∈ A has a unique decomposition a = a + a with a ∈ A , a ∈ A . If A is a Z graded C ∗ -algebra, the gradedcommutator [ a, b ] is defined as[ a, b ] = ab − ( − deg( a ) deg( b ) ba for homogeneous elements a, b ∈ A and it is defined on all of A by bilinear contin-uation.If A and B are two graded C ∗ -algebras, we define the graded algebraic tensorproduct A ⊙ gr B as the usual algebraic tensor product with graded multiplicationand involution given on elementary tensors of homogeneous elements by( a ⊗ b ) · ( a ⊗ b ) = ( − deg( b ) deg( a ) ( a a ⊗ b b )( a ⊗ b ) ∗ = ( − deg( a ) deg( b ) ( a ∗ ⊗ b ∗ ) . In what follows we write A ˆ ⊗ B for the minimal (or spatial) completion of A ⊙ gr B .We refer to [Bla86, 14.4] for more details of this construction. Example 3.1. (a)
For any C ∗ -algebra A there is a grading on M ( A ) given byconjugation with the symmetry J = (cid:18) − (cid:19) . This grading is called the standardeven grading on M ( A ). We then have M ( A ) = (cid:18) A A (cid:19) and M ( A ) = (cid:18) AA (cid:19) . (b) If A is a C ∗ -algebra, then the direct sum A L A carries a grading given by( a, b ) ( b, a ), which is called the standard odd grading. We then have (cid:16) A M A (cid:17) = { ( a, a ) : a ∈ A } and (cid:16) A M A (cid:17) = { ( a, − a ) : a ∈ A } . (c) Examples of nontrivially graded C ∗ -algebras which play an important rˆole inthe theory are the Clifford algebras Cl ( V, q ) where q : V × V → R is a (possiblydegenerate) symmetric bilinear form on a finite dimensional real vector space V . Cl ( V, q ) is defined as the universal C ∗ -algebra generated by the elements v ∈ V subject to the relations v = q ( v, v )1 ∀ v ∈ V and such that the embedding ι : V ֒ → Cl ( V, q ) is R -linear. Using the equation( v + v ′ ) − ( v − v ′ ) = 2( vv ′ + v ′ v ) we obtain the relations vv ′ + v ′ v = 2 q ( v, v ′ )1 ∀ v, v ′ ∈ V. If dim( V ) = n , then dim( Cl ( V, q )) = 2 n . The grading on Cl ( V, q ) is given as follows:the linear span of all products of the form v v · · · v m with m = 2 k even is the setof homogeneous elements of degree 0 and the linear span of all such products with m = 2 k − n ∈ N we write Cl n for Cl ( R n , h· , ·i ) where h· , ·i denotes the standard innerproduct on R n . Then Cl ∼ = C and Cl = C C e ∼ = C L C with the standardodd grading (sending λ µe to ( λ + µ, λ − µ ) ∈ C ). If n = 2 and if { e , e } is thestandard orthonormal basis of R , then there is an isomorphism of Cl ∼ = M ( C ), IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 9 equipped with the standard even grading, given by sending the generator e to (cid:18) (cid:19) and e to i (cid:18) −
11 0 (cid:19) .In general we have the formula Cl n ˆ ⊗ Cl m ∼ = Cl n + m as graded C ∗ -algebras, wherewe use the graded tensor product and the diagonal grading on the left hand side ofthis equation. Note that the isomorphism is given on the generators { v : v ∈ R n } and { w : w ∈ R m } by sending v ⊗ w to ( v, · (0 , w ) ∈ Cl n + m . In particular, Cl n can be constructed as the n th graded tensor product of Cl with itself.Note that for any C ∗ -Algebra A , it is an easy exercise to show that the gradedtensor product ( A L A ) ˆ ⊗ Cl , where A L A is equipped with the standard oddgrading, is isomorphic to M ( A ) equipped with the standard even grading. As aconsequence, it follows that Cl n ∼ = M n ( C ) and Cl n +1 ∼ = M n ( C ) M M n ( C )where the grading in the even case is given by conjugation with a symmetry J ∈ M n ( C ) (i.e., an isometry with J = 1) and the standard odd grading in the oddcase (e.g., see [Bla86, § ǫ E of a Hilbert B -module induces a grading Ad ǫ E on L B ( E )and K ( E ) in a canonical way and the morphism Φ : A → L B ( E ) in a Z -gradedcorrespondence has to be equivariant for the given grading on A and this gradingon L B ( E ). In what follows we want to suppress the grading in our notation and justkeep in mind that everything in sight will be Z graded. In most cases (except forClifford algebras), we shall consider the trivial grading ǫ A = id A for our C ∗ -algebras A , but we shall usually have non-trivial gradings on our Hilbert modules.3.2. Kasparov’s bivariant K -groups. In this section we are going to give thedefinition of Kasparovs’s G -equivariant bivariant K -groups. We refer to [Kas88]for the details (but see also [Bla86, Ska84]). We start with the definition of theunderlying KK -cycles: Definition 3.2.
Suppose that (
A, α ) and (
B, β ) are Z -graded G - C ∗ -algebras. A G -equivariant A - B Kasparov cycle is a quadruple ( E , u, Φ , T ) in which ( E , u, Φ) is a Z -graded ( A, α )-(
B, β ) correspondence and T ∈ L B ( E ) is a homogeneous elementwith deg( T ) = 1 such that(i) g Ad u g (Φ( a ) T ); G → L B ( E ) is continuous for all a ∈ A ;(ii) for all a ∈ A and g ∈ G we have( T − T ∗ )Φ( a ) , ( T − a ) , (Ad u g ( T ) − T )Φ( a ) , [ T, Φ( a )] ∈ K ( E )(where [ · , · ] denotes the graded commutator). Two Kasparov cycles ( E , u, Φ , T ) and( E ′ , u ′ , Φ ′ , T ′ ) are called isomorphic , if there exists an isomorphism W : E → E ′ of the correspondences ( E , u, Φ) and ( E ′ , u ′ , Φ ′ ) such that T ′ = W ◦ T ◦ W − . AKasparov cycle is called to be degenerate if( T − T ∗ )Φ( a ) , ( T − a ) , (Ad u g ( T ) − T )Φ( a ) , [ T, Φ( a )] = 0 for all a ∈ A and g ∈ G . We write E G ( A, B ) for the set of isomorphism classes ofall G -equivariant A - B Kasparov cycles and we write D G ( A, B ) for the equivalenceclasses of degenerate Kasparov cycles.
Example 3.3.
Every G -equivariant ∗ -homomorphism Φ : A → B determines a G -equivariant A - B Kasparov cycle (
B, β, Φ , B is considered as a Hilbert B -module in the obvious way. More generally, if ( E , u, Φ) is any Z graded ( A, α )-(
B, β ) correspondence such that Φ( A ) ⊆ K ( E ) (i.e., ( E , u, Φ) is a morphism in thecompact correspondence category in the sense of [Ech17, Definition 2.5.7], then itis an easy exercise to check that ( E , u, Φ ,
0) is a G -equivariant A - B Kasparov cycleas well. Note that the condition ( T − a ) ∈ K ( E ) for a Kasparov cycle impliesthat conversely, if ( E , u, Φ ,
0) is a Kasparov cycle, then Φ( A ) ⊆ K ( E ). A specialsituation of the above is the case in which A = B and Φ = id B which gives usthe Kasparov cycle ( B, β, id B , B, β ) is a Z -graded G - C ∗ -algebra, then we denote by B [0 , C ([0 , , B ) with point-wise G -action and grading. Suppose now that( E , u, Φ , T ) is a G -equivariant A - B [0 ,
1] Kasparov cycle. For each t ∈ [0 ,
1] let δ t : B [0 , → B ; δ t ( f ) = f ( t ) be evaluation at t . Then we obtain a G -equivariant A - B Kasparov cycle ( E t , u t , Φ t , T t ) by putting E t = E ⊗ B [0 , ,δ t B, u t = u ⊗ β, Φ t = Φ ⊗ , and T t = T ⊗ . Alternatively, we could define a B -valued inner product on E by h e, f i B := h e, f i B [0 , ( t )which factors through E t := E / ( E · ker δ t ). Then u, Φ , T factor uniquely throughsome action u t of G , a ∗ -homomorphism Φ t : A → L B ( E t ) and an operator T t ∈ L B ( E t ) such that ( E t , u t , Φ t , T t ) is a G -equivariant A - B Kasparov cycle. Itis isomorphic to the one constructed above. We call ( E t , u t , Φ t , T t ) the evaluation of ( E , u, Φ , T ) at t ∈ [0 , Definition 3.4 (Homotopy) . Two Kasparov cycles ( E , u , Φ , T ) and ( E , u , Φ , T )in E G ( A, B ) are said to be homotopic if there exists a G -equivariant A - B [0 , E , u, Φ , T ) such that ( E , u , Φ , T ) is isomorphic to the evaluationof ( E , u, Φ , T ) at 0 and ( E , u , Φ , T ) is isomorphic to the evaluation of ( E , u, Φ , T )at 1. We then write ( E , u , Φ , T ) ∼ h ( E , u , Φ , T ). We define KK G ( A, B ) := E G ( A, B ) / ∼ h . Remark 3.5.
Every degenerate Kasparov cycle ( E , u, Φ , T ) is homotopic to thezero-cycle (0 , , , E ⊗ C ([0 , , u ⊗ id , Φ ⊗ , T ⊗
1) where we view
E ⊗ C ([0 , ∼ = C ([0 , , E ) as a B [0 , E , u, Φ , T ) that ( E ⊗ C ([0 , , u ⊗ id , Φ ⊗ , T ⊗
1) is an A - B [0 ,
1] Kasparov cycle and it is straightforward to checkthat its evaluation at 0 coincides with ( E , u, Φ , T ) while its evaluation at 1 is thezero-cycle. Remark 3.6.
A special kind of homotopies are the operator homotopies whichare defined as follows: Assume that ( E , u, Φ , T ) and ( E , u, Φ , T ) are two A - B Kasparov cycles such that the underlying correspondence ( E , u, Φ) coincides forboth cycles. An operator homotopy between ( E , u, Φ , T ) and ( E , u, Φ , T ) is a IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 11 family of A - B Kasparov cycles ( E , u, Φ , T t ), t ∈ [0 , T t ) t ∈ [0 , is norm continuous and connects T with T . Such operator homotopydetermines a homotopy between ( E , u, Φ , T ) and ( E , u, Φ , T ) in which the A - B [0 , E ⊗ C [0 , , u ⊗ id , Φ ⊗ , ˜ T ) with (cid:0) ˜ T ( e ) (cid:1) ( t ) = T t e ( t ) for e ∈ E ⊗ C [0 ,
1] = C ([0 , , E ). Example 3.7.
Assume that ( E , u, Φ , T ) and ( E , u, Φ , T ) are two A - B Kasparovcycles. We then say that T is a compact perturbation of T if ( T − T )Φ( a ) ∈ K ( E )for all a ∈ A . In this case, the path T t = (1 − t ) T + tT gives an operator homotopybetween T and T , so both Kasparov cycles are homotopic.The following observation is due to Skandalis (see [Ska84]): Proposition 3.8.
The equivalence relation ∼ h on E G ( A, B ) coincides with theequivalence relation generated by operator homotopy together with adding degenerateKasparov cycles. Theorem 3.9 (Kasparov) . KK G ( A, B ) is an abelian group with addition definedby direct sum of Kasparov cycles: [ E , u , Φ , T ] + [ E , u , Φ , T ] = [ E M E , u M u , Φ M Φ , T M T ] , where [ E , u, Φ , T ] denotes the homotopy class of the Kasparov cycle ( E , u, Φ , T ) . Theinverse of a class [ E , u, Φ , T ] is given by the class [ E op , u, Φ ◦ ǫ A , − T ] in which E op denotes the module E with the opposite grading ǫ E op = − ǫ E . Functoriality:
Every G -equivariant ∗ -homomorphism Ψ : A → A induces agroup homomorphismΨ ∗ : KK G ( A , B ) → KK G ( A , B ) ; [ E , u, Φ , T ] [ E , u, Φ ◦ Ψ , T ]and every G -equivariant ∗ -homomorphism Ψ : B → B induces a group homomor-phismΨ ∗ : KK G ( A, B ) → KK G ( A, B ) ; [ E , u, Φ , T ] [ E ⊗ B B , u ⊗ β , Φ ⊗ , T ⊗ . Hence, KK G is contravariant in the first variable and covariant in the secondvariable. Direct sums. If A = L li =1 A i is a finite direct sum, then KK G ( B, A ) ∼ = L li =1 KK G ( B, A i ). The formula does not hold in general for (countable) infinitedirect sums (see [Bla86, 19.7.2]). On the other side, if A = L i ∈ I A i is a countabledirect sum of G - C ∗ -algebras, then KK G ( A, B ) ∼ = Q i ∈ I KK G ( A i , B ) for every G - C ∗ -algebra B . We leave it to the reader to construct these isomorphisms. The ordinary K -theory groups. Recall that for a trivially graded unital C ∗ -algebra B the ordinary K -group can be defined as the Grothendieck group gen-erated by the semigroup of all homotopy classes [ p ] of projections p ∈ M ∞ ( B ) = ∪ n ∈ N M n ( B ) with direct sum [ p ] + [ q ] = [ p L q ] as addition. If p ∈ M n ( B ), then p determines a ∗ -homomorphism Φ p : C → M n ( B ) ∼ = K ( B n ); λ λp , and hencea class [ B n , Φ p , ∈ KK ( C , B ). Note that all elements of the module B n arehomogeneous of degree 0. This construction preserves homotopy and direct sums and therefore induces a homomorphism of K ( B ) into KK ( C , B ), which, as shownby Kasparov, is actually an isomorphism of abelian groups. Thus we get KK ( C , B ) ∼ = K ( B ) . If B is not unital, we may first apply the above to the unitisation B and for C andthen use split-exactness to get the general case. If Φ : A → B is a ∗ -homomorphism,then the isomorphisms KK ( C , A ) ∼ = K ( A ) and KK ( C , B ) ∼ = K ( B ) intertwinesthe induced homomorphism Φ ∗ : KK ( C , A ) → KK ( C , B ) in KK -theory with themorphism of K -theory groups sending [ p ] to [Φ( p )].If we put the complex numbers C into the second variable, we obtain Kasparov’soperator theoretic K -homology functor K ( A ) := KK ( A, C ). The group KK ( C , C ) . Each element in KK ( C , C ) can be represented by a Kas-parov cycle of the form ( H , , T ) in which H = H L H is a graded Hilbert space, : C → L ( H ) is the action ( λ ) ξ = λξ and T is a self adjoint operator satisfying T − ∈ K ( H ). This follows from the standard simplifications as described in detailin [Bla86, § T = (cid:18) P ∗ P (cid:19) , then T = (cid:18) P ∗ P P P ∗ (cid:19) and the condition T − ∈ K ( H ) then implies that P is a Fredholm operator. We then obtain a welldefined mapindex : KK ( C , C ) → Z ; [ H , , T ] index( H , , T ) := index( P ) , where index( P ) = dim(ker( P )) − dim(ker( P ∗ )) denotes the Fredholmindex of P .The index map induces the isomorphism KK ( C , C ) ∼ = Z = K ( C ). (Compare thiswith the above isomorphism KK ( C , B ) ∼ = K ( B ) in case B = C .)3.3. The Kasparov product.
We are now coming to the main feature of Kas-parov’s KK -theory, namely the Kasparov product which is a pairing KK G ( A, B ) × KK G ( B, C ) → KK G ( A, C ) , where A, B, C are G - C ∗ -algebras. Starting with an A - B Kasparov cycle ( E , u, Φ , T )and a B - C Kasparov cycle ( F , v, Ψ , S ), the Kasparov product will be representedby a Kasparov cycle of the form ( E ⊗ B F , u ⊗ v, Φ ⊗ , R ), where all ingredientswith exception of the operator R are well known objects by now: E ⊗ B F denotesthe internal tensor product of E with F over B (with diagonal grading), u ⊗ v : G → Aut(
E ⊗ B F ) denotes the diagonal action, and Φ ⊗ A → L ( E ⊗ B F ) sends a ∈ A to the operator Φ( a ) ⊗ E ⊗ B F . But the construction of the operator R is, unfortunately, quite complicated and reflects the high complexity of Kasparov’stheory.As a first attempt one would look at the operator R = T ⊗ ⊗ S. But there are several problems with this choice. First of all, the operator 1 ⊗ S on the internal tensor product is not well defined in general (it only makes sense,if S commutes with Ψ( B ) ⊆ L ( F )). To resolve this, we need to replace 1 ⊗ S with a so-called S -connection , which we shall explain below. But even if 1 ⊗ S iswell-defined, the triple ( E ⊗ B F , u ⊗ v, Φ ⊗ , T ⊗ ⊗ S ) will usually fail to IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 13 be a Kasparov triple unless S = 0, and one needs to replace T ⊗ ⊗ S by acombination M / ( T ⊗
1) + N / (1 ⊗ S )where M, N ≥ M + N = 1 which can be obtained byan application of Kasparov’s technical theorem [Kas88, Theorem 1.4]. S -connection: For any ξ ∈ E define θ ξ : F → E ⊗ B F ; θ ξ ( η ) = ξ ⊗ η. Then θ ξ ∈ K ( F , F ⊗ B E ) with adjoint given by θ ∗ ξ ( ζ ⊗ η ) = Ψ( h ξ, ζ i B ) η. An operator S ∈ L ( E ⊗ B F ) is then called an S -connection, if for all homogeneouselements ξ ∈ E we have(3.1) θ ξ S − ( − deg( ξ ) deg S S θ ξ , θ ξ S ∗ − ( − deg( ξ ) deg S S ∗ θ ξ ∈ K ( F , E ⊗ B F )It is a good exercise to check that if S commutes with Ψ( B ) ⊆ L ( F ), then S =1 ⊗ S makes sense and is an S -connection in the above sense. Definition 3.10 (Kasparov product) . Suppose that
A, B, C are G - C ∗ -algebras,and that ( E , u, Φ , T ) is an A - B Kasparov cycle and ( F , v, Ψ , S ) is a B - C Kasparovcycle. Let S ∈ L ( E ⊗ B F ) be an S -connection as explained above. Then thequadruple ( E ⊗ B F , u ⊗ v, Φ ⊗ , S ) will be a Kasparov product for ( E , u, Φ , T ) and( F , v, Ψ , S ) if the following two conditions hold:(i) ( E ⊗ B F , u ⊗ v, Φ ⊗ , S ) is an A - C Kasparov cycle.(ii) For all a ∈ A we have (Φ( a ) ⊗
1) [ T ⊗ , S ] (Φ( a ∗ ) ⊗ ≥ K ( E⊗ B F ).In this case the class [ E ⊗ B F , u ⊗ v, Φ ⊗ , S ] ∈ KK G ( A, C ) is called a Kasparovproduct of [ E , u, Φ , T ] ∈ KK G ( A, B ) with [ F , v, Ψ , S ] ∈ KK G ( B, C ).We should note that the existence of an S -connection S which satisfies the con-ditions of the above definition follows from an application of Kasparov’s technicaltheorem [Kas88, Theorem 14]. The proof is quite technical and we refer to theliterature (see one of the references [Ska84, Kas88, Bla86]). Remark 3.11. (a)
One can show that the operator S in a Kasparov product isunique up to operator homotopy. (b) The following easy case is often very useful: Suppose that B acts on F bycompact operators, i.e., Ψ : B → L ( F ) takes its values in K ( F ). Then ( F , v, Ψ ,
0) isa B - C Kasparov cycle and the 0-operator on
E ⊗ B F is then clearly a 0-connection.Now if Φ : A → L ( E ) also takes values in K ( E ) and if T = 0, it follows that[ E ⊗ B F , u ⊗ v, Φ ⊗ ,
0] is a Kasparov product for [ E , u, Φ ,
0] and [ F , v, Ψ , Corr c ( G )(see [Ech17, Section 2.5.3]) – here we use countably generated Z -graded Hilbertmodules) into KK G given by ( E , u, Φ) [ E , u, Φ ,
0] which preserves multiplication.The details of the following theorem can be found in [Bla86] or [Kas88].
Theorem 3.12 (Kasparov) . Suppose that
A, B, C are separable G - C ∗ -algebras andlet x = [ E , u, Φ , T ] ∈ KK G ( A, B ) and y = [ F , v, Ψ , S ] ∈ KK G ( B, C ) . Then theKasparov product x ⊗ B y := [ E ⊗ B F , u ⊗ v, Φ ⊗ , S ] exists and induces a well-defined bilinear pairing ⊗ B : KK G ( A, B ) × KK G ( B, C ) → KK G ( A, C ) . Moreover, the Kasparov product is associative: If D is another G - C ∗ -algebra and z ∈ KK G ( C, D ) , then we have ( x ⊗ B y ) ⊗ C z = x ⊗ B ( y ⊗ C z ) ∈ KK G ( A, D ) . The elements A = [ A, α, id A , ∈ KK G ( A, A ) and B = [ B, β, id B , ∈ KK G ( B, B ) act as identities from the left and right on KK G ( A, B ) , i.e., we have A ⊗ A x = x = x ⊗ B B ∈ KK G ( A, B ) for all x ∈ KK G ( A, B ) . In particular, KK G ( A, A ) equipped with the Kasparovproduct is a unital ring. The following result is helpful for the computation of Kasparov products in someimportant special cases. For the proof we refer to [Bla86, 8.10.1].
Proposition 3.13.
Suppose that
A, B, C are G - C ∗ -algebras, ( E , u, Φ , T ) is an A - B Kasparov cycle and ( F , v, Ψ , S ) is a B - C Kasparov cycle. Suppose further that T = T ∗ and k T k ≤ . Let S ∈ L ( E ⊗ B F ) be a G -invariant S -connection of degreeone and let R := T ⊗ p − T ⊗ S . If [ R, Φ( A ) ⊗ ∈ K ( E ⊗ B F ) , then ( E ⊗ B F , u ⊗ v, Φ ⊗ , R ) ∈ E G ( A, C ) andrepresents the Kasparov product of [ E , u, Φ , T ] with [ F , v, Ψ , S ]We should notice that the conditions T = T ∗ and k T k ≤ x ∈ KK G ( A, B ).Associativity and the existence of neutral elements for the Kasparov product givesrise to an easy notion of KK G -equivalence for two G - C ∗ -algebras A and B : Assumethat there are elements x ∈ KK G ( A, B ) and y ∈ KK G ( B, A ) such that x ⊗ B y = 1 A ∈ KK G ( A, A ) and y ⊗ A x = 1 B ∈ KK G ( B, B ) . Then taking products with x from the left induces an isomorphism x ⊗ B · : KK G ( B, C ) → KK G ( A, C ); z x ⊗ B z with inverse given by y ⊗ A · : KK G ( A, C ) → KK G ( B, C ); w y ⊗ A w. This follows from the simple identities y ⊗ A ( x ⊗ B z ) = ( y ⊗ A x ) ⊗ B z = 1 B ⊗ B z = z for all z ∈ KK G ( B, C ) and similarly we have x ⊗ B ( y ⊗ A w ) = w for all w ∈ KK G ( A, C ). Of course, taking products from the right by x and y will giveus an isomorphism · ⊗ A x : KK G ( C, A ) → KK G ( C, B ) with inverse · ⊗ B y : KK G ( C, B ) → KK G ( C, A ). IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 15 Definition 3.14.
Suppose that x ∈ KK G ( A, B ) and y ∈ KK G ( B, A ) are as above.Then we say that x is a KK G -equivalence from A to B with inverse y . Two G - C ∗ -algebras A and B are called KK G -equivalent , if such elements x and y exist. Lemma 3.15.
Suppose that ( E , u, Φ) is a G -equivariant A - B Morita equivalence forthe G - C ∗ -algebras ( A, α ) and ( B, β ) and let ( E ∗ , u ∗ , Φ ∗ ) denote its inverse. Then x = [ E , u, Φ , ∈ KK G ( A, B ) is a KK G -equivalence with inverse y = [ E ∗ , u ∗ , Φ ∗ , .Proof. It follows from Remark 3.11 that the Kasparov product x ⊗ B y is representedby the Kasparov cycle [ E ⊗ B E ∗ , u ⊗ u ∗ , Φ ⊗ , E ⊗ B E ∗ , u ⊗ u ∗ , Φ ⊗
1) is isomorphic to (
A, α, id A ), it follows that x ⊗ B y = 1 A .Similarly we have y ⊗ B x = 1 B . (cid:3) Remark 3.16. If p ∈ M ( A ) is a full projection in a C ∗ -algebra A , then pA is a pAp - A Morita equivalence and hence, if ϕ : pAp → L ( pA ) denotes the canonicalmorphism, the element [ pA, ϕ, ∈ KK ( pAp, A ) is a KK -equivalence (it is a KK G -equivalence if A is a G - C ∗ -algebra and p ∈ M ( A ) is G -invariant). On the otherhand, if ψ : pAp → A denotes the canonical inclusion, then [ A, ψ,
0] determines theclass of the ∗ -homomorphism ψ . Both classes actually coincide, which follows fromthe simple fact that we can decompose the KK -cycle ( A, ψ,
0) as the direct sum( pA, ϕ, L ((1 − p ) A, ,
0) where the second summand is degenerate. In particular,it follows from this that ψ ∗ : K ∗ ( pAp ) → K ∗ ( A ) is an isomorphism.We are now going to describe a more general version of the Kasparov product. Forthis we first need to introduce a homomorphism · ˆ ⊗ D : KK G ( A, B ) → KK G ( A ˆ ⊗ D, B ˆ ⊗ D )which is defined for any G - C ∗ -algebra ( D, δ ) by[ E , u, Φ , T ] ˆ ⊗ D := [ E ˆ ⊗ D, u ˆ ⊗ δ, Φ ˆ ⊗ , T ˆ ⊗ . One can check that · ˆ ⊗ D is compatible with Kasparov products in the sense that( x ⊗ B y ) ˆ ⊗ D = ( x ˆ ⊗ D ) ⊗ B ˆ ⊗ D ( y ˆ ⊗ D )and it follows directly from the definition that 1 A ˆ ⊗ D = 1 A ˆ ⊗ D . In particular, itfollows that · ˆ ⊗ D sends KK G -equivalences to KK G -equivalences. Of course, in asimilar way we can define a homomorphism1 D ⊗ · : KK G ( A, B ) → KK G ( D ˆ ⊗ A, D ˆ ⊗ B ) . Remark 3.17.
By our conventions “ ˆ ⊗ ” denotes the minimal graded tensor prod-uct of the C ∗ -algebras A and B . But a similar map · ˆ ⊗ max D : KK G ( A, B ) → KK G ( A ˆ ⊗ max D, B ˆ ⊗ max D ) exists for the maximal graded tensor product. Theorem 3.18 (Generalized Kasparov product) . Suppose that ( A , α ) , ( A , α ) , ( B , β ) , ( B , β ) and ( D, δ ) are G - C ∗ -algebras. Then there is a pairing ⊗ D : KK G ( A , B ˆ ⊗ D ) × KK G ( D ˆ ⊗ A , B ) → KK G ( A ˆ ⊗ A , B ˆ ⊗ B ) given by ( x, y ) x ⊗ D y := ( x ˆ ⊗ A ) ⊗ B ˆ ⊗ D ˆ ⊗ A (1 B ˆ ⊗ y ) . This pairing is associative (in a suitable sense) and coincides with the ordinaryKasparov product if B = C = A . Moreover, in case D = C , the product ⊗ C : KK G ( A , B ) × KK G ( A , B ) → KK G ( A ˆ ⊗ A , B ˆ ⊗ B ) is commutative. Note that there are several other important properties of the generalized Kasparovproduct, which we don’t want to state here. We refer to [Kas88, Theorem 2.14] forthe complete list and their proofs. We close this section with a useful description ofKasparov cycles in terms of unbounded operators due to Baaj and Julg (see [BJ83]).For our purposes it suffices to restrict to the case where B = C , in which case wemay rely on the classical theory of unbounded operators on Hilbert spaces. But thepicture extends to the general case using a suitable theory of regular unboundedoperators on Hilbert modules. Lemma 3.19 (Baaj-Julg) . Suppose that A is a graded C ∗ -algebra and Φ : A →B ( H ) is a graded ∗ -representation of A on the graded separable Hilbert space H .Suppose further that D = (cid:18) d ∗ d (cid:19) is an unbounded selfadjoint operator on H ofdegree one such that(i) (1 + D ) − Φ( a ) ∈ K ( H ) for all a ∈ A , and(ii) the set of all a ∈ A such that [ D, Φ( a )] is densely defined and bounded isdense in A .Then ( H , Φ , T = D √ D ) is an A - C Kasparov cycle.
For the proof of this lemma, even in the more general context of A - B Kasparovcycles, we refer to [Bla86, Proposition 17.11.3]. Note that the operator T = D √ D is constructed via functional calculus for unbounded selfadjoint operators.3.4. Higher KK -groups and Bott-periodicity.Definition 3.20. Suppose that (
A, α ) and (
B, β ) are G - C ∗ -algebras. For each n ∈ N we define the (higher) KK G -group as KK Gn ( A, B ) := KK G ( A, B ˆ ⊗ Cl n ) and KK G − n ( A, B ) := KK G ( A ˆ ⊗ Cl n , B ) , where Cl n denotes the n th complex Clifford algebra with trivial G -action and grad-ing as defined in Section 3.With this definition of higher KK -groups it is easy to prove a (formal) version ofBott-periodicity. We need the following easy lemma: Lemma 3.21. If n ∈ N is even, then Cl n is Morita equivalent to C as graded C ∗ -algebras. If n is odd, then Cl n is Morita equivalent to Cl as graded C ∗ -algebra.Proof. Let n ∈ N . We know from Section 3 that Cl n ∼ = M n ( C ) with gradinggiven by cunjugation with a symmetry J ∈ M n ( C ). It is then easy to check thatthe Hilbert space C n equipped with the grading operator J and the canonicalleft action of Cl n on C n gives the desired Morita equivalence. Similarly, we have Cl n +1 ∼ = M n ( C ) L M n ( C ) ∼ = M n ˆ ⊗ Cl as graded C ∗ -algebras, which is Moritaequivalent to C ˆ ⊗ Cl = Cl . (cid:3) IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 17 Notations 3.22.
In what follows we shall denote by x n ∈ KK ( Cl n , C ) the(invertible) class of the Morita equivalence between Cl n and C as in the abovelemma and by x n +1 ∈ KK ( Cl n +1 , Cl ) the class of the Morita equivalence be-tween Cl n +1 and Cl . Proposition 3.23 (Formal Bott-periodicity) . For each n ∈ N there are canonicalisomorphisms KK G ( A ˆ ⊗ Cl n , B ) ∼ = KK G ( A, B ) ∼ = KK G ( A, B ˆ ⊗ Cl n ) and KK G ( A ˆ ⊗ Cl n +1 , B ) ∼ = KK G ( A ˆ ⊗ Cl , B ) ∼ = KK G ( A, B ˆ ⊗ Cl ) ∼ = KK G ( A, B ˆ ⊗ Cl n +1 ) . As a consequence, we have KK Gl ( A, B ) ∼ = KK Gl +2 ( A, B ) for all l ∈ Z .Proof. In the even case, the isomorphisms follow by taking Kasparov productswith the KK -equivalences 1 A ⊗ x − n ∈ KK G ( A, A ˆ ⊗ Cl n ) and 1 B ⊗ x − n ∈ KK G ( B, B ˆ ⊗ Cl n ) from the left and right, respectively (where we con-sider the trivial action of G on the Clifford algebras). The same argumentwill provide isomorphisms KK G ( A ˆ ⊗ Cl n +1 , B ) ∼ = KK G ( A ˆ ⊗ Cl , B ) and KK G ( A, B ˆ ⊗ Cl ) ∼ = KK G ( A, B ˆ ⊗ Cl n +1 ), respectively.To finish off, one checks that the composition KK G ( A ˆ ⊗ Cl , B ) y y ⊗ Cl −→ KK G ( A ˆ ⊗ Cl ˆ ⊗ Cl , B ˆ ⊗ Cl ) (1 A ⊗ x − ) ⊗· ∼ = KK G ( A, B ˆ ⊗ Cl ) is an isomorphism with inverse given by the composition KK G ( A, B ˆ ⊗ Cl ) y y ⊗ Cl −→ KK G ( A ˆ ⊗ Cl , B ˆ ⊗ Cl ˆ ⊗ Cl ) ·⊗ (1 B ⊗ x ) ∼ = KK G ( A ˆ ⊗ Cl , B ) , where we use that Cl ˆ ⊗ Cl ∼ = Cl ∼ M C . (cid:3) Of course we would like to have a version of Bott-periodicity showing that, alter-natively, we could define the higher KK -groups via suspension. For this we aregoing to construct a KK -equivalence between Cl and C ( R ). Indeed, we shalldo this by first constructing a KK -equivalence between C ( R ) ˆ ⊗ Cl ∼ = C ( R , Cl )with C . Since we consider the trivial G -action on C ( R ) and Cl it suffices to dothis for the trivial group G = { e } . In what follows next we write D := C ( R ) ˆ ⊗ Cl and D := C ∞ c ( R ) ˆ ⊗ Cl ⊆ D , where C ∞ c ( R ) denotes the dense subalgebra of C ( R )consisting of smooth functions with compact supports. A typical element of D canbe written as f + f e with f , f ∈ C ( R ), where we identify f with f Cl andwhere e = e denotes the generator of Cl with e = 1. The Dirac element:
We define an element α = [ H , Φ , T ] ∈ KK ( D , C )as follows: We let H = L ( R ) L L ( R ) be equipped with the grading induced bythe operator J = (cid:18) − (cid:19) . We define T = D √ D with D = (cid:18) − dd (cid:19) , where d : L ( R ) → L ( R ) denotes the densely defined operator d = ddt . Then D isan essentially selfadjoint operator on the dense subspace C ∞ c ( R ) L C ∞ c ( R ) of H and therefore extends to a densely defined selfadjoint operator on H (we refer to[HK01, Chapter 10] for details). Let Φ : D → L ( H ) be given by(3.2) Φ( f + f e ) · (cid:18) ξ ξ (cid:19) = (cid:18) f f f f (cid:19) · (cid:18) ξ ξ (cid:19) = (cid:18) f · ξ + f · ξ f · ξ + f · ξ (cid:19) . In order to check that ( H , Φ , T ) is a D - C Kasparov cycle we need to check theconditions of Lemma 3.19 for the triple ( H , Φ , D ). Notice that D = diag(∆ , ∆),where ∆ = − d dt denotes the (positive) Laplace operator on R . By [RS78, XIII.4Example 6] (or [HK01, 10.5.1]) the operator (1 + ∆) − M ( f ) is a compact operatorfor all f ∈ C ∞ c ( R ) (where M : C b ( R ) → B ( L ( R )) denotes the representation asmultiplication operators). Hence(1 + D ) − ◦ Φ( f + f e ) = (cid:18) (1 + ∆) − M ( f ) (1 + ∆) − M ( f )(1 + ∆) − M ( f ) (1 + ∆) − M ( f ) (cid:19) ∈ K ( H )for all f + f e ∈ D = C ∞ c ( R ) L C ∞ c ( R ). Since D is dense in D and since(1 + D ) − Φ( f + f e ) depends continuously on ( f , f ), this proves condition (i)of Lemma 3.19. To see condition (ii) we first observe that for all f ∈ C ∞ c ( R ) theoperator [ d, M ( f )] is defined for all ξ ∈ C ∞ c ( R ) ⊆ L ( R ) and we have[ d, M ( f )] ξ = ddt ( f ξ ) − f · ( ddt ξ ) = ( ddt f ) · ξ, Hence [ d, M ( f )] extends to a bounded operator on L ( R ) and[ D, Φ( f + f e )] = (cid:18) − [ d, M ( f )] − [ d, M ( f )][ d, M ( f )] [ d, M ( f )] (cid:19) is densely defined and bounded for all f + f e ∈ D . The dual-Dirac element:
Choose any odd continuous function ϕ : R → [ − , ϕ ( x ) > x > x →∞ ϕ ( x ) = 1. For instance we could take ϕ ( x ) = x √ x or ϕ ( x ) = (cid:26) sin( x/ | x | ≤ π x | x | | x | ≥ π (cid:27) . We then define an element β = [ D , , S ] ∈ KK ( C , D ) as follows: We consider D as graded Hilbert D -module in the canonical way, and we put 1( λ ) a = λa for all λ ∈ C and a ∈ D . The operator S ∈ M ( D ) is defined via multiplication with theelement ϕe ∈ C b ( R ) ˆ ⊗ Cl ⊆ M ( D ). To check that ( D , Φ , S ) is a Kasparov cycleit suffices to check that S − ∈ K ( D ) = D . But this follows from the fact that S − ϕ − D since lim ± x →∞ ϕ ( x ) = 1 by conditions (i) and (ii) for ϕ . Since S = S ∗ all otherconditions of Definition 3.2 are trivial.Note that the class β does not depend on the particular choice of the function ϕ : R → R . Indeed, if two functions ϕ , ϕ are given which satisfy conditions (i)and (ii), we can define for each t ∈ [0 ,
1] a function ϕ t : R → R by ϕ t ( x ) = tϕ ( x ) + (1 − t ) ϕ ( x ) . Then each ϕ t satisfies the requirements (i) and (ii) and if S t denotes the corre-sponding operators it follows that t S t is an operator homotopy joining S with S . IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 19 Notations 3.24.
The element β ∈ KK ( C , C ( R ) ˆ ⊗ Cl ) = KK ( C , C ( R )) con-structed above is called the Bott class . The Kasparov product β ⊗ D α ∈ KK ( C , C ) . We are now going to show that β ⊗ D α = 1 C ∈ KK ( C , C ). For this we first claim that β ⊗ D α is represented bythe triple ( H , , T ′ ) with(3.3) T ′ = (cid:18) M ( ϕ ) M ( ϕ ) 0 (cid:19) − M ( p − ϕ ) − d √ d √ ! , with H = L ( R ) L L ( R ) as above. Indeed, since Φ : D → L ( H ) is a non-degenerate representation we obtain an isomorphism(3.4) D ⊗ D H ∼ = H ; a ⊗ ξ Φ( a ) ξ. Let T denote the operator on D ⊗ D H corresponding to T = − d √ d √ ! under this isomorphism. We claim that T is a T -connection. Recall that forany a ∈ D the operator θ a : H → D ⊗ D H is given by ξ a ⊗ ξ . Composedwith the above isomorphism we get the operator ξ Φ( a ) ξ on H . Condition (3.1)follows then for T from the fact that [ T, Φ( a )] ∈ K ( H ) for all a ∈ D . We now useProposition 3.13 to see that the Kasparov product β ⊗ D α is represented by thetriple ( D ⊗ D H , ⊗ , S ⊗ p − S ⊗ T ) . We leave it as an exercise to check that this operator corresponds to the operator R of (3.3) under the isomorphism (3.4).Hence to see that β ⊗ D α = 1 C ∈ KK ( C , C ) we only need to show that the Fredholmindex of the operator F := M ( ϕ ) + M (cid:16)p − ϕ (cid:17) d √ L ( R ) → L ( R )is one (see the discussion at the end of Section 3.2).Recall that ϕ : R → R can be any function satisfying the conditions (i) and (ii) asstated in the construction of β . Thus we may choose ϕ ( x ) = (cid:26) sin( x/
2) if x ∈ [ − π, π ] x | x | if | x | > π (cid:27) . To do the computation we want to restrict the operator to the interval [ − π, π ]. Forthis consider the orthogonal projection Q : L ( R ) → L [ − π, π ]. Since Q commuteswith M (1 − ϕ ) and since [ d √ , M ( ψ )] ∈ K ( L ( R )) for all ψ ∈ C ( R ) (whichfollows from [ T, Φ( a )] ∈ K ( H ) for all a ∈ D ), we have M (1 − ϕ ) d √ ∼ M ( p − ϕ ) d √ M ( p − ϕ )= M ( p − ϕ ) Q d √ M ( p − ϕ ) Q ∼ M (1 − ϕ ) d √ Q, where ∼ denotes equality up to compact operators. Thus we may replace F by theoperator F := M ( ϕ ) + M (cid:16)p − ϕ (cid:17) d √ Q : L ( R ) → L ( R )Decomposing L ( R ) as the direct sum L [ − π, π ] L L (( −∞ , − π ) ∪ ( π, ∞ )), we seethat the operator F fixes both summands and acts as the identity on the secondsummand. Hence for computing the index we may restrict our operator to thesummand L [ − π, π ] on which it acts by F := M (sin( x/ M (cos( x/ d √ . Now there comes a slightly tricky point and we need to appeal to some computationsgiven in [HR00, Chapter 10]. We want to replace the operator d √ by the opera-tor ˜ d √ − ∆ T : L ( T ) → L ( T ) (identifying L [ − π, π ] with L ( T )), where ˜ d : L ( T ) → L ( T ) denotes the operator given by the differential ddt on the smooth 2 π -periodicfunctions on R and where ∆ T = − ˜ d denotes the corresponding Laplace opera-tor. Although the operators d and ˜ d clearly coincide on C ∞ c ( − π, π ) the functionalcalculus which has been applied for producing the operators d √ and ˜ d √ − ∆ T depends on the full domains of the selfadjoint extensions of these operators, whichclearly differ. The solution of this problem is implicitly given in [HR00, Lemma10.8.4]: Recall that d √ is equal to − iχ ( id ) where we apply the functional cal-culus for unbounded selfadjoint operators for the function χ ( x ) = x √ x to the(unique) selfadjoint extension of id . Let ψ ∈ C ∞ c ( − π, π ) be any fixed function.Choose a positive function µ ∈ C ∞ c ( R ) with supp µ ∈ ( − π, π ) such that µ ≡ U := supp ψ + ( − δ, δ ) for some suitable δ >
0. Let d µ = M ( µ ) ◦ d ◦ M ( µ ). It followsthen from [HR00, Corollary 10.2.6] that id µ is an essentially selfadjoint operatorwhich coincides with id on U . It then follows from [HR00, Lemma 10.8.4] that M ( ψ ) χ ( id ) ∼ M ( ψ ) χ ( id µ ) on L [ − π, π ], where, as above, ∼ denotes equality upto compact operators. Applying the same argument to the canonical inclusion ofthe interval ( − π, π ) into T shows that M ( ψ ) χ ( id µ ) ∼ M ( ψ ) χ ( i ˜ d ). Together we seethat M ( ψ ) χ ( id ) ∼ M ( ψ ) χ ( i ˜ d ) on L [ − π, π ] for all ψ ∈ C ∞ c ( − π, π ) and then alsofor all ψ ∈ C ( − π, π ). Applying this to ψ ( x ) = cos( x/
2) gives the desired result.Thus we may replace the operator F by the operator F := M (sin( x/ M (cos( x/ d √ T . Multiplying F from the left with the invertible operator M (2 ie i x ) does not changethe Fredholm index, so we compute the index of the operator F = M (2 ie i x sin( x/ M (2 ie i x cos( x/ d √ T = M ( e ix −
1) + iM ( e ix + 1) ˜ d √ T . In what follows let { e n : n ∈ Z } denote the standard othonormal basis of ℓ ( Z )and let U : ℓ ( Z ) → ℓ ( Z ) denote the bilateral shift operator U ( e n ) = e n +1 . Using IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 21 Fourier transform and the Plancherel isomorphism L [ − π, π ] ∼ = ℓ ( Z ) the operator F transforms to the operator c F : ℓ ( Z ) → ℓ ( Z ) given by c F = ( U − id) + i ( U + id) R where R : ℓ ( Z ) → ℓ ( Z ) is given by R ( e n ) = in √ n e n . Let sign( n ) = (cid:26) n | n | if n = 00 if n = 0 (cid:27) and let R ′ ( e n ) = i sign( n ) e n for n ∈ Z and let us write c F := ( U − id) + i ( U + id) R ′ . Since (cid:12)(cid:12)(cid:12) in √ n − i sign( n ) (cid:12)(cid:12)(cid:12) → | n | → ∞ wehave R − R ′ ∈ K ( ℓ ( Z )), which implies that c F − c F ∈ K ( ℓ ( Z )) and henceindex( c F ) = index( c F ). Applying c F to some basis element e n gives c F ( e n ) = ( U − id) + i ( U + id) R ′ ( e n )= ( e n +1 − e n ) − sign( n )( e n +1 + e n ) = e n +1 if n < e − e if n = 0 − e n if n > . It follows from this that c F is surjective and ker( c F ) = C ( e + 2 e + e − ). Henceindex( β ⊗ D α ) = index( c F ) = 1 . Let us state as a lemma what we have proved so far:
Lemma 3.25.
Let D = C ( R ) ˆ ⊗ Cl = C ( R ) L C ( R ) with the standard odd grad-ing and let α ∈ KK ( D , C ) and β ∈ KK ( C , D ) as above. Then β ⊗ D α = 1 C ∈ KK ( C , C ) . In order to show that α and β are KK -equivalences, we also need to check thatthe product α ⊗ C β = 1 D ∈ KK ( D , D ). For this we shall use a rotation trick whichoriginally goes back to Atiyah, and which has been adapted very successfully tothis situation by Kasparov. Recall that by Theorem 3.18 the Kasparov productover C is commutative, i.e., we have α ⊗ C β = β ⊗ C α = ( β ⊗ D ) ⊗ D ˆ ⊗D (1 D ⊗ α )= ( β ⊗ D ) ⊗ D ˆ ⊗D (cid:0) Σ D , D ⊗ D ˆ ⊗D ( α ⊗ D ) (cid:1) , where Σ D , D : D ˆ ⊗D → D ˆ ⊗D denotes (the KK -class of) the flip homomorphism x ⊗ y ( − deg( x ) deg( y ) y ⊗ x . If we can show that there is an invertible class η ∈ KK ( D , D ) such that Σ D , D = 1 D ˆ ⊗ η ∈ KK ( D ˆ ⊗D , D ˆ ⊗D ) the result will followfrom the following reasoning: α ⊗ C β = ( β ⊗ D ) ⊗ D ˆ ⊗D (cid:0) Σ D , D ⊗ D ˆ ⊗D ( α ⊗ D ) (cid:1) = ( β ⊗ D ) ⊗ D ˆ ⊗D (cid:0) (1 D ⊗ η ) ⊗ D ˆ ⊗D ( α ⊗ D ) (cid:1) = ( β ⊗ D ) ⊗ D ˆ ⊗D (cid:0) Σ D , D ◦ ( η ⊗ D ) ◦ Σ − D , D (cid:1) ⊗ D ˆ ⊗D (cid:0) Σ D , D ⊗ D ˆ ⊗D (1 D ⊗ α ) (cid:1) = ( β ⊗ D ) ⊗ D ˆ ⊗D (cid:0) Σ D , D ◦ ( η ⊗ D ) (cid:1) ⊗ D ˆ ⊗D (1 D ⊗ α )= ( β ⊗ D ) ⊗ D ˆ ⊗D Σ D , D ⊗ D ˆ ⊗D ( η ⊗ C α )= ( β ⊗ D ) ⊗ D ˆ ⊗D ( α ⊗ C η )= (cid:0) ( β ⊗ D ) ⊗ D ˆ ⊗D ( α ⊗ D ) (cid:1) ⊗ C ˆ ⊗D (1 C ⊗ η )= η. Since η is invertible in KK ( D , D ) this implies that α has a right KK -inverse γ ,say. But then γ = β , since β = β ⊗ D ( α ⊗ C γ ) = ( β ⊗ D α ) ⊗ C γ = γ. To see that Σ D , D = 1 D ˆ ⊗ η ∈ KK ( D ˆ ⊗D , D ˆ ⊗D ) for a suitable invertible KK -class η we consider the isomorphism D ˆ ⊗D = (cid:0) C ( R ) ˆ ⊗ Cl (cid:1) ˆ ⊗ (cid:0) C ( R ) ˆ ⊗ Cl (cid:1) ∼ = C ( R ) ˆ ⊗ Cl . If τ : R n → R n is any orthogonal transformation, it induces an automorphism τ ∗ : C ( R n ) → C ( R n ) by τ ∗ ( f )( x ) = f ( τ − ( x )) and an automorphism ˜ τ of Cl n byextending, via the universal property of Cl n , the map ˜ τ : R n → Cl n ; v ι ◦ τ ( v )to all of Cl n , where ι : R n → Cl n denotes the canonical inclusion. We then get anautomorphism Φ τ := τ ∗ ˆ ⊗ ˜ τ : C ( R n ) ˆ ⊗ Cl n → C ( R n ) ˆ ⊗ Cl n . Moreover, a homotopy of orthogonal transformations of R n between τ and τ clearly induces a homotopy between the automorphisms Φ τ and Φ τ . In particular,for any orthogonal transformation which is homotopic to id R n we get[Φ τ ] = 1 C ( R n , Cl n ) ∈ KK ( C ( R n , Cl n ) , C ( R n , Cl n )) . It is not difficult to check that under the isomorphism D ˆ ⊗D ∼ = C ( R ) ˆ ⊗ Cl theflip automorphism Σ D , D corresponds to Φ σ : C ( R ) ˆ ⊗ Cl → C ( R ) ˆ ⊗ Cl with σ : R → R ; σ ( x, y ) = ( y, x ) . Since det( σ ) = −
1, it is, unfortunately, not homotopic to id R . But the orthogonaltransformation ρ : R → R ; ρ ( x, y ) = ( − y, x ) is homotopic to id R via the path oftransformations ρ t , t ∈ [0 , π/
2] with ρ t ( x, y ) = (cos( t ) x − sin( t ) y, sin( t ) x + cos( t ) y ) . One checks that Φ ρ corresponds to Σ D , D ◦ (id D ⊗ Φ − id ) , where − id : R → R , x
7→ − x is the flip on R . Hence, if η = [Φ − id ] ∈ KK ( D , D ), we haveΣ D , D ⊗ D ˆ ⊗D (1 D ⊗ η ) = [Σ D , D ◦ (id D ⊗ Φ − id )] = [Φ ρ ] = 1 D ˆ ⊗D , where, by abuse of notation, we identify Φ ρ with the corresponding automorphismof D ˆ ⊗D . Since Σ D , D = Σ − D , D it follows that 1 D ⊗ η = Σ D , D ∈ KK ( D ˆ ⊗D , D ˆ ⊗D )and we are done. Corollary 3.26.
Let α ∈ KK ( C ( R ) , Cl ) and β ∈ KK ( Cl , C ( R )) be theimages of α and β under the isomorphisms KK ( C ( R ) ⊗ Cl , C ) ∼ = KK ( C ( R ) , Cl ) and KK ( C , C ( R ) ⊗ Cl ) ∼ = KK ( Cl , C ( R )) of Proposition 3.23. Then α is a KK -equivalence with inverse β . As a consequence, for all G -algebras A and B ,there are canonical Bott-isomorphisms KK G ( A ˆ ⊗ C ( R ) , B ) ∼ = KK G ( A, B ) ∼ = KK G ( A, B ˆ ⊗ C ( R )) . More generally, for all n, m ∈ N we get Bott-isomorphisms KK G ( A ˆ ⊗ C ( R n ) , B ˆ ⊗ C ( R m )) ∼ = KK Gn + m ( A, B ) . IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 23 Proof.
The proof is a straightforward consequence of the results and techniquesexplained above and is left to the reader. (cid:3) Kasparov actually proved a more general version of the above KK -theoretic Bott-periodicity theorem, which provides a KK G -equivalence between C ( V ) and theClifford algebra Cl ( V, h· , ·i ), in which G is a (locally) compact group which acts bya continuous orthogonal representation ρ : G → O( V ) on the finite dimensionaleuclidean vector space V , and h· , ·i is any G -invariant inner product on V . Theaction of G on Cl ( V ) is the unique action which extends the given action of G on V ⊆ Cl ( V ). Identifying V with R n equipped with the standard inner product,this KK -equivalence is constructed as in the above special case where n = 1 via a KK G -equivalence between C ( R n ) ˆ ⊗ Cl n and C . In case of trivial actions, this resultfollows from the case n = 1, using the fact that the n -fold graded tensor productof C ( R ) ˆ ⊗ Cl with itself is isomorphic to C ( R n ) ˆ ⊗ Cl n . We refer to Kasparov’soriginal papers [Kas75, Kas81] for the proof of the general case.3.5. Excision in KK -theory. Recall that every short exact sequence 0 → I ι → A q → A/I → C ∗ -algebras induces a six-term exact sequence in K -theory K ( I ) ι −−−−→ K ( A ) q −−−−→ K ( A/I ) δ x y exp K ( A/I ) ←−−−− ι K ( A ) ←−−−− q K ( I )which happens to be extremely helpful for the computation of K -theory groups.To some extend we get similar six-term sequences in KK -theory, but one has toimpose some extra conditions on the short exact sequence: Definition 3.27.
Let G be a locally compact group. A short exact sequence ofgraded G - C ∗ -algebras 0 → I ι → A q → A/I → G -equivariantly semisplit if there exists a G -equivariant completely positive,normdecreasing, grading preserving cross setion φ : A/I → A for the quotient map q : A → A/I . We then also say that A is a G -semisplit extension of A/I by I .By an important result of Choi-Effros [CE76] a short exact sequence 0 → I ι → A q → A/I → G -action) is always semisplit if A is nuclear. But there aremany other important cases of semisplit extensions.Every G -semisplit extension determines a unique class in KK G ( A/I, I ) which playsan important rˆole in the construction of the six-term exact sequences in KK -theory.The non-equivariant version is well documented (e.g., see [Kas81, Ska85, CS86] and[Bla86, Section 19.5]). But the details of the equivariant version, which we shallneed below, are somewhat scattered in the literature. The main ingredients areexplained in [BS89, Remarques 7.5] (see also the proof of [CE01a, Lemma 5.17]).We summarise the important steps as follows: (i) If 0 → I ι → A q → A/I → G -equivariant semisplit extension, thenthe canonical embedding e : I → C q := C ([0 , , A ) /C ((0 , , I ), whichsends a ∈ I to the equivalence class of (1 − t ) a in C q , determines a KK G -equivalence [ e ] ∈ KK G ( I, C q ).(ii) View the Bott-class β ∈ KK ( C , C (0 , KK G ( C , C (0 , G -actions everywhere and let i : C ((0 , , A/I ) → C q denotethe canonical map. Let c ∈ KK G ( A/I, I ) be the class defined via theKasparov product c = ( β ⊗ A/I ) ⊗ C ((0 , ,A/I ) [ i ] ⊗ C q [ e ] − , where [ e ] − ∈ KK G ( C q , I ) denotes the inverse of [ e ]. We call c ∈ KK G ( A/I, I ) the class attached to the equivariant semisplit ex-tension 0 → I ι → A q → A/I → G -equivariant versions of Stinespring’s theoremand of Kasparov’s stabilisation theorem ([MP84]). Theorem 3.28.
Suppose that → I ι → A q → A/I → is a G -equivariant semisplitshort exact sequence of C ∗ -algebras. Then for every G - C ∗ -algebra B , we have thefollowing two six-term exact sequences: KK G ( B, I ) ι ∗ −−−−→ KK G ( B, A ) q ∗ −−−−→ KK G ( B, A/I ) ∂ x y ∂ KK G ( B, A/I ) ←−−−− q ∗ KK G ( B, A ) ←−−−− ι ∗ KK G ( B, I ) and KK G ( A/I, B ) q ∗ −−−−→ KK G ( A, B ) ι ∗ −−−−→ KK G ( I, B ) ∂ x y ∂ KK G ( I, B ) ←−−−− ι ∗ KK G ( A, B ) ←−−−− q ∗ KK G ( A/I, B ) , where the boundary maps are all given by taking Kasparov product with the class c ∈ KK G ( A/I, I ) of the given extension. The Baum-Connes conjecture
The universal proper G -space. In what follows, for a locally compact group G , a G -space will mean a locally compact space X together with a homomorphism h : G → Homeo( X ) such that the map G × X → X ; ( s, x ) s · x := h ( s )( x )is continuous. A G -space X is called proper , if the map ϕ : G × X → X × X ; ( s, x ) ( s · x, x )is proper in the sense that inverse images of compact sets are compact. Equivalently, X is a proper G -space, if every net ( s i , x i ) in G × X such that ( s i · x i , x i ) → ( y, x )for some ( y, x ) ∈ X × X has a convergent subnet. We also say that G acts properly IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 25 on X . Proper G -spaces have an extremely nice behaviour and they are very closelyconnected to actions by compact groups.Let us state some important properties: Lemma 4.1.
Suppose that X is a proper G -space. Then the following hold:(i) For every x ∈ X the stabiliser G x = { s ∈ G : s · x = x } is compact.(ii) The orbit space G \ X equipped with the quotient topology is a locally compactHausdorff space.(iii) If X is a G -space, Y is a proper G -space and φ : X → Y is a G -equivariantcontinuous map, then X is a proper G -space as well.Proof. The first assertion follows from G x × { x } = ϕ − ( { ( x, x ) } ), if ϕ : G × X → X × X is the structure map.For the second assertion we first observe that the quotient map q : X → G \ X isopen since for any open subset U ⊆ X we have q − ( q ( U )) = G · U is open in X .This then easily implies that G \ X is locally quasi-compact. We need to show that G \ X is Hausdorff. For this assume that there is net ( x i ) such that the net of orbits( G ( x i )) converges to two orbits G ( x ), G ( y ). We need to show that y = s · x forsome s ∈ G . Since the quotient map q : X → G \ X is open, we may assume, afterpassing to a subnet if necessary, that x i → x and s i · x i → y for some suitable net( s i ) in G . Hence ( s i · x i , x i ) → ( y, x ), and by properness we may assume, afterpassing to a subnet if necessary, that ( s i , x i ) → ( s, x ) in G × X for some s ∈ G .But then s i · x i → s · x which implies y = s · x .For the third assertion let K ⊆ X be compact. If ( s · x, x ) ∈ K × K it followsthat φ × φ ( s · x, x ) ∈ φ ( K ) × φ ( K ), hence, by properness of Y , ( s, φ ( x )) lies in thecompact set ϕ − Y ( φ ( K ) × φ ( K )) of G × Y . If C ⊆ G denotes the compact projectionof this set in G , we see that ϕ − X ( K × K ) ⊆ C × K is compact as well. (cid:3) Example 4.2. (a) If G is compact, then every G -space X is proper, since for all C ⊆ X compact, we have that ϕ − ( C × C ) ⊆ G × C is compact. (b) Suppose that H ⊆ G is a closed subgroup of G and assume that Y is a proper H -space. The induced G -space G × H Y is defined as the quotient H \ ( G × Y ) withrespect to the H -action h · ( s, y ) = ( sh − , h · y ). This action is proper by part (iii)of the above lemma, hence G × H Y is a locally compact Hausdorff space. We let G act on G × H Y by s · [ t, y ] = [ st, y ]. We leave it as an exercise to check that thisaction is proper as well. (c) It follows as a special case of (b) that whenever K ⊆ G is a compact subgroupof G and Y is a K -space, then the induced G -space G × K Y is a proper G -space.Indeed, by a theorem of Abels (see [Abe78, Theorem 3.3]) every proper G -space islocally induced from compact subgroups. More precise, if X is a proper G -spaceand x ∈ X , then there exists a G -invariant open neighbourhood U of x such that U ∼ = G × K Y as G -space for some compact subgroup K of G (depending on U ) andsome K -space Y . In particular, if G does not have any compact subgroup, thenevery proper G -space is a principal G -bundle. (d) If M is a finite dimensional manifold, then the action of the fundamental group G = π ( M ) on the universal covering f M by deck transformations is a (free and)proper action. Definition 4.3.
A proper G -space Z is called a universal proper G -space if forevery proper G -space X there exists a continuous G -map φ : X → Z which isunique up to G -homotopy. We then write Z =: EG . Note that EG is unique up to G -homotopy equivalence.The following result is due to Kasparov and Skandalis (see [KS91, Lemma 4.1]): Proposition 4.4.
Let X be a proper G -space and let M ( X ) denote the set of finiteRadon measures on X with total mass in ( , equipped with the weak-* topologyas a subset of the dual C ( X ) ′ of C ( X ) and equipped with the action induced bythe action of G on C ( X ) . Then M ( X ) is a universal proper G -space. Since the restriction of a proper G -action to a closed subgroup H is again proper,it follows that the restriction of the G -space M ( X ) of the above proposition to anyclosed subgroup H is a universal proper H -space as well. By uniqueness of EG upto G -homotopy we get Corollary 4.5.
Suppose that H is a closed subgroup of G and let Z be a universalproper G -space. Then Z is also a universal proper H -space if we restrict the given G -action to H . Example 4.6. (a) If G is compact, then the one-point space { pt } with the trivial G -action is a universal proper G -space. Similarly, every contractible space Z withtrivial G -action is universal. Hence EG = { pt } and EG = Z . (b) We have E R n = R n : Since R n has no compact subgroups it follows thatevery proper G -space is a principal R n -bundle. On the other hand, since R n iscontractible, it follows that every principal R n -bundle is trivial. It follows thatevery proper R n -space X is isomorphic to R n × Y with trivial action on Y andtranslation action on R n . Hence the projection p : X ∼ = R n × Y → R n maps X equivariantly into R n . If φ , φ : X → R n are two such maps, then φ t : X → R n : φ t ( x ) = tφ ( x ) + (1 − t ) φ ( x )is a G -homotopy of equivariant maps between them. Thus R n is universal. (c) It follows from (b) together with Corollary 4.5 that E Z n = R n . (d) If G is a torsion free discrete group, then, as explained above, every proper G -space is a principal G -bundle. It follows that EG = EG , the universal principal G -bundle. (e) If G is an almost connected group (i.e., the quotient G/G of G by the connectedcomponent G of the identity in G is compact), then G has a maximal compactsubgroup K ⊆ G . It is then shown by Abels in [Abe74] that G/K is a universalproper G -space. (f ) It follows from (e) and Corollary 4.5 that for every closed subgroup H of analmost connected group G , we have EH = G/K , with K a maximal compactsubgroup of G . In particular, we have E SL( n, Z ) = SL( n, R ) / SO( n ). IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 27 The Baum-Connes assembly map.
The
Baum-Connes conjecture with co-efficients in a C ∗ -algebra A (denoted BCC for short) states, that for every G - C ∗ -algebra A a certain assembly map µ ( G,A ) : K G ∗ ( EG ; A ) → K ∗ ( A ⋊ α G )is an isomorphism of abelian groups. Here K G ∗ ( EG ; A ), often called the topo-logical K -theory of G with coefficients in A , can be regarded as the equivariant K -homology of EG with coefficients in A . We give a precise definition of thisgroup and of the assembly map below. In case A = C we get the Baum-Connesconjecture with trivial coefficients (BC for short), which relates the equivariant K -homology K G ∗ ( EG ) := K G ∗ ( EG ; C ) with K ∗ ( C ∗ r ( G )), the K -theory of the reducedgroup algebra of G .It is well known by work of Higson, Lafforgue and Skandalis ([HLS02]) that thenow often called Gromov Monster group G fails the conjecture with coefficients.But there is still no counterexample for the conjecture with trivial coefficients. Onthe other hand we know by work of Higson and Kasparov [HK01] that (even a verystrong version of) BCC holds for all a- T -menable groups – a large class of groupswhich contains all amenable groups. We give a more detailed discussion of thisin Section 4.3 below. The relevance of the Baum-Connes conjecture comes from anumber of facts:(i) It implies many other important conjectures, like the Novikov conjecturein topology, the Kaplansky conjecture on idempotents in group algebras,and the Gromov-Lawson conjecture on positiv scalar curvatures in Differ-ential Geometry. So the validity of the conjecture for a given group G hasmany positive consequences. We refer to [BCH94, Val02] for more detaileddiscussions on these applications.(ii) At least in the case of trivial coefficients the left hand side K G ∗ ( EG ) iscomputable (at least in principle) by classical techniques from algebraictopology like excision, taking direct limits, and such. These methods areusually not available for the computation of K ∗ ( C ∗ r ( G )) (or K ∗ ( A ⋊ r G )).(iii) As we shall see further down, the conjecture allows a certain flexibility forthe coefficients in a number of interesting cases, which makes it possible toperform explicit K -theory computations for certain crossed products and(twisted) group algebras.Before we go on with this general discussion, we now want to explain the ingredientsof the conjecture. For this we let G be a locally compact group and X a proper G -space. Let us further assume that X is G -compact , which means that G \ X iscompact. Then there exists a continuous function c : X → [0 , ∞ ) with compactsupport such that for all x ∈ X we have Z G c ( s − · x ) ds = 1 . For the construction, just choose any compactly supported positive function ˜ c on X such that for each x ∈ X there exists s ∈ G with ˜ c ( s · x ) = 0, divide this functionby the strictly positive function d ( x ) := R G ˜ c ( s − · x ) ds and then put c = q ˜ cd . Weshall call such function c : X → [0 , ∞ ) a cut-off function for ( X, G ). For such c consider the function p c : G × X → [0 , ∞ ); p c ( s, x ) = ∆ G ( s ) − / c ( x ) c ( s − · x ) , ∀ ( s, x ) ∈ G × X. It follows from the properness of the action and the fact c has compact supportthat p c ∈ C c ( G × X ) ⊆ C c ( G, C ( X )). Thus p c can be regarded as an element ofthe reduced (or full) crossed product C ( X ) ⋊ r G = C c ( G, C ( X )). In fact, p c is aprojection in C ( X ) ⋊ r G : For each ( s, x ) ∈ G × X we have p c ∗ p c ( s, x ) = Z G p c ( t, x ) p c ( t − s, t − · x ) dt = Z G ∆ G ( s ) − / c ( x ) c ( t − · x ) c ( s − · x ) dt = p c ( s, x ) · Z G c ( t − · x ) dt = p c ( s, x )and it is trivial to check that p ∗ c = p c . Thus p c determines a class [ p c ] ∈ K ( C ( X ) ⋊ r G ) = KK ( C , C ( X ) ⋊ r G ). Note that this class does not depend onthe particular choice of the cut-off function c , for if ˜ c is another cut-off function,then c t = p tc + (1 − t )˜ c , t ∈ [0 , c with ˜ c , and then p c t is a path of projectionsjoining p c with p ˜ c . We call [ p c ] the fundamental K -theory class of C ( X ) ⋊ r G .Recall that Kasparov’s descent homomorphism J G : KK G ∗ ( A, B ) → KK ∗ ( A ⋊ r G, B ⋊ r G )is defined by sending a class x = [ E , Φ , γ, T ] ∈ KK G ( A, B ) to the class J G ( x ) =[ E ⋊ r G, Φ ⋊ r G, ˜ T ] ∈ KK ( A ⋊ r G, B ⋊ r G ), where [ E , Φ , γ ] [ E ⋊ r G, Φ ⋊ r G ] isthe descent in the correspondence categories as described in [Ech17, § T on E ⋊ r G is given on the dense subspace C c ( G, E ) by( ˜ T ξ )( s ) = T ( ξ ( s )) ∀ ξ ∈ C c ( G, E ) , s ∈ G. A similar descent also exists if we replace the reduced crossed products by fullcrossed products.Now, if A is a G - C ∗ -algebra, we can consider the following chain of maps µ X : KK G ∗ ( C ( X ) , A ) J G −→ KK ∗ ( C ( X ) ⋊ r G, A ⋊ r G ) [ p c ] ⊗· −→ KK ∗ ( C , A ⋊ r G ) ∼ = K ∗ ( A ⋊ r G ) , where J G denotes Kasparov’s descent homomorphism. If X and Y are two G -compact proper G -spaces and if ϕ : X → Y is a continuous G -equivariant map, thenone can check that ϕ : X → Y is automatically proper, i.e., inverse images of com-pact sets are compact, and therefore it induces a G -equivariant ∗ -homomorphism ϕ ∗ : C ( Y ) → C ( X ); f f ◦ ϕ. Moreover, if c : Y → [0 , ∞ ) is a cut-off function for ( G, Y ), then ϕ ∗ ( c ) : X → [0 , ∞ )is a cut-off function for ( G, X ) such that p ϕ ∗ ( c ) = ( ϕ ∗ ⋊ r G )( p c ). Using this fact, it IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 29 is easy to check that the diagram KK G ∗ ( C ( X ) , A ) −−−−→ J G KK ∗ ( C ( X ) ⋊ r G, A ⋊ r G ) −−−−−→ [ p ϕ ∗ c ] ⊗· KK ∗ ( C , A ⋊ r G ) [ ϕ ∗ ] ⊗· y [ ϕ ∗ ⋊ r G ] ⊗· y y = KK G ∗ ( C ( Y ) , A ) −−−−→ J G KK ∗ ( C ( Y ) ⋊ r G, A ⋊ r G ) −−−−→ [ p c ] ⊗· KK ∗ ( C , A ⋊ r G )commutes. Hence if we define the topological K -theory of G with coefficients in A as K G ∗ ( EG ; A ) := lim X ⊆ EGX is G -compact KK G ∗ ( C ( X ) , A )where the G -compact subsets of EG are ordered by inclusion, we get a well definedhomomorphism µ ( G,A ) : K G ∗ ( EG ; A ) = lim X KK G ∗ ( C ( X ) , A ) lim X µ X −→ K ∗ ( A ⋊ r G ) . This is the Baum-Connes assembly map for the system (
A, G, α ). We say that G satisfies BC for A , if this map is bijective. Remark 4.7.
We should remark that almost the same construction yields an as-sembly map µ full ( G,A ) : K G ∗ ( EG ; A ) → K ∗ ( A ⋊ α G )for the full crossed product A ⋊ α G such thatΛ ∗ ◦ µ full ( G,A ) = µ ( G,A ) , where Λ : A ⋊ α G → A ⋊ α,r G denotes the regular representation. The only differenceis that we then use Kasparov’s descent J fullG : KK G ( C ( X ) , A ) → KK ( C ( X ) ⋊ G, A ⋊ G ) for the full crossed products. Note that, since G acts properly (and henceamenably) on X , we have C ( X ) ⋊ G ∼ = C ( X ) ⋊ r G (see Remark 4.16 below), sothat we can use the same product of the fundamental class [ p c ] ∈ K ∗ ( C ( X ) ⋊ G )as for the reduced assembly map. But it is well known that the full analogue of theconjecture must fail for all lattices Γ in any almost connected Lie group G which hasKazhdan’s property (T). But for a large class of groups (including the K -amenablegroups of Cuntz and Julg-Valette [Cun83,JV84]), the regular representation inducesan isomorphism in K -theory for the full and the reduced crossed products, and thenthe full assembly map coincides up to this isomorphism with the reduced one. Example 4.8 ( The Green-Julg Theorem). If G is a compact group withnormed Haar measure we have EG = { pt } and hence K G ∗ ( EG ; A ) = KK G ∗ ( C , A ) isthe G -equivariant K -theory K G ∗ ( A ) of A . The isomorphism K G ∗ ( A ) ∼ = K ∗ ( A ⋊ G )is the content of the Green-Julg theorem (see [Jul81]). Let us briefly look at thespecial form of the assembly map in this situation: First of all, if G is compact,we may realise KK G ( C , A ) as the set of homotopy classes of triples ( E , γ, T ) inwhich E is a graded Hilbert A -module, γ : G → Aut( E ) is a compatible action and T ∈ L ( E ) is a G -invariant operator such that T ∗ − T, T − ∈ K ( E ) . (If T is not G -invariant, it may be replaced by the G -invariant operator ˜ T = R G Ad γ s ( T ) ds ). The cut-off function of the one-point space is simply the function which sends this point to 1, and the projection p ∈ C ( G ) ⊆ C ∗ ( G ) is the constantfunction 1 G . It acts on ξ ∈ C ( G, E ) ⊆ E ⋊ G via ( pξ )( s ) = R G γ t ( ξ ( t − s )) dt .Now, given ( E , γ, T ) ∈ KK G ( C , A ) the assembly map sends this class to the classof the Kasparov cycle ( E ⋊ G, p, ˜ T ), with ( ˜ T ξ )( s ) = T ( ξ ( s )) for all s ∈ G . Since T is G -invariant, a short computation shows that ˜ T commutes with p , and hence wecan decompose ( E ⋊ r G, p, ˜ T ) = ( p ( E ⋊ G ) , , ˜ T ) L ((1 − p )( E ⋊ G ) , , ˜ T ) in whichthe second summand is degenerate. Thus we get µ ([ E , γ, T ]) = [ p ( E ⋊ G ) , ˜ T ] ∈ KK ( C , A ⋊ G ) . Note that a function ξ ∈ C ( G, E ) lies in p ( E ⋊ G ) if and only if ξ ( s ) = γ s ( ξ ( e )) forall s ∈ G , and it is clear that such functions are dense in p ( E ⋊ G ). Using this, themodule p ( E ⋊ G ) can be described alternatively as follows: We equip E with the A ⋊ G valued inner product and left action of A ⋊ G given by h e , e i A ⋊ G ( s ) = h e , γ s ( e e ) i A and e · f = Z G γ s ( e · f ( s − )) ds for f ∈ C ( G, A ) ⊆ A ⋊ G . We denote by E A ⋊ G the completion of E as a Hilbert A ⋊ G -module with this action and inner product. It is then easy to check thatevery G -invariant operator S ∈ L ( E ) extends to an operator S G ∈ L ( E A ⋊ G ). Ashort computation shows that the map Φ : p ( E ⋊ G ) → E A ⋊ G given by Φ( ξ ) = ξ ( e )for ξ ∈ C ( G, E ) ∩ p ( E ⋊ G ) is an isomorphism of Hilbert A ⋊ G -modules whichintertwines ˜ T with T G ∈ L ( E A ⋊ G ). Using this, we get the following description ofthe assembly map µ : KK G ( C , A ) → KK ( C , A ⋊ G ); µ ([ E , γ, T ]) = [ E A ⋊ G , T G ] . This map has a direct inverse given as follows: Let L ( G, A ) be the Hilbert A -module with A -valued inner product given by h f, g i A = R G α t − ( f ( t ) ∗ g ( t )) dt andright A -action given by ( f · a )( s ) = f ( s ) α s ( a ). Then A ⋊ G acts on L ( G, A ) viathe regular representation given by convolution. There is a canonical α -compatibleaction σ : G → Aut( L ( G, A )) given by right translation σ s ( f )( t ) = f ( ts ). It isthen not difficult to check that ν : KK ( C , A ⋊ G ) → KK G ( C , A ); [ F , S ] [ F ⊗ A ⋊ G L ( G, A ) , id ⊗ σ, S ⊗ µ . For a few more details on these computations see [Ech08]. Example 4.9. If G is a discrete torsion free group, then EG = EG , the universalprincipal G -bundle of G . Since G acts freely and properly on EG it follows from atheorem of Green [Gre77] that C ( EG ) ⋊ G is Morita equivalent to C ( G \ EG ) = C ( BG ), where BG = G \ EG is the classifying space of G . Now, for any discretegroup and any G - C ∗ -algebra A we have a canonical isomorphism KK G ( A, C ) ∼ = KK ( A ⋊ G, C )which sends the class of an equivariant A − C KK -cycle ( H , Φ , γ, T ) to the classof the A ⋊ G − C KK -cycle ( H , Φ ⋊ γ, T ). Note that in this situation (Φ , γ ) is acovariant representation of ( A, G, α ) on the Hilbert space H , and hence sums upto a representation of A ⋊ G by the universal property of the full crossed product.Since G is discrete, one checks that condition (ii) in Definition 3.2 for ( H , Φ ⋊ γ, T ) IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 31 is equivalent to the corresponding condition for ( H , Φ , γ, T ). Thus, if in addition EG is G -compact, we get K G ∗ ( EG, C ) ∼ = KK G ∗ ( C ( EG ) , C ) ∼ = KK ∗ ( C ( EG ) ⋊ G, C ) Morita-eq. ∼ = KK ∗ ( C ( BG ) , C ) = K ∗ ( BG ) . Hence in this situation the left hand side of the Baum-Connes conjecture is thetopological K -homology of the classifying space of G . If EG is not G -compact, asimilar argument gives K G ∗ ( EG, C ) = lim C ⊆ BG K ∗ ( C ) , where C runs through the compact subsets of BG , which is the K -homology of BG with compact supports. Hence the Baum-Connes conjecture relates the K -theory of the (often quite complicated) C ∗ -algebra C ∗ r ( G ) to the K -homology of theclassifying space BG of G , which can be handled by methods of classical algebraictopology (but can still be difficult to compute).We close this section with an exercise: Exercise 4.10.
Suppose that A and B are G - C ∗ -algebras and let x ∈ KK G ( A, B ).Then x induces a map · ⊗ A x : K G ∗ ( EG ; A ) → K G ∗ ( EG ; A )given on the level of KK G ( C ( X ) , A ) for some G -compact subset X ⊆ EG via themap KK G ( C ( X ) , A ) → KK G ( C ( X ) , B ); y y ⊗ A x. On the other hand, we have a map · ⊗ A ⋊ r G j G ( x ) : KK ( C , A ⋊ r G ) → KK ( C , B ⋊ r G )between the K -theory groups of the crossed products.Show that the map · ⊗ A x : K G ∗ ( EG ; A ) → K G ∗ ( EG ; A ) is well-defined and that thediagram K G ∗ ( EG ; A ) µ ( G,A ) −−−−→ K ∗ ( A ⋊ r G ) ·⊗ A x y y ·⊗ A ⋊ rG J G ( x ) K G ∗ ( EG ; B ) µ ( G,B ) −−−−→ K ∗ ( B ⋊ r G )commutes. Show that it follows from this that if A and B are KK G -equivalent,then µ ( G,A ) is an isomorphism if and only if µ ( G,B ) is an isomorphism. Check thata similar result holds for the full assembly maps µ full ( G,A ) and µ full ( G,B ) of Remark 4.7.4.3. Proper G -algebras and the Dirac dual-Dirac method. As an extensionof the Green-Julg theorem one can prove that the Baum-Connes assembly map isalways an isomorphism if the coefficient algebra A is a proper G - C ∗ -algebra in thesense of Kasparov, which we are now going to explain. Recall that if X is a locallycompact space, then a C ∗ -algebra A is called a C ( X )-algebra, if there exists anon-degenerate ∗ -homomorphismΦ : C ( X ) → ZM ( A ) , the center of the multiplier algebra of A . If A is a C ( X )-algebra, then A canbe realized as an algebra of C -sections of a (upper semicontinuous) bundle of C ∗ -algebras { A x : x ∈ X } , where each fibre A x is given by A x = A/I x with I x = ( C ( X r { x } ) · A ), where we write f · a := Φ( f ) a for f ∈ C ( X ), a ∈ A . Werefer to [Wil07, Appendix C] for a detailed discussion of C ( X )-algebras. Definition 4.11.
Suppose that G is a locally compact group and A is a G - C ∗ -algebra. Suppose further that A is a C ( X )-algebra such that the structure mapΦ : C ( X ) → ZM ( A ) is G -equivariant. We then say that A is an X ⋊ G - C ∗ -algebra.If A is an X ⋊ G - C ∗ -algebra for some proper G -space X , then A is called a proper G - C ∗ -algebra .Note that in the definition of a proper G - C ∗ -algebra we may always assume X tobe a realisation of EG : Since if ϕ : X → EG is a G -equivariant continuous map,we get a non-degenerate G -equivariant ∗ -homomorphism ϕ ∗ : C ( EG ) → C b ( X ) = M ( C ( X )); ϕ ∗ ( f ) = f ◦ ϕ and then the composition Φ ◦ ϕ ∗ : C ( EG ) → ZM ( A ) makes A into an EG ⋊ G -algebra. Recall from our discussion of proper G -spaces that proper actions behavevery much like actions of compact groups since they are locally induced from actionsof compact subgroups. It is therefore not very surprising that an analogue of theGreen-Julg theorem should hold also for proper G - C ∗ -algebras. Theorem 4.12.
Suppose that A is a proper G - C ∗ -algebra. Then the Baum-Connesassembly map µ : K G ∗ ( EG, A ) → K ∗ ( A ⋊ r G ) is an isomorphism. However, the proof of this result is much harder than the proof of the Green-Julgtheorem shown in the previous section. In what follows we want to indicate at leastsome ideas towards this result. On the way we discuss some useful results aboutinduced dynamical systems and their applications to the Baum-Connes conjecture.In what follows suppose that H is a closed subgroup of the locally compact group G and that β : H → Aut( B ) is an action of H on the C ∗ -algebra B . Recall from[Ech17, Section 2.6] that the induced C ∗ -algebra Ind GH B is defined as the algebraInd GH B := (cid:26) f ∈ C b ( G, B ) : f ( sh ) = β h − ( f ( s )) for all s ∈ G and h ∈ H and ( sH
7→ k f ( s ) k ) ∈ C ( G/H ) (cid:27) . This is a C ∗ -subalgebra of C b ( G, B ) which carries an action Ind β : G → Aut(Ind GH B )) given by (cid:0) Ind β s ( f ) (cid:1) ( t ) = f ( s − t ) . If Y is a locally compact H -space, then Ind GH C ( Y ) ∼ = C ( G × H Y ) as G -algebras.Hence the above procedure extends the procedure of inducing G -spaces as discussedin Example 4.2 above. The following result is quite useful when working withinduced algebras. For the formulation recall that for any G - C ∗ -algebra A we havea continuous action of G on the primitive ideal space Prim( A ) given by ( s, P ) α s ( P ). Theorem 4.13.
Suppose that A is a G - C ∗ -algebra and let H be a closed subgroupof G . Then the following are equivalent: IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 33 (i) There exists an H -algebra B such that A ∼ = Ind GH B as G -algebras.(ii) A carries the structure of a G/H ⋊ G - C ∗ -algebra.(iii) There exist a continuous G -equivariant map φ : Prim( A ) → G/H .Proof. (i) ⇔ (iii) is [Ech17, Theorem 2.6.2]. The proof of (iii) ⇔ (ii) follows fromthe general correspondence between continuous maps φ : Prim( A ) → X and non-degenerate ∗ -homomorphisms Φ : C ( X ) → ZM ( A ) given by the Dauns-HofmannTheorem. We refer to [Wil07, Appendix C] for a discussion of this correspondence. (cid:3) Let us briefly indicate how the objects in (i) and (ii) of the above theorem arerelated to each other. If A = Ind GH B , then the G -equivariant ∗ -homomorphismΦ : C ( G/H ) → ZM (Ind GH B ) is simply given by (cid:0) Φ( g ) f (cid:1) ( s ) = g ( sH ) f ( s ) , g ∈ C ( G/H ) , f ∈ Ind GH B. Conversely, if Φ : C ( G/H ) → ZM ( A ) is given, let B := A eH = A/ ( C ( G/H r { eH } ) · A )be the fibre of A over the coset eH . Since Φ : C ( G/H ) → ZM ( A ) is G -equivariant,it follows that the ideal C ( G/H r { eH } ) · A is H -invariant for the restriction of α to H . Thus α | H induces an action β : H → Aut( A eH ) = Aut( B ). The G -isomorphismΨ : A → Ind GH B is then given byΨ( a )( s ) = q ( α s − ( a )) , where q : A → A eH denotes the quotient map.The induction of H -algebras to G -algebras extends to an induction mapInd GH : KK H ( B, C ) → KK G (Ind GH B, Ind GH C )given as follows: If [ E , Φ , γ, T ] ∈ KK H ( B, C ), then we define the induced HilbertInd GH C -module Ind GH E asInd GH E = (cid:26) ξ ∈ C b ( G, E ) : ξ ( sh ) = γ h − ( ξ ( s )) for all s ∈ G and h ∈ H and ( sH
7→ k f ( s ) k ) ∈ C ( G/H ) (cid:27) with Ind GH C -valued inner product and left Ind GH C -action given as follows: h ξ, η i Ind GH C ( s ) = h ξ ( s ) , η ( s ) i C and ( ξ · f )( s ) = ξ ( s ) · f ( s ) . Similarly, if Φ : B → L ( E ) is a left action of B on E , then we get an actionInd Φ : Ind GH B → L (Ind GH E ) by(Ind Φ( g ) ξ )( s ) = Φ( g ( s )) ξ ( s ) . Finally, we define the operator ˜ T ∈ L (Ind GH E ) via ( ˜ T ξ )( s ) = T ( ξ ( s )). It is notdifficult to check that (ind GH E , Ind Φ , Ind β, ˜ T ) is a G -equivariant Ind GH B − Ind GH C Kasparov cycle and Kasparov’s induction map in KK -theory is then defined asInd GH ([ E , Φ , γ, T ]) = [ind GH E , Ind Φ , Ind β, ˜ T ] ∈ KK G (Ind GH B, Ind GH C ) . We want to use this map to define an induction mapInd GH : K H ∗ ( EH ; B ) → K G ∗ ( EG ; Ind GH B ) for every H -algebra B . For this suppose that Y ⊆ EH is an H -compact subset.Then the induced G -space G × H Y is proper and G -compact and therefore mapsequivariantly into EG via some continuous map j : G × H Y → EG whose image isa G -compact subset X ( Y ) ⊆ EG . One can check that the composition of maps KK G ( C ( Y ) , B ) Ind GH −→ KK G ( C ( G × H Y ) , Ind GH B ) j ∗ −→ KK G ( C ( X ( Y )) , Ind GH B )is compatible with taking limits and therefore induces a well-defined induction map I GH : K H ( EH ; B ) → K G ( EG ; Ind GH B ) . Replacing B by B ⊗ Cl (or B ⊗ C ( R )) gives an analogous map from K H ( EH ; B )to K G ( EG ; Ind GH B ). We then have the following theorem, which has been shown in[CE01a, Theorem 2.2] (for G discrete and H finite the result has first been obtainedearlier in [GHT00]): Theorem 4.14.
Suppose that H is a closed subgroup of G . Then the inductionmap I GH : K H ∗ ( EH ; B ) → K G ∗ ( EG ; Ind GH B ) is an isomorphism of abelian groups for every H - C ∗ -algebra B . Now Green’s imprimitivity theorem (see [Ech17, Theorem 2.6.4]) says that thefull (resp. reduced) crossed products B ⋊ β, ( r ) G and Ind GH B ⋊ Ind β, ( r ) G are Moritaequivalent via a canonical B ⋊ ( r ) H − Ind GH B ⋊ ( r ) G equivalence bimodule X GH ( B ) ( r ) .Since Morita equivalences provide KK -equivalences, we obtain the following dia-gram of maps K H ∗ ( EH ; B ) µ H −−−−→ K ∗ ( B ⋊ r H ) I GH y y ⊗ [ X GH ( B ) r ] K G ∗ ( EG,
Ind GH B ) −−−−→ µ G K ∗ (Ind GH B ⋊ r G )in which both vertical arrows are isomorphisms. It is shown in [CE01a, Proposition2.3] that this diagram commutes. As a corollary we get Corollary 4.15.
Suppose that H is a closed subgroup of G and B is an H -algebra.Then the assembly map µ H : K H ∗ ( EH ; B ) → K ∗ ( B ⋊ r H ) is an isomorphism if and only if µ G : K G ∗ ( EG,
Ind GH B ) → K ∗ (Ind GH B ⋊ r G ) is an isomorphism. In particular, if G satisfies BCC, then so does H . A similarresult holds for the assembly maps into the K -theories of the full crossed products B ⋊ H and Ind GH B ⋊ G , respectively (see Remark 4.7). We now come back to general proper G -algebras A . So suppose that X is a proper G -space and that A is an X ⋊ G - C ∗ -algebra. Since every proper G -space is locallyinduced from a compact subgroup, we find for each x ∈ X an open G -invariantneighbourhood U ⊆ X such that U ∼ = G × K Y for some compact subgroup K of G and some K -space Y . Then C ( G × K Y ) ∼ = Ind GH C ( Y ) is a G/K ⋊ G -algebraby Theorem 4.13. Let A ( U ) := Φ( C ( U )) A ⊆ A . Then A ( U ) is a G -invariant IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 35 ideal of A and carries the structure of a U ⋊ G -algebra in the canonical way. Thecomposition C ( G/H ) → C b ( G × K Y ) ∼ = C b ( U ) Φ −→ ZM ( A ( U ))then gives A ( U ) the structure of a G/K ⋊ G -algebra. Thus it follows from Theorem4.13 that A ( U ) ∼ = Ind GK B for some K -algebra B . By the Green-Julg theoremwe know that K satisfies BCC and hence it follows from Corollary 4.15 that theassembly map µ U : K G ∗ ( EG, A ( U )) → K ∗ ( A ( U ) ⋊ r G )is an isomorphism. Thus we see that for proper G -algebras the Baum-Connesconjecture holds locally . Now every G -invariant open subset W ⊆ X with G -compact closure W can be covered by a finite union of open sets U , . . . , U l suchthat each U i is isomorphic to some induced space G × K i Y i for some compactsubgroup K i ⊆ G . Using six-term sequences and induction on the number l ofopen sets in this covering, we then conclude that µ W : K G ∗ ( EG, A ( W )) → K ∗ ( A ( W ) ⋊ r G )for all such W . Now, taking inductive limits indexed by W , one can show that theassembly map µ : K G ∗ ( EG, A ) → K ∗ ( A ⋊ r G )is an isomorphism as well. This then finishes the proof of Theorem 4.12. (We referto [CEM01] for the original proof and further details.) Remark 4.16.
An application of Green’s imprimitivity theorem also implies thatfor any proper G - C ∗ -algebra A the full and reduced crossed products coincide. Tosee this, let A be an X ⋊ G -algebra for some proper G -space X . Let { U i : i ∈ I } bean open cover of X consisting of G -invariant open sets such that each of these setsis induced by some compact subgroup K of G . Suppose now that π ⋊ U : A ⋊ G →B ( H ) is any irreducible representation of A ⋊ G . We claim that there exists at leastone i ∈ I such that π does not vanish on the ideal A ( U i ) of A , and hence π ⋊ U doesnot vanish on the ideal A ( U i ) ⋊ G of the crossed product. Indeed, since { U i : i ∈ I } is a covering of X it is an easy exercise, using a partition of unity argument, toshow that P i ∈ I A ( U i ) is a dense ideal in A . The claim then follows since π = 0.It now suffices to show that A ( U i ) ⋊ G = A ( U i ) ⋊ r G ⊆ A ⋊ r G , since this impliesthat every irreducible representation of the ideal A ( U i ) ⋊ G corresponds to anirreducible representation of A ⋊ r G . To see this recall that A ( U i ) ∼ = Ind GK B forsome compact subgroup K of G and some K -algebra B . Since K is compact,hence amenable, we have B ⋊ K = B ⋊ r K . Since the B ⋊ r K − Ind GK B ⋊ r G -equivalence bimodule X GH ( B ) r is the quotient of the B ⋊ K − Ind GK B ⋊ G -equivalencebimodule X GK ( B ) by the submodule (ker Λ ( B,K ) ) · X GK ( B ) = { } , it follows from theRieffel-correspondence ([Ech17, Proposition 2.5.4]) that A ( U i ) ⋊ G ∼ = Ind GK B ⋊ G =Ind GK B ⋊ r G ∼ = A ( U i ) ⋊ r G , which finishes the proof.We are now coming to Kasparov’s Dirac-dual Dirac method for proving the Baum-Connes conjecture. As we shall discuss below, this has been the most successfulmethod so far for proving that the conjecture holds for certain classes of groups.Since, as we saw above, the Baum-Connes conjecture always holds for proper G - C ∗ -algebras as coefficients, the basic idea is to show that for a given group G every G - C ∗ -algebra B is KK G -equivalent to a proper G - C ∗ -algebra. Since by Exercise4.10 the validity of the Baum-Connes conjecture is invariant under passing to KK G -equivalent coefficient algebras, this would result in a proof that the group G satisfiesBCC, i.e., Baum-Connes for all coefficients. Indeed, we need less: Definition 4.17.
Suppose that G is a second countable locally compact group andassume that there is a proper G - C ∗ -algebra D together with elements α ∈ KK G ( D , C ) and β ∈ KK G ( C , D )such that γ := β ⊗ D α = 1 C ∈ KK G ( C , C ) . We then say that G has γ -element equal to one . If, in addition α ⊗ C β = 1 D , thenwe say that G satisfies the strong Baum-Connes conjecture .Note that in almost all cases where G has γ -element equal to one there is also aproof of strong BC.If G has γ -element equal to one and B is any other G - C ∗ -algebra, it follows that( β ˆ ⊗ B ) ⊗ D ˆ ⊗ B ( α ˆ ⊗ B ) = γ ˆ ⊗ B = 1 B ∈ KK G ( B, B )and similarly, since the descent KK G ( A, B ) → KK ( A ⋊ r G, B ⋊ r G ) is compatiblewith Kasparov products, we get J G ( β ˆ ⊗ B ) ⊗ ( D ˆ ⊗ B ) ⋊ r G J G ( α ˆ ⊗ B ) = J G (1 B ) = 1 B ⋊ r G ∈ KK ( B ⋊ r G, B ⋊ r G ) . Moreover, it follows from Exercise 4.10 that the following diagram commutes(4.1) K G ∗ ( EG, B ) µ ( G,B ) −−−−→ K ∗ ( B ⋊ r G ) ·⊗ ( β ⊗ B ) y y ·⊗ J G ( β ⊗ B ) K G ∗ ( EG, D ˆ ⊗ B ) µ ( G, D ˆ ⊗ B ) −−−−−−→ ∼ = K ∗ (( D ˆ ⊗ B ) ⋊ r G ) ·⊗ ( α ⊗ B ) y y ·⊗ J G ( α ⊗ B ) K G ∗ ( EG, B ) µ ( G,B ) −−−−→ K ∗ ( B ⋊ r G ) . Since D ˆ ⊗ B is a proper G -algebra (via the composition of Φ : C ( X ) → ZM ( D )with the canonical map of M ( D ) to M ( D ˆ ⊗ B )), the middle horizontal map is anisomorphism of abelian groups, and by the above discussion it follows that thecompositions of the vertical maps on either side are isomorphisms as well. It thenfollows by an easy diagram chase that the upper horizontal map is injective and thelower horizontal map is surjective, hence µ ( G,B ) is an isomorphism as well. Thuswe get Corollary 4.18. If G has γ -element equal to one, then G satisfies the Baum-Connes conjecture with coefficients (BCC). Remark 4.19.
In diagram (4.1) we can replace all reduced crossed products bythe full ones and the (reduced) assembly map by the full assembly map to see thatwhenever G has γ -element equal to one, then the full assembly map µ full ( G,B ) : K G ∗ ( EG ; B ) → K ∗ ( B ⋊ β G ) IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 37 is an isomorphism as well. Moreover, if Λ : B ⋊ β G → B ⋊ β,r G is the regularrepresentation, then the diagram K G ∗ ( EG ; B ) µ ( G,B ) ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ µ full ( G,B ) / / K ∗ ( B ⋊ G ) Λ ∗ (cid:15) (cid:15) K ∗ ( B ⋊ r G )commutes. Since both assembly maps are isomorphisms, it follows that the regularrepresentation induces an isomorphism in K -theory between the maximal and thereduced crossed products by G . Indeed, it is shown by Tu in [Tu99b] that G is K -amenable in the sense of Cuntz [Cun83] and Julg-Valette [JV84] (which actuallyimplies that Λ is a KK -equivalence) whenever G has γ -element equal to one. Remark 4.20. If G has γ -element equal to one, then so does every closed subgroupof G . Indeed, if α ∈ KK G ( D , C ) and β ∈ KK G ( C , D ) are as in the definition ofstrong BC, then the action of G on D restricts to a proper action of H on D .Moreover, for every pair of G - C ∗ -algebras A, B we have a natural homomorphismres GH : KK G ( A, B ) → KK H ( A, B )which is given by simply restricting all actions on algebras and Hilbert modulesfrom G to H . It is easy to see that this restriction map is compatible with theKasparov product, so that we getres GH ( β ) ⊗ D res GH ( α ) = res GH ( β ⊗ D α ) = res GH (1 C ) = 1 C ∈ KK H ( C , C ) . Example 4.21.
As a sample, we want to show that R and Z satisfy strong BC(and, in particular, have γ = 1 C ). For this recall the construction of the Dirac anddual Dirac elements in the proof of the Bott periodicity theorem in Section 3.4:Let D = C ( R ) ˆ ⊗ Cl . We constructed elements α ∈ KK ( D , C ) and β ∈ KK ( C , D )which are inverse to each other in KK . Let τ : R → Aut( C ( R )) denote thetranslation action ( τ s ( f ))( x ) = f ( x − s ). Then D becomes a proper R -algebra viathe action τ ˆ ⊗ id Cl . Now recall that the classes α and β have been given by α = (cid:20) H = L ( R ) M L ( R ) , Φ , T = D √ D (cid:21) and β = [ D , , S ] , in which D = (cid:18) − ddtddt (cid:19) , Φ : D → L ( H ) is given as in (3.2), and S = S ϕ is givenby pointwise multiplication with a function ϕ : R → [ − , ϕ ( x ) = 0 ⇔ x = 0 and lim t →∞ ϕ ( t ) = 1. With the given R -action on D we may view β as a class β = [ D , τ ˆ ⊗ id Cl , , S ] ∈ KK R ( C , D ) . The only extra condition to check is the condition that Ad τ ⊗ id( s ) ( S ) − S ∈ K ( D ) = D , which follows from the fact that for any function ϕ as above, we have τ s ( ϕ ) − ϕ ∈ C ( R ) for all s ∈ R . Similarly, if we equip H = L ( R ) L L ( R ) with the represen-tation λ L λ , where (cid:0) λ s ( ξ ) (cid:1) ( x ) = ξ ( x − s ) denotes the regular representation of R ,we obtain a class α = [ H , λ, Φ , T ] ∈ KK R ( D , C ) . As above, the only extra condition to check is that (Ad λ s ( T ) − T )Φ( d ) ∈ K ( H ) forall s ∈ R and d ∈ D , which we leave as an exercise for the reader. We claim that(4.2) β ⊗ D α = 1 C ∈ KK R ( C , C ) . Note that this would be true if we equipped everything with the trivial R -actioninstead of the translation action, since then it would be a direct consequence ofthe product β ⊗ D α = 1 C ∈ KK ( C , C ), which we proved in Section 3.4. Weshow equation (4.2) by a simple trick, showing that the translation action of R is homotopic to the trivial action in the following sense: We consider the algebra D [0 ,
1] = D ˆ ⊗ C [0 ,
1] equipped with the R -action ˜ τ ˆ ⊗ id Cl where˜ τ : R → Aut( C ( R × [0 , (cid:0) ˜ τ s ( f ) (cid:1) ( x, t ) = f ( x − ts, t ) . We then consider the class ˜ α ∈ KK R ( D [0 , , C [0 , C [0 ,
1] carries thetrivial R -action, given by˜ α = (cid:2) H ˆ ⊗ C [0 , , ˜ λ M ˜ λ, Φ ˆ ⊗ , T ˆ ⊗ (cid:3) , where the R -action ˜ λ L ˜ λ on H ˆ ⊗ C [0 ,
1] is given by a formula similar to the one for˜ τ . On the other hand, we consider the class˜ β = (cid:2) D [0 , , ˜ τ ˆ ⊗ , , S ˆ ⊗ (cid:3) ∈ KK R ( C , D [0 , . If we evaluate the class ˜ β ⊗ D [0 , ˜ α ∈ KK R ( C , C [0 , β with α equipped with trivial R -actions, which is 1 C ∈ KK R ( C , C ) by the proofof Bott periodicity. If we evaluate at 1, we obtain the product β ⊗ α with respectto the proper translation action on D . Hence both classes are homotopic whichproves (4.2).Hence we see that the Dirac dual-Dirac method applies to R , and by Remark 4.20it then also applies to Z ⊆ R and both have γ -element equal to one. Using an easyproduct argument, this proof also implies that R n and Z n have γ -element equal toone. We leave it as an exercise to check that α ⊗ C β = 1 D ∈ KK R ( D , D ), i.e., that R n and Z n do satisfy the strong Baum-Connes conjecture.Extending Example 4.21 to higher dimensions, one can use Kasparov’s equivariantBott periodicity theorem as discussed in the last paragraph of Section 3.4 to showthat the Dirac dual-Dirac method works for all groups which act properly andisometrically by affine transformations on a finite dimensional euclidean space. Thishas already been pointed out by Kasparov in his conspectus [Kas95]. Later, in[Kas88], he extended this to show that the method works for all amenable Liegroups (and their closed subgroups) and, together with Pierre Julg in [JK95], theyshowed that the method works for the Lie groups SO ( n,
1) and SU ( n,
1) and theirclosed subgroups. But the most far reaching positive result which includes all casesmentioned above has been obtained by Higson and Kasparov in [HK01]:
Theorem 4.22 (Higson-Kasparov) . Suppose that the second countable locally com-pact group acts continuously and metrically properly by isometric affine transforma-tions on a separable real Hilbert space H . Then G satisfies the strong Baum-Connesconjecture. Note that the action of G on H is called metrically proper if for any ξ ∈ H and R > C ⊆ G such that k s · ξ k > R for all s ∈ G \ C . IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 39 The basic idea of the proof is to construct the proper G -algebra as an inductivelimit of algebras C ( V ) ˆ ⊗ Cl ( V ), where V ⊆ H runs through the finite dimensionalsubspaces of the Hilbert space H . But the precise construction of the algebra D and the classes α and β is very complex and we refer to the original work [HK01]of Higson and Kasparov for more details. A detailed exposition of certain aspectsof the proof can be found in the very recent paper [Nis16]. We should also notethat the original proof of Higson and Kasparov uses E -theory, a variant of KK -theory introduced by Connes and Higson in [CH90] (see also [Bla86, Chapter 25]).A groupoid version of the above theorem has been shown by Tu in [Tu99a].The class of groups which satisfies the conditions of the Higon-Kasparov theoremhas been studied first by Gromov who called them a- T -menable groups. A secondcountable group G is a- T -menable if and only if it satisfies the Haagerup approxi-mation property which says that the trivial representation 1 G can be approximateduniformly on compact sets by a net of positive definite functions ( ϕ i ) on G suchthat each ϕ i vanishes at ∞ on G . We refer to [CCJ +
01] for a detailed expositionon the class of a- T -menable groups. As a consequence of the theorem we get Corollary 4.23.
Every amenable second countable locally compact group satisfiesstrong BC. Also, the free groups F n in n generators, n ∈ N ∪ {∞} and all closedsubgroups of the Lie groups SU( n, and SO( n, satisfy strong BC. All groups in the above corollary satisfy the Haagerup property.The Dirac dual-Dirac method can also be used in cases, in which the element γ = β ⊗ D α ∈ KK G ( C , C )is not necessarily equal to 1 C , but where it satisfies the following weaker condition: Definition 4.24 (Kasparov’s γ -element) . Suppose that D is a proper G -algebra, α ∈ KK G ( D , C ) and β ∈ KK G ( C , D ). Then γ = β ⊗ D α ∈ KK G ( C , C ) is called a γ -element for G iff γ ⊗ C ( X ) = 1 C ( X ) ∈ KK G ( C ( X ) , C ( X ))for every proper G -space X .The class of groups which admit a γ -element is huge. It has been shown by Kas-parov in [Kas95, Kas88] that it contains all almost connected groups (i.e., groupswith co-compact connected component of the identity) and it is clear that the ex-istence of a γ -elements passes to closed subgroups. In [KS91, KS03] Kasparov andSkandalis proved existence of γ -elements for many other groups. Note that theabove definition of a γ -element is slightly weaker than Kasparov’s original defini-tion, in which he requires that γ ⊗ C ( X ) = 1 C ( X ) in the X ⋊ G -equivariant group KK X ⋊ G ( C ( X ) , C ( X )), where X ⋊ G denotes the transformation groupoid for the G -space X . Since the above definition suffices for our purposes and since we wantto avoid to talk about equivariant KK -theory for groupoids, we use it here. Wehave: Theorem 4.25 (Kasparov) . Suppose that G is a second countable group whichadmits a γ -element. Then for every G - C ∗ -algebra B the Baum-Connes assemblymap µ ( G,B ) : K G ∗ ( EG ; B ) → K ∗ ( B ⋊ r G ) is split injective (the same holds true for the full assembly map µ full ( G,B ) ).Proof. Going back to diagram (4.1), we see that for proving split injectivity itsuffices to show that the composition of the left vertical arrows of the diagram is theidentity map. So we need to check that the map · ⊗ γ : K G ∗ ( EG ; B ) → K G ∗ ( EG ; B ),which is given on the level of any G -compact subset X ⊆ EG by the map KK G ∗ ( C ( X ) , B ) → KK G ∗ ( C ( X ) , B ); x x ⊗ B (1 B ⊗ γ ) , is the identity on KK G ∗ ( C ( X ) , B ). But by commutativity of the Kasparov productover C we get x ⊗ (1 B ⊗ γ ) = x ⊗ C γ = γ ⊗ C x = ( γ ⊗ C ( X ) ) ⊗ C ( X ) x = 1 C ( X ) ⊗ C ( X ) x = x. (cid:3) The above proof relies heavily on Theorem 4.12, which in turn relies on Theorem4.14. In the course of proving that theorem in [CE01a] the authors made heavyuse of Kasparov’s result that all almost connected groups have a γ -element andthat this implies (without using BC for proper coefficients) that the Baum-Connesassembly map is injective whenever G has a γ -element in the stronger sense ofKasparov. Remark 4.26.
It is shown by Kasparov in [Kas95, Kas88] that for a discrete group G the rational injectivity (i.e., injectivity after tensoring both sides with Q ) of theassembly map µ G : K G ∗ ( EG ; C ) → K ∗ ( C ∗ r ( G ))implies the famous Novikov-conjecture from topology. We do not want to discussthis conjecture here (e.g., see [Kas88] for the formulation), but we want to mentionthat Theorem 4.25 shows that every group which admits a γ -element also satisfiesthe Novikov conjecture. This fact leads to the following notation: A group G issaid to satisfy the strong Novikov conjecture with coefficients , if the assembly map µ ( G,B ) is injective for every G - C ∗ -algebras B . Remark 4.27. If G has a γ -element in the sense of Definition 4.24 and if B is anygiven G - C ∗ -algebra, then the assembly map µ ( G,B ) : K G ∗ ( EG ; B ) → K ∗ ( B ⋊ r G ) issurjective if and only if the map F γ : K ∗ ( B ⋊ r G ) → K ∗ ( B ⋊ r G ); x x ⊗ B ⋊ r G J G (1 B ⊗ γ )coincides with the identity map. This follows easily from the proof of Theorem4.25 together with diagram (4.1). Indeed, more generally, we may conclude fromthe lower square of diagram (4.1) that every element in the image of F γ lies in theimage of the assembly map, and then the outer rectangle of (4.1) implies that weactually have µ ( G,B ) ( K G ∗ ( EG ; B )) = F γ ( K ∗ ( B ⋊ r G )) . Moreover, it follows also from (4.1) that F γ is idempotent, so it is surjective if andonly if it is the identity. Kasparov calls F γ ( K ∗ ( B ⋊ r G )) the γ -part of K ∗ ( B ⋊ r G ).So one strategy for proving the Baum-Connes conjecture for given coefficients isto show that F γ is the identity on K ∗ ( B ⋊ r G ). This method has been used quiteeffectively by Lafforgue in [Laf02] for proving the Baum-Connes conjecture withtrivial coefficients for a large class of groups (including all real or p -adic reductive IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 41 linear groups). For doing this he first introduced a Banach version of KK -theoryin oder to show that the γ -element induces an isomorphism in K -theory of certainBanach algebras S ( G ), which can be viewed as algebras of Schwartz-functions,which admit an embedding as dense subalgebras of C ∗ r ( G ) such that the inclusion S ( G ) ֒ → C ∗ r ( G ) induces an isomorphism in K -theory. As a result, the map F γ isthe identity on K ∗ ( C ∗ r ( G )) which proves BC.Extending his methods, Lafforgue later showed that all Gromov-hyperbolic groupssatisfy the Baum-Connes conjecture with coefficients (see [Laf12, Pus14]).We close this section with a short argument of how Connes’s Thom isomorphismfor the K -theory of crossed products by R and the Pimsner-Voiculescu sequencefor the K -theory of crossed products by Z can be deduced quite easily from theDirac-dual Dirac method for actions of R as worked out in Example 4.21. Corollary 4.28 (Connes’s Thom isomorphism) . Let α : R → Aut( A ) be an actionof R on the C ∗ -algebra A . Then the crossed product A ⋊ α R is KK -equivalent to A ˆ ⊗ Cl . In particular, there is a canonical isomorphism K ∗ ( A ⋊ α R ) ∼ = K ∗ ( A ˆ ⊗ Cl ) = K ∗ +1 ( A ) . Proof.
To construct the KK -equivalence, let β ∈ KK R ( C , C ( R ) ⊗ Cl ) be asin Example 4.21. Then 1 A ⊗ β ∈ KK R ( A, A ˆ ⊗ C ( R ) ˆ ⊗ Cl ) is an R -equivariant KK -equivalence between A and A ˆ ⊗ C ( R ) ˆ ⊗ Cl and its descent J R (1 A ⊗ β ) ∈ KK ( A ⋊ α R , ( A ˆ ⊗ C ( R ) ˆ ⊗ Cl ) ⋊ α ⊗ τ ⊗ id Cl R ) is a KK -equivalence as well. But( A ˆ ⊗ C ( R ) ˆ ⊗ Cl ) ⋊ α ⊗ τ ⊗ id Cl R is isomorphic to A ˆ ⊗K ( L ( R )) ˆ ⊗ Cl by an applica-tion of [Ech17, Lemma 2.4.1] and [Ech17, Example 2.6.6] (2). This finishes theproof. (cid:3) Theorem 4.29 (Pimsner-Voiculescu) . Let α be a fixed automorphism of the C ∗ -algebra A and let n α n be the corresponding action of Z on A . Then there is asix-term exact sequence K ( A ) id − α ∗ −−−−→ K ( A ) ι ∗ −−−−→ K ( A ⋊ α Z ) ∂ x y ∂ K ( A ⋊ α Z ) ←−−−− ι ∗ K ( A ) ←−−−− id − α ∗ K ( A ) where ι : A → A ⋊ α Z denotes the canonical inclusion.Scetch of proof. By Green’s theorem [Ech17, Theorem 2.6.4] the crossed product A ⋊ α Z is Morita equivalent to the crossed product Ind RZ A ⋊ Ind α R where theinduced algebra Ind RZ A is isomorphic to the mapping cone C α ( A ) = { f : [0 , → A : f (0) = α ( f (1)) } . Thus, by Connes’s Thom isomophism, we get K ∗ ( A ⋊ α Z ) ∼ = K ∗ ( C α ( A ) ⋊ R ) ∼ = K ∗ +1 ( C α ( A )) . The mapping cone C α ( A ) fits into a canonical short exact sequence0 → C (0 , ⊗ A → C α ( A ) → A → , where the quotient map is given by evaluation at 1, say. This gives the six-termexact sequence K ( C (0 , ⊗ A ) −−−−→ K ( C α ( A )) −−−−→ K ( A ) x y K ( A ) ←−−−− K ( C α ( A )) ←−−−− K ( C (0 , ⊗ A ) . Using K ∗ ( C (0 , ⊗ A ) ∼ = K ∗ +1 ( A ) and K ∗ +1 ( C α ( A )) ∼ = K ∗ ( A ⋊ α Z ) this turns intothe six-term sequence of the theorem. (However, it is not completely trivial to checkthat the maps in the sequence coincide with the ones given in the theorem). (cid:3) The above method of proof of the Pimsner-Voiculescu theorem is taken from Black-adar’s book [Bla86]. The original proof of Pimsner and Voiculescu in [PV80] wasindependent of Connes’s Thom isomorphism and used a certain Toeplitz extensionof A ⋊ α Z .4.4. The Baum-Connes conjecture for group extensions.
Suppose that N isa closed normal subgroup of the second countable locally compact group G . Then,if A is a G - C ∗ -algebra, we would like to relate the Baum-Connes conjecture for G to the Baum-Connes conjecture for N and G/N . In order to do so, we first need towrite the crossed product A ⋊ r G as an iterated crossed product ( A ⋊ r N ) ⋊ r G/N for a suitable action of
G/N on A ⋊ r N . Unfortunately, this is not possible ingeneral if we restrict ourselves to ordinary actions, but it can be done by usingtwisted actions as discussed in [Ech17, Section 2.8]. In what follows we shall simplywrite A ⋊ r G/N for the reduced crossed product of a twisted action of the pair(
G, N ) in the sense of Green. We then get the desired isomorphism A ⋊ r G ∼ = ( A ⋊ r N ) ⋊ r G/N (and similarly for the full crossed products). Recall that by [Ech17, Theorem2.8.9] every Green-twisted (
G, N )-action is equivariantly Morita equivalent to anordinary action of
G/N , which allows to cheaply extend many results known forordinary crossed products to the twisted case. In [CE01b] the authors extended theBaum-Connes assembly map to the category of twisted (
G, N )-actions, and theyconstructed a partial assembly map(4.3) µ ( G,N )( N,B ) : K G ∗ ( EG ; B ) → K G/N ∗ ( E ( G/N ) , B ⋊ r N )such that the following diagram commutes K G ∗ ( EG ; B ) µ ( G,B ) −−−−→ K ∗ ( B ⋊ r G ) µ ( G,N )( N,B ) y y ∼ = K G/N ∗ ( E ( G/N ) , B ⋊ r N ) −−−−−−−−−→ µ ( G/N,B ⋊ rN ) K ∗ (( B ⋊ r N ) ⋊ r G/N )As a consequence, if the partial assembly map (4.3) is an isomorphism, then G satisfies BC for B if and only if G/N satisfies BC for B ⋊ r N . Using these ideas,the following result has been shown in [CEOO04, Theorem 2.1] extending someearlier results of [CE01b, CE01a, Oyo01]: IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 43 Theorem 4.30.
Suppose that N is a closed normal subgroup of the second countablelocally compact group G and let B be a G - C ∗ -algebra. Assume further, that thefollowing condition (A) holds:(A) Every closed subgroup L ⊆ G such that N ⊆ L and L/N is compact satisfiesthe Baum-Connes conjecture for B .Then G satisfies BC for G if and only if G/N satisfies BC for B ⋊ r N . Of course, the idea is that one should show that condition (A) implies that thepartial assembly map (4.3) is an isomorphism. This has been the approach in[CE01b,CE01a], but in [CEOO04] a slightly different version of the partial assemblymap has been used instead. Since every compact extension N ⊆ L of an a- T -menable group N is a- T -menable (see [CCJ + T -menablegroup satisfies the Baum-Connes conjecture with coefficients, we get the followingcorollary: Corollary 4.31.
Suppose that N is a closed normal subgroup of the second count-able locally compact group G such that N is a- T -menable. Suppose further that G is any G - C ∗ -algebra. Then G satisfies BC for B if and only if G/N satisfies BCfor B ⋊ r N . In particular, if → N → G → G/N → is a short exact sequence of second countable groups such that G and G/N are botha- T -menable, then G satisfies the Baum-Connes conjecture with coeficients. Note that it is not true in general that G is a- T -menable if N and G/N are a- T -menable. For counter examples see [CCJ + G is any almost connected group, thenone can use structure theory for such groups to see that there exists an amenablenormal subgroup of N of G such that G/N is a reductive Lie-group. Since amenablegroups are also a- T -menable, we can apply Corollary 4.31 to see that G satisfiesBC with trivial coefficients if and only G/N satisfies BC with coefficient C ∗ r ( N ).Now by Lafforgue’s results we know that the reductive group G/N satisfies BCwith trivial coefficients and we somehow need to find good arguments which give usBC for the coefficient algebra C ∗ r ( N ) instead. It is this point where the argumentsbecome quite complicated and we refer to [CEN03] for the details of the proof.5. The Going-Down (or restriction) principle and applications
The Going-Down principle.
In this section we want to discuss an appli-cation of the Baum-Connes conjecture which helps, among other things, to giveexplicit K -theory computations in some interesting cases. Assume we have two G - C ∗ -algebras A and B and a G -equivariant ∗ -homomorphism φ : A → B . Thismap descents to a map φ ⋊ r G : A ⋊ r G → B ⋊ r G. Suppose we want to prove that this map induces an isomorphism in K -theory. If G satisfies the Baum-Connes conjecture for A and B , this problem is equivalent tothe problem that the map φ ∗ : K G ∗ ( EG ; A ) → K G ∗ ( EG ; B )is an isomorphism (use Exercise 4.10). The restriction (or Going-Down) principleallows us to deduce the isomorphism on the level of topological K -theory from thebehaviour on compact subgroups of G . Let us state the theorem: Theorem 5.1 (Going-Down principle) . Suppose that G is a second countable locallycompact group, A and B are G - C ∗ -algebras, and x ∈ KK G ( A, B ) such that for allcompact subgroups K ⊆ G the class res GK ( x ) ∈ KK K ( A, B ) induces an isomorphism · ⊗ A res GK ( x ) : KK K ∗ ( C , A ) ∼ = → KK K ∗ ( C , B ) . Then the map · ⊗ A x : K G ∗ ( EG ; A ) → K G ∗ ( EG ; B ) , which is given on the level of KK G ∗ ( C ( X ) , A ) for a G -compact X ⊆ EG by Kas-parov product with x , is an isomorphism. As a consequence, if G satisfies theBaum-Connes conjecture for A and B , then the class x induces an isomorphism · ⊗ A ⋊ r G J G ( x ) : K ∗ ( A ⋊ r G ) ∼ = −→ K ∗ ( B ⋊ r G ) . Remark 5.2.
There are many interesting groups G for which the trivial subgroup isthe only compact subgroup (e.g., G = R n , Z n or the free group F n in n generators).For those groups the condition on compact subgroups in the theorem reduces tothe single condition that · ⊗ A x : K ∗ ( A ) → K ∗ ( B )is an isomorphism. In many applications, this condition comes for free. Remark 5.3.
Instead of asking that · ⊗ A res GK ( x ) : KK K ∗ ( C , A ) ∼ = → KK K ∗ ( C , B ) isan isomorphism for all compact subgroups K of G , we could alternatively requirethat · ⊗ A ⋊ K J K (res GK ( x )) : K ∗ ( A ⋊ K ) → K ∗ ( B ⋊ K )is an isomorphism for all such K . This follows from the commutativity of thediagram KK K ∗ ( C , A ) ·⊗ A res GK ( x ) −−−−−−−→ KK K ∗ ( C , B ) µ ( K,A ) y y µ ( K,B ) K ∗ ( A ⋊ K ) −−−−−−−−−−−−−→ ·⊗ A ⋊ K J K (res GK ( x )) K ∗ ( B ⋊ K ) , in which the vertical arrows are the isomorphisms of the Green-Julg theorem (seeExample 4.8).The proof of Theorem 5.1 in the above version is given in [ELPW10, Proposition1.6.], but it relies very heavily on a more general Going-Down principle obtainedby Chabert, Echterhoff and Oyono-Oyono in [CEOO04, Theorem 1.5]. In thatpaper we also show how Theorem 4.30 on the Baum-Connes conjecture for groupextensions can be obtained as a consequence of the (more general) Going-Downprinciple. In what follows below we shall present the proof in the case where G IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 45 is discrete. In this case the proof becomes much easier, but still reveals the basicideas. Note that most of the relevant details for the discrete case first appeared (inthe language of E -theory) in [GHT00, Chapter 12].If G is discrete, then the first observation is that each G -compact proper G -spacemaps continuously and G -equivariantly into the geometric realisation of a G -finite G -simplicial complex. For this let F ⊆ G be any finite subset of G which containsthe identity of G . We then define M F = (cid:8) f ∈ C + c ( G ) : X g ∈ G f ( g ) = 1 and ∀ g, h ∈ supp( f ) : g − h ∈ F (cid:9) . Then M F is the geometric realisation of a locally finite simplicial complex withvertices { g : g ∈ G } and { g , . . . , g n } is an n -simplex if and only if g i = g j for i = j and g − i g j ∈ F for all 1 ≤ i, j ≤ n . It follows directly from the definition that forany simplex { g , . . . , g n } we have g − { g , . . . , g n } ⊆ F , hence M F is G -finite in thesense that there exists a finite set S of simplices such that every other simplex is atranslate of one in S . Note that this implies that for all f ∈ M F the formula(5.1) 1 = X g ∈ G f ( g ) = X g ∈ G g · f ( e )holds. Note also that if F ⊆ F ′ for some finite set F ′ ⊆ G , then there is a canonicalinclusion M F ⊆ M F ′ . With this we get: Lemma 5.4.
Suppose that G is a discrete group and let X be a G -compact proper G -space. Suppose further that c : X → [0 , is a cut-off function for X as in Section4.2. Then there exists a finite subset F ⊆ G such that g (supp( c )) ∩ supp( c ) = ∅ forall g / ∈ F and a continuous G -map ϕ c : X → M F ; ϕ c ( x ) = [ g c ( g − x )] . Moreover, for any other continuous G -map ψ : X → M F there is a finite set F ′ ⊆ G containing F such that ψ is G -homotopic to ϕ in M F ′ .Proof. Existence of the finite set F as in the lemma follows from compactness ofthe set { ( g, x ) : ( gx, x ) ∈ supp( c ) × supp( c ) } ⊆ G × X . It is compact since G acts properly on X . It is then straightforward to check that ϕ c is a continuous G -map. Suppose now that ψ : X → M F is any other continuous G -map. Wedefine ˜ c : X → [0 ,
1] as ˜ c ( x ) := p ψ ( x )( e ). It follows then from (5.1) that ˜ c is acut-off function as well and that ψ = ϕ ˜ c . Now let d : X × [0 , → [0 ,
1] be given by d ( x, t ) := p (1 − t ) c ( x ) + t ˜ c ( x ). Then there exists a finite set F ′ ⊇ F such that g (supp( d )) ∩ supp( d ) = ∅ for all g / ∈ F ′ . The continuous G -map ϕ d : X × [0 , →M F ′ then evaluates to ϕ c at t = 0 (using M F ⊆ M F ′ ) and to ϕ ˜ c = ψ at t = 1. (cid:3) Lemma 5.5.
Let G be a discrete group. Then for every G - C ∗ -algebra A we have K G ∗ ( EG ; A ) = lim F KK G ∗ ( C ( M F ) , A ) , where F runs through all finite subsets of G and the limit is taken with respect tothe canonical inclusion M F ⊆ M F ′ if F ⊆ F ′ .Proof. This follows from the definition K G ∗ ( EG ; A ) = lim X KK G ∗ ( C ( X ) , A ) , where X runs through the G -compact subsets of EG and Lemma 5.4: By theuniversal property of EG there are G -compact subsets X F ⊆ EG and G -continuousmaps M F → X F ⊆ EG , which, up to a possible enlargement of X F , are uniqueup to G -homotopy. On the other hand, Lemma 5.4 provides maps X F → M F ′ forsome F ′ ⊇ F which, up to passing to a bigger set F ′′ if necessary, is also unique upto G -homotopy. Thus we get a zigzag diagram KK G ∗ ( C ( M F ) , A ) (cid:15) (cid:15) / / KK G ∗ ( C ( M F ′ ) , A ) (cid:15) (cid:15) / / KK G ∗ ( C ( M F ′′ ) , A ) .....KK G ∗ ( C ( X ) , A ) ♠♠♠♠♠♠♠♠♠♠♠♠♠ / / KK G ∗ ( C ( X F ) , A ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ / / KK G ∗ ( C ( X F ′ ) , A ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ / / ..... which commutes sufficiently well to induce an isomorphism of the inductive limits. (cid:3) The next lemma gives the crucial point in the proof Theorem 5.1 in case of discrete G . It has first been shown in the setting of E -theory in [GHT00, Lemma 12.11]. Amore general version for arbitrary open subgroups H of a second countable locallycompact group G has been shown in [CE01a, Proposition 5.14]. Lemma 5.6.
Suppose that K ⊆ G is a finite subgroup of the discrete group G .Then, for every G - C ∗ -algebra B , there is a well defined compression isomorphismcomp K : KK G ∗ ( C ( G/K ) , B ) → KK K ∗ ( C , B ) given as the composition of the maps KK G ∗ ( C ( G/K ) , B ) res GK −→ KK K ∗ ( C ( G/K ) , B ) ι ∗ −→ KK K ∗ ( C , B ) , where ι : C ֒ → C ( G/K ) denotes the inclusion λ λδ eK with δ eK the characteristicfunction of the open one-point set { eK } ⊆ G/K .Proof.
We construct an inverseind GK : KK K ∗ ( C , B ) → KK G ∗ ( C ( G/K ) , B )for the compression map. We may restrict ourselves to the case of the K -groups,the K -case then follows from passing from B to B ⊗ C ( R ). Let ( E , , γ, T ) be arepresentative for a class x ∈ KK K ( C , B ) where T ∈ L ( E ) is a K -invariant operatorsuch that T ∗ − T, T − ∈ K ( E ). We then define a Hilbert B -module ind GK E asInd GK E = (cid:26) ξ : G → E : s.t. ξ ( gk ) = γ k − ( ξ ( g )) for all g ∈ G, k ∈ K and P g ∈ G β g ( h ξ ( g ) , ξ ( g ) i B ) < ∞ (cid:27) where P g ∈ G β g ( h ξ ( g ) , ξ ( g ) i ) < ∞ just means that the sum converges in the norm-topology of B . The grading on Ind GK E is given by the grading of E applied pointwiseto the elements of Ind GK E . We define the B -valued inner product and the right B -action on Ind GK E by h ξ, η i B = 1 | K | X g ∈ G β g ( h ξ ( g ) , η ( g ) i B ) and ( ξ · b )( g ) = ξ ( g ) β g − ( b )for all ξ, η ∈ Ind GK E , b ∈ B and g ∈ G . Moreover, we define a ∗ -homomorphism M : C ( G/K ) → L (Ind GK E ); ( M ( f ) ξ )( g ) := f ( gK ) ξ ( g ) IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 47 and an operator ˜ T ∈ L (Ind GK E ) by ( ˜ T ξ )( g ) = T ξ ( g ). Finally, the G -action Ind γ : G → Aut(Ind GK E ) is given by (Ind γ g ξ )( h ) = ξ ( g − h ) for g, h ∈ G .It is then an easy exercise to check that (Ind GK E , M, Ind γ, ˜ T ) is a G -equivariant C ( G/K ) − B Kasparov cycle such thatcomp K (cid:0) [Ind GK E , M, Ind γ, ˜ T ] (cid:1) = [ E , , γ, T ] . For the converse, averaging over K , we may first assume that for a given class x = [ F , Φ , ν, S ] ∈ KK G ( C ( G/K ) , B ) the operator S is K -invariant and thatΦ : C ( G/K ) → L ( F ) is non-degenerate. Let ˜ S = P gK ∈ G/K Φ( δ gK ) S Φ( δ gK ). Weclaim that ˜ S is a compact perturbation of S , i.e.,( S − ˜ S )Φ( f ) = X gK ∈ G/K ( S − Φ( δ gK ) S )Φ( δ gK ) f ( gK ) ∈ K ( F )for all f ∈ C ( G/K ). For this we first observe that, since [ S, Φ( δ gK )] ∈ K ( F ) for all gK ∈ G/K , each summand lies in K ( F ). Moreover, since f ∈ C ( G/K ), the sumconverges in norm, which proves the claim. Thus, replacing S by ˜ S if necessary, wemay assume that [ S, Φ( f )] = 0 for all f ∈ C ( G/K ). In particular, if p := Φ( δ eK ),it follows that S = pSp + (1 − p ) S (1 − p ).The class comp K ( x ) is represented by the KK -cycle [ F , Φ | C δ eK , ν | K , S ]. For p =Φ( δ eK ), let E := p F , T = pSp and γ : K → Aut( E ) be the restriction of ν | K to the summand E of F . Since S = pSp + (1 − p ) S (1 − p ) and since [(1 − p ) F , Φ | C δ eK , ν | K , (1 − p ) S (1 − p )] is degenerate, we see that comp K ( x ) = [ E , , γ, T ].It is then straightforward to check that U : Ind GK E → F ; U ξ = 1 | K | X g ∈ G ν g ( ξ ( g ))is a an isomorphism of Hilbert- B -modules which induces an isomorphism betweenthe KK -cycles (Ind GK E , M, Ind γ, ˜ T ) and ( F , Φ , ν, S ). This finishes the proof. (cid:3) Suppose now that X is a proper G -space, U ⊆ X is an open G -invariant subset of X , and Y := X r U . Since C ( X ) is nuclear, there exists a completely positivecontractive section Φ : C ( Y ) → C ( X ) for the restriction homomorphism res Y : C ( X ) C ( Y ) : f f | Y . By properness of the action, we may average Φ withthe help of a cut-off function c : X → [0 , ∞ ) to get the G -equivariant completelypositive and contractive sectionΦ G ( ϕ )( x ) := Z G c ( g − x )Φ( ϕ )( x ) dg for ϕ ∈ C ( Y ). It follows then from Theorem 3.28 that for every G -algebra B thereexists a six-term exact sequence KK G ( C ( Y ) , B ) res ∗ Y −−−−→ KK G ( C ( X ) , B ) ι ∗ −−−−→ KK G ( C ( U ) , B ) ∂ x y ∂ KK G ( C ( U ) , B ) ←−−−− ι ∗ KK G ( C ( X ) , B ) ←−−−− res ∗ K KK G ( C ( Y ) , B ) . We are now ready for
Proof of Theorem 5.1 for G discrete. By Lemma 5.5 it suffices to show that forevery (geometric realisation) of a G -finite G -simplicial complex X , the map · ⊗ A x : KK G ∗ ( C ( X ) , A ) → KK G ∗ ( C ( X ) , B )given by taking Kasparov product with the class x ∈ KK G ( A, B ) is an isomor-phism. We do the proof by induction on the dimension of X . Suppose first thatdim( X ) = 0. In that case X is discrete and therefore decomposes into a finite unionof G -orbits X = G ( x ) ˙ ∪ G ( x ) ˙ ∪ · · · ˙ ∪ G ( x l )for suitable elements x , . . . , x l in X . Then we have C ( X ) ∼ = L li =1 C ( G ( x i )) and KK G ∗ ( C ( X ) , A ) = Q li =1 KK G ∗ ( C ( G ( x i )) , A ) (and similarly for KK G ∗ ( C ( X ) , B )).Thus it suffices to show that · ⊗ A x : KK G ∗ ( C ( G ( x i )) , A ) ∼ = → KK ∗ G ( C ( G ( x i )) , B )for all 1 ≤ i ≤ l . But G ( x i ) ∼ = G/G x i as a G -space, where G x i = { g ∈ G : gx i = x i } denotes the stabiliser of x i . By properness, we have G x i finite for all i . We thenget a commutative diagram KK G ∗ ( C ( G/G x i ) , A ) ·⊗ A x −−−−→ KK G ∗ ( C ( G/G x i ) , B ) comp Gxi y y comp
Gxi KK G xi ∗ ( C , A ) ·⊗ A res GGxi ( x ) −−−−−−−−−→ KK G xi ∗ ( C , B )in which all vertical arrows are isomorphisms by Lemma 5.6 and the lower horizontalarrow is an isomorphism by the assumption of the theorem. Hence the upperhorzontal arrow is an isomorphism as well.Suppose now that dim( X ) = n . After performing a baricentric subdivision of X , if necessary, we may assume that the action of G on X satisfies the followingcondition: If ∆ is a simplex in X , then an element g ∈ G either fixes all of ∆ or g · int(∆) ∩ int(∆) = ∅ , where int(∆) denotes the interior of ∆. Now let e X denotethe union of the interiors of all n -dimensional simplices in X . Then X n − := X r e X is an n − G -simplicial complex and by the induction assumption wehave KK G ∗ ( C ( X n − ) , A ) ∼ = KK G ∗ ( C ( X n − ) , B ) via taking Kasparov product with x . We now show that KK G ∗ ( C ( e X ) , A ) ∼ = KK G ∗ ( C ( e X ) , B ) as well. If this is done,then the five-Lemma applied to the diagram KK G ∗− ( C ( e X ) , A ) ∂ −−−−−→ KK G ∗ ( C ( X n − ) , A ) res ∗ −−−−−→ KK G ∗ ( C ( X ) , A ) ·⊗ A x y ·⊗ A x y ·⊗ A x y KK G ∗− ( C ( e X ) , B ) ∂ −−−−−→ KK G ∗ ( C ( X n − ) , B ) res ∗ −−−−−→ KK G ∗ ( C ( X ) , B ) ι ∗ −−−−−→ KK G ∗ ( C ( e X ) , A ) ∂ −−−−−→ KK G ∗ +1 ( C ( X n − ) , A ) y ·⊗ A x y ·⊗ A xι ∗ −−−−−→ KK G ∗ ( C ( e X ) , B ) ∂ −−−−−→ KK G ∗ +1 ( C ( X n − ) , B ) shows that KK G ∗ ( C ( X ) , A ) ∼ = KK ∗ G ( C ( X ) , B ).To see that KK G ∗ ( C ( e X ) , A ) ∼ = KK G ∗ ( C ( e X ) , B ) we first observe that e X is a finiteunion of orbits of open simplices int(∆ ) , . . . , int(∆ k ) for some k ∈ N . Via thecorresponding product decomposition of the KK -groups, we may then assume that IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 49 e X = G · int(∆) for a single open n -simplex ∆. By our assumption on the action of G on X , we have G · int(∆) ∼ = G/G ∆ × int(∆)where G ∆ = { g ∈ G : g · ∆ = ∆ } denotes the (finite!) stabiliser of ∆ and wherethe G -action on G/G ∆ × int(∆) is given by left translation on the first factor. Wethen get a diagram KK G ∗ ( C ( G/G ∆ × int(∆)) , A ) Bott −−−−−→ ∼ = KK G ∗ + n ( C ( G/G ∆ ) , A ) comp G ∆ −−−−−−→ ∼ = KK G ∆ ∗ + n ( C , A ) ·⊗ A x y ·⊗ A x y ·⊗ A res GG ∆ ( x ) y KK G ∗ ( C ( G/G ∆ × int(∆)) , B ) Bott −−−−−→ ∼ = KK G ∗ + n ( C ( G/G ∆ ) , B ) comp G ∆ −−−−−−→ ∼ = KK G ∆ ∗ + n ( C , B ) . Since, by assumption, the last vertical arrow is an isomorphism, the result follows. (cid:3)
We should note that for groups which satisfy the strong Baum-Connes conjecturein the sense of Definition 4.17 a stronger version of Theorem 5.1 has been shownby Meyer and Nest in [MN06, Theorem 9.3]:
Theorem 5.7 (Meyer-Nest) . Suppose that the second countable group G satisfiesthe strong Baum-Connes conjecture (e.g., this is satisfied if G is a- T -menable or,in particular, if G is amenable) and assume that x ∈ KK G ( A, B ) such that forevery compact subgroup K of G the class J K (res GK ( x )) ∈ KK ( A ⋊ K, B ⋊ K ) is a KK -equivalence. Then J G ( x ) ∈ KK ( A ⋊ r G, B ⋊ r G ) is KK -equivalence as well. Applications of the Going-Down principle.
We are now going to give anumber of applications. The first one is the proof that every exact locally compactgroup satisfies the strong Novikov conjecture. Recall that a locally compact groupis called exact (in the sense of Kirchberg and Wassermann) if for every short exactsequence of G - C ∗ -algebras 0 → I ι → A q → A/I → → I ⋊ r G ι ⋊ r G → A ⋊ r G q ⋊ r G → A/I ⋊ r G → X . This means that the transformation groupoid X ⋊ G is topologically amenable in the sense of [ADR00]. Very recently the result ofOzawa has been generalised by Brodzki, Cave and Li ([BCL16]) to second countablelocally compact groups. The following result has been shown first for discrete G byHigson in [Hig00]. The result has been extended in [CEOO04] to the case of secondcountable locally compact groups acting amenably on a compact space. Theorem 5.8.
Let G be an exact second countable locally compact group. Then G satisfies the strong Novikov conjecture with coefficients, i.e., for each G - C ∗ -algebra B the assembly map µ ( G,B ) : K G ∗ ( EG ; B ) → K ∗ ( B ⋊ r G ) is split injective. A similar statement holds true for the full assembly map µ full ( G,B ) . Proof.
By [BCL16], being exact is equivalent to the condition that there exists acompact amenable G -space X . Following the arguments given by Higson in [Hig00,Lemma 3.5 and Lemma 3.6] we may as well assume that X is a metrisable convexspace and G acts by affine transformations. In particular, X is K -equivariantlycontractible for every compact subgroup K of X . It then follows that the inclusionmap ι : C → C ( X ) is a KK K -equivalence for every compact subgroup K of G – it’s inverse is given by the map C ( X ) → C ; f f ( x ) for any K -fixed point x ∈ X . It then follows, that for every G - C ∗ -algebra B , the ∗ -homomorphism B → B ⊗ C ( X ); b b ⊗ X is a KK K -equivalence as well. Thus it follows fromTheorem 5.1 that Φ ∗ : K G ∗ ( EG ; B ) → K G ∗ ( EG ; B ⊗ C ( X ))is an isomorphism. Moreover, by Tu’s extension of the Higson-Kasparov theoremto groupoids (see [Tu99a]), the assembly map µ ( X ⋊ G,B ⊗ C ( X )) : K X ⋊ G ∗ ( E ( X ⋊ G ) , B ⊗ C ( X )) → K ∗ (( B ⊗ C ( X )) ⋊ r G )is an isomorphism, since X ⋊ G is an amenable, and hence a- T -menable groupoid.Moreover, it has been shown in [CEOO03] that the forgetful map F : K X ⋊ G ∗ ( E ( X ⋊ G ) , B ⊗ C ( X )) → K G ∗ ( EG, B ⊗ C ( X ))is an isomorphism and that the diagram K X ⋊ G ∗ ( E ( X ⋊ G ) , B ⊗ C ( X )) µ ( X ⋊ G,B ⊗ C ( X )) −−−−−−−−−−→ K ∗ (( B ⊗ C ( X )) ⋊ r G ) F y y = K G ∗ ( EG, B ⊗ C ( X )) µ ( G,B ⊗ C ( X )) −−−−−−−−→ K ∗ (( B ⊗ C ( X )) ⋊ r G )commutes. The result then follows from the commutative diagram K G ∗ ( EG, B ) µ ( G,B ⊗ C ( X )) −−−−−−−−→ K ∗ ( B ⋊ r G ) Φ ∗ y y Φ ⋊ r G ∗ K G ∗ ( EG, B ⊗ C ( X )) µ ( G,B ⊗ C ( X )) −−−−−−−−→ K ∗ (( B ⊗ C ( X )) ⋊ r G )in which the left vertical arrow and the bottom horizontal arrow are isomorphisms. (cid:3) The main application of the Going-Down principle in [CEOO04] was the proof ofa version of the K¨unneth formula for K G ∗ ( EG, B ) with applications for the Baum-Connes conjecture with trivial coefficients. We don’t want to go into the detailshere. But we would like to mention some other useful applications. By a homotopybetween two actions α , α : G → Aut( A ) we understand a path of actions α t : G → Aut( A ), t ∈ [0 , (cid:0) α g ( f ) (cid:1) ( t ) := α tg ( f ( t )) ∀ f ∈ A [0 , , g ∈ G, t ∈ [0 , A [0 ,
1] = C ([0 , , A ). The following is of course a directconsequence of Theorem 5.1: Corollary 5.9.
Suppose that α : G → Aut( A [0 , is a homotopy between theactions α , α : G → Aut( A ) and assume that G satisfies BC for ( A [0 , , α ) and ( A, α t ) for t = 0 , . Suppose further that for t = 0 , and for every compact subgroup IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 51 K of G the evaluation map ǫ t : A [0 , → A ; f f ( t ) induces an isomorphism ǫ t ⋊ K ∗ : K ∗ ( A [0 , ⋊ α K ) ∼ = → K ∗ ( A ⋊ α t K ) . Then ǫ t ⋊ r G ∗ : K ∗ ( C ([0 , , A ) ⋊ α,r G ) → K ∗ ( A ⋊ α t ,r G ) is an isomorphism as well. In particular, we have K ∗ ( A ⋊ α ,r G ) ∼ = K ∗ ( A ⋊ α ,r G ) . Of course, the condition on the compact subgroups in the above corollary is quiteannoying. However, for those groups which have no compact subgroups other thanthe trivial group, the corollary becomes very nice, since the evaluation maps ǫ t : A [0 , → A ; f f ( t ) are always KK -equivalences. Corollary 5.10.
Suppose that α : G → Aut( A [0 , is a homotopy between theactions α , α : G → Aut( A ) and assume that G satisfies BC for ( A [0 , , α ) and ( A, α ) , ( A, α ) . If { e } is the only compact subgroup of G , then K ∗ ( A ⋊ α ,r G ) ∼ = K ∗ ( A ⋊ α ,r G ) . In [ELPW10] Corollary 5.9 has been used to show that for groups G which sat-isfy BC for suitable coefficients, the K -theory of reduced twisted group algebras C ∗ r ( G, ω ), where ω : G × G → T is a Borel 2-cocycle on G , only depends on the ho-motopy class of the 2-cocycle ω (with a suitable definition of homotopy). We don’twant to go into the details here, but we want to mention that if ( ω t ) t ∈ [0 , is such ahomotopy of 2-cocycles, it induces a homotopy α : G → Aut( K [0 , G on the compact operators K = K ( ℓ ( N )) such that K ⋊ α t ,r G ∼ = K ⊗ C ∗ r ( G, ω t )for all t ∈ [0 , K on K must be exterior equivalent to a constant path of actions,hence K [0 , ⋊ K ∼ = ( K ⋊ K )[0 , G , from which it follows that the evaluation maps ǫ t ⋊ K : K [0 , ⋊ K → K ⋊ K are KK -equivalences for all K . Thus, if G satisfies BC for K and K [0 ,
1] (for therelevant actions), it follows from Corollary 5.9 that K ∗ ( C ∗ r ( G, ω )) ∼ = K ∗ ( K ⋊ α ,r G ) ∼ = K ∗ ( K ⋊ α ,r G ) ∼ = K ∗ ( C ∗ r ( G, ω )) . Note that this result extends earlier results of Elliott ([Ell81]) for the case of finitelygenerated abelian groups G and of Packer and Raeburn [PR92] for a class of solvablegroups G . The main application in [ELPW10] was given for the computation ofthe K -theory of the crossed products A θ ⋊ F of the non-commutative 2-tori A θ , θ ∈ [0 ,
1] with finite subgroups F ⊆ SL(2 , Z ) acting canonically on A θ . It turnedout that A θ ⋊ F ∼ = C ∗ r ( Z ⋊ F, ω θ ) for some cocycles ω θ which depend continuouslyon the parameter θ . Since Z ⋊ F is amenable it satisfies strong BC, and then itfollows from the above results that K ∗ ( A θ ⋊ F ) ∼ = K ∗ ( C ∗ r ( Z ⋊ F, ω θ )) ∼ = K ∗ ( C ∗ r ( Z ⋊ F, ω )) = K ∗ ( C ( T ) ⋊ F ) . The last group can be computed by methods from classical topology. We referto [ELPW10] for further details on this. Note that in this situation we can alsouse Theorem 5.7 to deduce that all algebras A θ ⋊ F , θ ∈ [0 , KK -equivalent. Crossed products by actions on totally disconnected spaces.
We nowwant to apply our techniques to certain crossed products of groups G acting “nicely”on totally disconnected spaces Ω. The main application of this will be given forreduced semigroup algebras and crossed products by certain semigroups, which isdescribed in detail in [Cun17] and [Li17].If Ω is a totally disconnected locally compact space we denote by U c (Ω) the collec-tion of all compact open subsets of Ω. This set is countable if and only if Ω has acountable basis of its topology, i.e., Ω is second countable. For any set V ⊆ U c (Ω)we say that V generates U c (Ω), if every subset U ⊆ U c (Ω) which contains V and isclosed under finite intersections, finite unions, and taking differences U r W with U, W ∈ U , must coincide with U c (Ω).Let C ∞ c (Ω) denotes the dense subalgebra of C (Ω) consisting of locally constantfunctions with compact supports on Ω. Then C ∞ c (Ω) = span { U : U ∈ U c (Ω) } , where 1 U denotes the indicator function of U ⊆ Ω. The straight-forward proof ofthe following lemma is given in [CEL13, Lemma 2.2]:
Lemma 5.11.
Suppose that V is a family of compact open subsets of the totallydisconnected locally compact space Ω . Then the following are equivalent:(i) The set { V : V ∈ V} of characteristic functions of the elements in V generates C (Ω) as a C ∗ -algebra.(ii) The set V generates U c (Ω) in the sense explained above.If, in addition, V is closed under finite intersections, then (i) and (ii) are equivalentto (iii) span { V : V ∈ V} = C ∞ c (Ω) . We see in particular, that the commutative C ∗ -algebra C (Ω) is generated as a C ∗ -algebra by a (countable) set of projections. The converse is also true: If D is anycommutative C ∗ -algebra which is generated by a set of projections { e i : i ∈ I } ⊆ D and if Ω = Spec( D ) is the Gelfand spectrum of D , then Ω is totally disconnectedand the sets V = { supp(ˆ e i ) : i ∈ I } , where, for any d ∈ D , ˆ d ∈ C (Ω) denotes the Gelfand transform of d , is a familyof compact open subsets of Ω which generates U c (Ω). For a proof see [CEL13,Lemma 2.3]. Thus there is an equivalence between studying sets of projectionswhich generate D or sets of compact open subsets of Ω which generate U c (Ω). Lemma 5.12.
Suppose that { e i : i ∈ I } is a set of projections in the commutative C ∗ -algebra D . Then for each finite subset F ⊆ I there exists a smallest projection e ∈ D such that e i ≤ e for every i ∈ F . We then write e := ∨ i ∈ F e i .Proof. By the above discussion we may assume that D = C (Ω) for some totallydisconnected space Ω. For each i ∈ F let V i := supp( e i ). Then e = 1 V with V = ∪ li =1 V i . (cid:3) IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 53 The independence condition given in the following definition is central for the resultsof this section:
Definition 5.13.
Suppose that { X i : i ∈ I } is a family of subsets of a set X . Wethen say that { X i : i ∈ I } is independent if for any finite subset F ⊆ I and for anyindex i ∈ I we have X i = ∪ i ∈ F X i ⇒ i ∈ F. Similarly, a family { e i : i ∈ I } of projections in the commutative C ∗ -algebra D iscalled independent if for any finite subset F ⊆ I and every i ∈ I we have e i = ∨ i ∈ F e i ⇒ i ∈ F. Of course, if D = C (Ω) then { e i : i ∈ I } is an independent family of projectionsin D if and only if { supp( e i ) : i ∈ I } is an independent family of compact opensubsets of Ω. The following lemma is [CEL13, Lemma 2.8]. The proof follows from[Li13, Proposition 2.4]: Lemma 5.14.
Suppose that { e i : i ∈ I } is an independent set of projections inthe commutative C ∗ -algebra D which is closed under finite multiplication up to .Then { e i : i ∈ I } is independent if and only if it is linearly independent. Definition 5.15.
Suppose that Ω is a totally disconnected locally compact Haus-dorff space. An independent family V of non-empty compact open subsets of Ω iscalled a regular basis (for the compact open subsets of Ω) if it generates U c (Ω) andif V ∪ {∅} is closed under finite intersections.A family of projections { e i : i ∈ I } in the commutative C ∗ -algebra D is called a regular basis for D , if it is (linearly) independent, closed under finite multiplicationup to 0 and generates D as a C ∗ -algebra.The following lemma is a consequence of the above discussions. We leave the detailsto the reader. Lemma 5.16.
A family of projections { e i : i ∈ I } in the commutative C ∗ -algebra D is a regular basis for D if and only if the set V = { supp ˆ e i : i ∈ I } is a regularbasis for the compact open subsets of Ω = Spec( D ) . Conversely, V is a regularbasis for the compact open subsets of the locally compact space Ω if and only if { V : V ∈ V} is a regular basis for C (Ω) . It is not difficult to see that every totally disconnected locally compact space Ωhas a regular basis for its compact open sets. A formal proof is given in [CEL13,Proposition 2.12]. The following example shows the existence for the Cantor set:
Example 5.17.
Let Ω = { , } Z denote the direct product of copies of { , } over Z equipped with the product topology. For each finite subset F ⊆ Z let V F = { ( ǫ n ) n ∈ Ω : ǫ n = 0 ∀ n ∈ F } be the corresponding cylinder set in Ω. It is then an easy exercise to check that thecollection V = { V F : F ⊆ Z finite } is a regular basis for the compact open subsetsof Ω. Assume now that G is a second countable locally compact group and Ω is a secondcountable totally disconnected G -space such that there exists a G -invariant regularbasis V = { V i : i ∈ I } for the compact open subsets of Ω. For all i ∈ I let e i = 1 V i be the characteristic function of V i . Then, since V is G -invariant, theaction of G on Ω induces an action of G on I . Consider the unitary representation U : G → U ( ℓ ( I )); ( U g ξ )( i ) = ξ ( g − i ) and let Ad U : G → Aut( K ( ℓ ( I ))) denotethe corresponding adjoint action. For each i ∈ I let δ i denote the Dirac functionat i and let d i : ℓ ( I ) → C δ i denote the orthogonal projection. Then there exists aunique G -equivariant ∗ -homomorphismΦ : C ( I ) → C (Ω) ⊗ K ( ℓ ( I )) such that Φ( δ i ) = e i ⊗ d i , for all i ∈ I , where the action of G on C ( I ) is induced by the action on I andthe action on C (Ω) ⊗ K ( ℓ ( I )) is given by the diagonal action τ ⊗ Ad U , where τ : G → Aut( C (Ω)) denotes the given action of G on C (Ω). More generallyif α : G → Aut( A ) is an action of G on a C ∗ -algebra A , then there exists a G -equivariant ∗ -homomorphismΦ A : C ( I ) ⊗ A → C (Ω) ⊗ A ⊗ K ( ℓ ( I )) s.t. Φ A ( δ i ⊗ a ) = e i ⊗ a ⊗ d i . Note that the action τ ⊗ α ⊗ Ad U of G on C (Ω) ⊗ A ⊗K ( ℓ ( I )) is Morita equivalentto the action τ ⊗ α of G on C (Ω) ⊗ A via the G -equivariant equivalence bimodule E := ( C (Ω) ⊗ A ⊗ ℓ ( I ) , τ ⊗ α ⊗ U ). Thus we obtain a KK G -class x = [Φ A ] ⊗ C (Ω) ⊗ A ⊗K E ∈ KK G ( C ( I, A ) , C (Ω , A )) . The following is the main result of this section:
Theorem 5.18 (cf. [CEL13]) . Suppose that { e i : i ∈ I } is a G -equivariant reg-ular basis for C (Ω) , A is any G - C ∗ -algebra, and G satisfies the Baum-Connesconjecture for C ( I, A ) and C (Ω , A ) . Then the descent J G ( x ) ∈ KK ( C ( I, A ) ⋊ r G, C (Ω , A ) ⋊ r G ) of the class x ∈ KK G ( C ( I, A ) , C (Ω , A )) constructed above induces an isomor-phism K ∗ ( C ( I, A ) ⋊ r G ) ∼ = K ∗ ( C (Ω , A ) ⋊ r G ) .If, moreover, G satsfies the strong Baum-Connes conjecture and if A is type I, then J G ( x ) ∈ KK ( C ( I, A ) ⋊ r G, C (Ω , A ) ⋊ r G ) is a KK -equivalence. The above theorem has originally been shown in [CEL13, § K -theorycalculations. The main reason is due to the relatively easy structure of crossedproducts by groups acting on discrete spaces I . If such action is given (as inthe situation of our theorem) and if A is any other G - C ∗ -algebra, we obtain a G -equivariant direct sum decomposition C ( I, A ) ∼ = M [ i ] ∈ G \ I C ( G · i ) ⊗ A, in which G · i = { g · i : g ∈ G } denotes the G -orbit of the representative i of theclass [ i ] ∈ G \ I . Let G i := { g ∈ G : g · i = i } denote the stabiliser of i in G . Then G i is open in G and we have a G -equivariant bijection G/G i ∼ = → G · i ; gG i g · i. IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 55 Moreover, by Green’s imprimitivity theorem ([Ech17, Theorem 2.6.4]; see also[Ech17, Remark 2.6.9]), there are natural Morita equivalences C ( G/G i , A ) ⋊ r G ∼ = A ⋊ r G i . Putting things together, we therefore get C ( I, A ) ⋊ r G ∼ = M [ i ] ∈ G \ I C ( G/G i , A ) ⋊ r G ∼ M M [ i ] ∈ G \ I A ⋊ r G i . Since Morita equivalences are KK -equivalences, we get K ∗ ( C ( I, A ) ⋊ r G ) ∼ = M [ i ] ∈ G \ I K ∗ ( A ⋊ r G i ) . Thus
Corollary 5.19.
Suppose that G, Ω , A and { e i : i ∈ I } are as in Theorem 5.18.Then there is an isomorphism K ∗ ( C (Ω , A ) ⋊ r G ) ∼ = M [ i ] ∈ G \ I K ∗ ( A ⋊ r G i ) . In particular, if A = C , there is an isomorphism K ∗ ( C (Ω) ⋊ r G ) ∼ = M [ i ] ∈ G \ I K ∗ ( C ∗ r ( G i )) . If G satisfies strong BC and A is type I, the isomorphism is induced by a KK -equivalence between C (Ω , A ) ⋊ r G and L [ i ] ∈ G \ I A ⋊ r G i . In many interesting examples coming from the theory of C ∗ -semigroup algebras andcrossed products of semigroups by automorphic actions of semigroups P ⊆ G , thestabilisers for the action of G on I have very easy structure, so that the K -theorygroups of the crossed products A ⋊ r G are computable. This is in particular truein the case A = C . The applications of Theorem 5.18 to C ∗ -semigroup algebras isdiscussed in more detail in [Li17, Section 5.10]) and [Cun17, Section 6.5]. Example 5.20.
To illustrate the usefulness of our approach we considerthe group algebra of the lamplighter group Z / ≀ Z which is the semi-directproduct (cid:0) L Z Z / (cid:1) ⋊ Z , where the action is given via translation of thesummation index. Since the dual group of L Z Z / Q Z { , − } = { , − } Z the group algebra C ∗ ( Z / ≀ Z ) is isomor-phic to C (Ω) ⋊ Z . Moreover, by Example 5.17 there exists a regular basis V = { V F : F ⊆ Z finite } for the compact open subsets of Ω consisting of the cylin-der sets V F = { ( ǫ n ) n ∈ Ω : ǫ n = 1 ∀ n ∈ F } attached to the finite subsets F ⊆ Z .This basis is clearly Z -invariant, hence our theorem applies to the correspondingregular basis { e F = 1 V F : F ⊆ Z finite } of C (Ω). Let F = { F ⊆ Z : F finite } denote the index set of this basis and let F ∗ = F r {∅} . The action of Z on F fixes ∅ and acts freely on F ∗ . Hence, our theorem gives K ∗ ( C ∗ ( Z / ≀ Z )) = K ∗ ( C (Ω) ⋊ Z ) ∼ = K ∗ ( C ( F ) ⋊ Z ) ∼ = K ∗ ( C ∗ ( Z )) M M [ F ] ∈ Z \F ∗ K ∗ ( C ) . Since C ∗ ( Z ) ∼ = C ( T ) and K ( C ( T )) = Z = K ( C ( T )) we get K ( C ∗ ( Z / ≀ Z ) ∼ = M [ F ] ∈ G \F Z and K ( C ∗ ( Z / ≀ Z )) = Z . Of course, the result can easily be extended to more general wreath products Z / ≀ G = L G Z / ⋊ G , where G is a countable discrete group which satisfies appropriateversions of the Baum-Connes conjecture. Proof of Theorem 5.18.
For the sake of presentation, let us assume that G is count-able discrete (which is the case in most applications). Since the (strong) Baum-Connes conjecture is invariant with respect to KK G -equivalent actions, and since G -equivariant Morita equivalences are KK G -equivalences, it suffices to proof thatthe descent Φ A ⋊ r G : C ( I, A ) ⋊ r G → (cid:0) C (Ω , A ) ⊗ K ( ℓ ( I )) (cid:1) ⋊ r G of the homomorphism Φ A induces an isomorphism in K -theory (resp. a KK -equivalence in case where G satisfies strong BC and A is type I). To see that this isthe case we want to exploit the Going-Down principle of the previous section, i.e.,we need to show that for every finite subgroup F ⊆ G , the map(5.2) Φ A ⋊ F : C ( I, A ) ⋊ F → (cid:0) C (Ω , A ) ⊗ K ( ℓ ( I )) (cid:1) ⋊ F induces an isomorphism in K -theory. Note that if A is type I, the same is truefor (cid:0) C (Ω , A ) ⊗ K ( ℓ ( I )) (cid:1) ⋊ F by [Ech17, Corollary 2.8.21] and therefore it followsfrom the Universal-Coefficient-Theorem of KK -Theory (e.g., see [Bla86, Chapter23]) that Φ A ⋊ F being an isomorphism already implies that it is a KK -equivalence.Thus, in this situation, the stronger result that Φ A ⋊ r G is a KK -equivalence willthen follow from the Meyer-Nest Theorem 5.7.Using the Green-Julg theorem, the map Φ A ⋊ F of (5.2) being an isomorphism isequivalent to (Φ A ) ∗ : K F ∗ ( C ( I, A )) → K F ∗ (cid:0) C (Ω , A ) ⊗ K ( ℓ ( I )) (cid:1) being an isomorphism.So in what follows let us fix a finite subgroup F of G . Let J ⊆ I be any finite F -invariant subset such that { e i : i ∈ J } is closed under multiplication (up to 0).Then D J := span { e i : i ∈ J } is a finite dimensional commutative C ∗ -subalgeba of C (Ω) of dimension dim( D J ) = | J | . Consider the map(5.3) Φ J : C ( J ) → D J ⊗ K ( ℓ ( J )); Φ J ( δ i ) = e i ⊗ d i . We want to show that Φ J is invertible in KK F (cid:0) C ( J ) , D J ⊗ K ( ℓ ( J )) (cid:1) ∼ = KK F ( C ( J ) , D J ). If this happens to be true, then Φ A,J := Φ J ⊗ id A : C ( J, A ) → D J ⊗ A ⊗ K ( ℓ ( J )) will be KK F -invertible as well, and the desired result then IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 57 follows from the following commutative diagram K F ∗ ( C ( J, A )) (Φ A,J ) ∗ −−−−−→ ∼ = K F ∗ (cid:0) D J ⊗ A ⊗ K ( ℓ ( J )) (cid:1) ι ∗ y y ι ∗ lim J K F ∗ ( C ( J, A )) lim J (Φ A,J ) ∗ −−−−−−−−→ ∼ = lim J K F ∗ (cid:0) D J ⊗ A ⊗ K ( ℓ ( J )) (cid:1) ∼ = y y ∼ = K F ∗ ( C ( I, A )) (Φ A ) ∗ −−−−→ ∼ = K F ∗ (cid:0) C (Ω) ⊗ A ⊗ K ( ℓ ( I )) (cid:1) The KK F -invertibility of Φ J in (5.3) will be a consequence of a UCT-type result forfinite dimensional F -algebras, which we are now going to explain: Suppose that C and D are commutative finite dimensional F -algebras with dim( C ) = n, dim( D ) = m (in our application, C will be C ( J ), D = D J , and n = m = | J | ). Let { c , . . . , c n } and { d , . . . , d m } be choices of pairwise orthogonal projections, which then form abasis of C and D , respectively. Then we have isomorphisms Z n ∼ = K ( C ) and Z m ∼ = K ( D ) sending the j th unit vector e j to [ c j ] (resp. [ d j ]). If we ignore the F -action, the UCT-theorem for KK implies that(5.4) KK ( C, D ) ∼ = Hom( K ( C ) , K ( D )) ∼ = M ( m × n, Z ) , where the first isomorphism is given by sending x ∈ KK ( C, D ) to the map · ⊗ C x : K ( C ) → K ( D ) and the second map is given via the above identifications of K ( C ) ∼ = Z n and K ( D ) ∼ = Z m . Suppose now that C and D are F -algebras suchthat F acts via permutations of the basis elements in { c , . . . , c n } and { d , . . . , d m } ,respectively. Note that the actions of F on C and D are determined by two homo-morphisms τ : F → S n and σ : F → S m such that g · c i = c τ g ( i ) and g · d j = d σ g ( j ) for all i, j . Then the equivariant version of (5.4) does not give an isomorphism ingeneral, but we get a homomorphism(5.5) Ψ C,D : KK F ( C, D ) → Hom F ( K ( C ) , K ( D )) ∼ = M F ( m × n, Z ) : x Γ x where Hom F ( K ( C ) , K ( D )) denotes the F -equivariant homomorphisms (with F acting on the basis elements [ c i ] and [ d i ] of K ( C ) and K ( D ), repectively) and M F ( m × n, Z ) denotes the set of all m × n -matrices Γ = ( γ ij ) over Z which satisfy(5.6) γ ij = γ τ g ( i ) ,σ g ( j ) ∀ g ∈ F. We need to construct a section M F ( m × n, Z ) → KK F ( C, D ); Γ x Γ for Ψ C,D which is compatible with taking Kasparov-products. For this let Γ = ( γ ij ) ∈ M F ( m × n, Z ) be given. Let E ij = C | γ ij | ⊗ C d i viewed as a Hilbert D -modulein the canonical way. Let ϕ ij : C → K ( E ij ) be the ∗ -homomorphism such that ϕ ij ( c j ) = 1 E ij and ϕ ij ( c k ) = 0 for all k = i . Let E +Γ = M γ ij > E ij and ϕ + = M γ ij > ϕ ij : C → K ( E + ) , and, similarly, E − Γ = M γ ij < E ij and ϕ − = M γ ij < ϕ ij : C → K ( E − ) . Because of (5.6) there are canonical actions of F on E + , E − such that g · E ij = E τ g ( i ) σ g ( j ) for all i, j and such that ( E +Γ , ϕ + ) and ( E − Γ , ϕ − ) become F -equivariant C − D correspondences. Finally, let E Γ = E +Γ L E − Γ with Z / (cid:18) − (cid:19) and let ϕ = (cid:18) ϕ + ϕ − (cid:19) . Since ϕ takes value in K ( E ), we get aclass x Γ := [ E Γ , ϕ, ∈ KK F ( C, D ) . The proof of the following lemma is left as an exercise for the reader (or see [CEL13,Lemma A.2]):
Lemma 5.21.
Suppose that
B, C, D are finite dimensional commutative F -algebrassuch that { b , . . . , b k } , { c , . . . , c n } , and { d , . . . , d m } are F -invariant bases con-sisting of orthogonal projections in B, C, D , respectively. Then, for all matrices Λ ∈ M F ( n × k, Z ) and Γ ∈ M F ( m × n, Z ) we get x Λ ⊗ C x Γ = x Γ · Λ ∈ KK F ( B, D ) . In particular, if n = dim( C ) = dim( D ) and Γ ∈ M F ( n × n, Z ) is invertible over Z ,then x Γ ∈ KK F ( C, D ) is invertible as well. Let us come back to the class Φ J : C ( J ) → D J ⊗ K ( ℓ ( J )) which sends δ i to e i ⊗ d i , where d i denotes the orthogonal projection onto C δ i . The following lemmagives the crucial point of how independence of the family { e i : i ∈ I } of the basiselements of C (Ω) enters the picture: Lemma 5.22 (cf [CEL13, Lemma 3.8]) . Let D be a commutative C ∗ -algebra gen-erated by a multplicatively closed (up to ) and independent finite set of projections { e i : i ∈ J } . For each i ∈ J let e ′ i := e i − ∨ e j 0] as constructed in the above lemma with respect to the basis { δ i : i ∈ J } of C ( J ) and the basis { e ′ i : i ∈ J } of D J . A combination ofLemma 5.22 with Lemma 5.21 then implies that [Φ J ] is invertible. Indeed, since IVARIANT KK -THEORY AND THE BAUM-CONNES CONJECURE 59 γ ij = 0 or 1, it follows that E Γ = E +Γ = L i ∈ J (cid:0) L j ∈ J,γ ij =1 C e ′ i (cid:1) embeds as a directsummand into D J ⊗ ℓ ( J ) such that Φ J : C ( J ) → D J ⊗ K ( ℓ ( J )) ∼ = K ( D J ⊗ ℓ ( J ))decomposes as ϕ L (cid:3) As remarked before, the main applications for Theorem 5.18 are given in case ofcomputing the K -theory of reduced semigroup algebras C ∗ λ ( P ), where e ∈ P ⊆ G isa sub-semigroup of the countable group G . In case where P ⊆ G satisfies a certainToeplitz condition (which is discussed in detail in [Li17]), there exists a totallydisconnected G -space Ω P ⊆ G such that C ∗ λ ( P ) can be realised as a full corner in thecrossed product C (Ω P ⊆ G ) ⋊ r G , hence K ∗ ( C ∗ λ ( P )) ∼ = K ∗ ( C (Ω P ⊆ G ) ⋊ r G ). Now,the existence of a G -invariant regular basis for C (Ω P ⊆ G ) will follow from a certainindependence condition for the inclusion P ⊆ G , which, somewhat surprisingly, issatisfied in a large number of interesting cases. Again, we refer to the [Li17] and[Cun17] for more details on this.Unfortunately, a G -invariant regular basis { e i : i ∈ I } for C (Ω), as required forthe proof of Theorem 5.18, does not exist in general. In fact, we have the followingresult, which excludes a large number of interesting cases from our theory: Proposition 5.23 ([CEL13, Proposition 3.18]) . 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