The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups

THE CONNES-KASPAROV CONJECTURE FOR ALMOST CONNECTED GROUPS AND LINEAR p-ADIC GROUPS JÉRÔME CHABERT, SIEGFRIED ECHTERHOFF, AND RYSZARD NEST Abstract. Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C ∗ -algebra of G is an isomorphism. The same is shown for the groups of k-rational points of any linear algebraic group over a local field k of characteristic zero. Contents 1. Introduction and statement of results 2. Some preliminaries and first reductions 2.1. Reduction of the proof of Theorem 1.1 3. Baum-Connes for continuous fields of C ∗ -algebras 4. The semi-simple case 5. The general case of almost connected groups 6. The p-adic case 7. Relations to the K-theory of the maximal compact subgroup References 1 4 8 9 14 19 27 29 32 1. Introduction and statement of results In this paper we give a proof of the Baum-Connes conjecture for almost connected groups and for linear algebraic groups over local fields of characteristic zero. To be more precise, we prove the following theorem: Theorem 1.1. Let G be a second countable almost connected group (i.e., G/G0 is compact, where G0 denotes the connected component of G) or the group of k-rational points of a linear algebraic group over a local field k of characteristic zero (i.e., k = R, C or a finite extension of the p-adic numbers Qp ). Then G satisfies the Baum-Connes conjecture with trivial coefficients C, i.e., if Ktop ∗ (G) denotes the topological K-theory of G, then the Baum-Connes assembly map ∗ µ : Ktop ∗ (G) → K∗ (Cr (G)) is an isomorphism. The case of almost connected groups is known in a slightly different formulation as the Connes-Kasparov conjecture (see [3] and §7 below for a discussion). So the above theorem gives a complete solution to that conjecture. It was already shown by Kasparov in [26] that This research has been supported by the Deutsche Forschungsgemeinschaft (SFB 478). 1 2 CHABERT, ECHTERHOFF, AND NEST the result for almost connected groups is true if G is amenable. In fact, by a more recent result of Higson and Kasparov, we know that the Baum-Connes conjecture with arbitrary coefficients holds for any amenable group. By work of A. Wassermann [47], we also know that Theorem 1.1 is true for all connected reductive linear Lie groups. More recently, Lafforgue used quite different methods to give a proof in the case where G is a real reductive group whose semi-simple part has finite centre (see [33, §4.2] – we should point out that this does not cover all real reductive groups) or where G is a reductive p-adic group. The main idea of the proofs of our general results is to use the Mackey-machine approach, as outlined in [10], in order to reduce to the reductive case. The strategy for doing this rests heavily on some ideas presented in Pukánszky’s recent book [43] where he reports on his deep analysis of the representation theory of connected groups. In particular the methods of his proof that locally algebraic connected real Lie groups are type I, presented on the first four pages of his book, were most enlightening. The result on almost connected groups in Theorem 1.1 is actually a special case of a more general result which we shall explain below. If G is a second countable locally compact group, then by a G-algebra A we shall always understand a C ∗ -algebra A equipped with a strongly continuous action of G by ∗-automorphisms of A. Let E(G) denote a locally compact universal proper G-space in the sense of [28] (we refer to [11] for a discussion about the relation to the notion of universal proper G-space as introduced by Baum, Connes and Higson in [3]). If A is a G-algebra, the topological K-theory of G with coefficients in A is defined as G Ktop ∗ (G, A) = lim KK∗ (C0 (X), A), X where X runs through the G-compact subspaces of E(G) (i.e., X/G is compact) ordered by inclusion, and KKG ∗ (C0 (X), A) denotes Kasparov’s equivariant KK-theory. If A = C, we (G) for Ktop simply write Ktop ∗ (G, C). ∗ The construction of Baum, Connes and Higson presented in [3, §9] determines a homomorphism µA : Ktop ∗ (G, A) → K∗ (A or G), usually called the assembly map. We say that G satisfies BC for A (i.e., G satisfies the Baum-Connes conjecture for the coefficient algebra A), if µA is an isomorphism. The result on almost connected groups in Theorem 1.1 is then a special case of Theorem 1.2. Suppose that G is any second countable locally compact group such that G/G0 satisfies BC for arbitrary coefficients, where G0 denotes the connected component of G. (By the results of Higson and Kasparov [22] this is in particular true if G/G0 is amenable or, more general, if G/G0 satisfies the Haagerup property.) Then G satisfies BC for K(H), H a separable Hilbert space, with respect to any action of G on K(H). It is well known that in case of almost connected groups, the topological K-theory has a very nice description in terms of the maximal compact subgroup L of G. In fact, under some mild extra conditions on G, the group Ktop ∗ (G, A) can be computed by means of the K-theory of the crossed product A o L. We give a brief discussion of these relations in §7 below. As was already pointed out in [44], our results have interesting Ktop ∗ (G, A) CONNES-KASPAROV CONJECTURE 3 applications to the study of square-integrable representations. In fact, combining our results with [44, Theorem 4.6] gives Corollary 1.3 (cf [44, Corollary 4.7]). Let G be a connected unimodular Lie group. Then all square-integrable factor representations of G are type I. Moreover, G has no square-integrable factor representations if dim(G/L) is odd, where L denotes the maximal compact subgroup of G. The paper is outlined as follows: In our preliminary section, §2, we recall the main results from [9, 10] on the permanence properties of the Baum-Connes conjecture which are needed in this work. We will also use these results to perform some first reductions of the problem. In fact, we end the section by reducing the proof of Theorem to the following statement. Proposition 1.4. Assume that G is a Lie group with finitely many components and let α : G → Aut(K) be an action of G on the compact operators on some separable Hilbert space H. Then G satisfies BC for K. In §3 we prove a result on continuous fields of actions, showing under some mild conditions on the group G and the base space X of the field, that G satisfies the Baum-Connes conjecture with coefficients in the algebra of C0 -sections of the field if it satisfies the conjecture for all fibres. This result will be another basic tool for the proof of our main theorem. In §4 we are concerned with the reductive groups. Using (and slightly extending) some recent results of Lafforgue [32] on the case of semi-simple groups with finite center, we will use the results on continuous fields obtained in §3 to show that the Baum-Connes conjecture with trivial coefficients holds for all reductive real groups without any extra conditions. In §5 we use Pukánzsky’s methods in combination with an extensive use of the permanence properties for BC to give the final steps for the proof of Theorem 1.1 in case of almost connected groups. In §6 we give the proof of Theorem 1.1 for linear algebraic groups over local fields k which are finite extensions over Qp . Since algebraic groups over R or C are almost connected, this will complete the proof of Theorem 1.1. Finally, in §7 we discuss the connection between the K-theory of the reduced group algebra of an almost connected group G with the representation ring of the maximal compact subgroup L of G as given in the original formulation of the Connes-Kasparov conjecture. Note that §7 does not contain any new material, except for the conclusions drawn out of our main theorem. Acknowledgments. The authors profited quite a lot from valuable discussions with several friends and colleagues. We are most grateful to Alain Connes who suggested (after an Oberwolfach lecture on the permanence properties for BC) to use the results of [9, 10] to attack the Connes-Kasparov conjecture with trivial coefficients. We are also grateful to Bachir Bekka, Ludwig Bröcker, Guennadi Kasparov, Guido Kings, Hervé Oyono-Oyono, Jörg Schürmann for some useful discussions or comments. But, most important, we would like to thank Peter Slodowy for checking some of our arguments on quotients of real algebraic group actions. 4 CHABERT, ECHTERHOFF, AND NEST We dedicate this paper to the memory of Peter Slodowy, who lost his fight against cancer in November 2002. 2. Some preliminaries and first reductions Let us collect some general facts which were presented in [10] – for the definitions of twisted actions and twisted equivariant KK-theory we refer to [9]. Assume that G is a second countable group and let B be a G-C ∗ -algebra. We say that G satisfies BC with coefficients in B if the assembly map µB : Ktop ∗ (G, B) → K∗ (B or G) is an isomorphism. If N is a closed normal subgroup of G, there exists a twisted action of (G, N ) on B or N such that the twisted crossed product (B or N ) or (G, N ) is canonically isomorphic to B or G. Moreover, we can use the twisted equivariant KK-theory of [9] to define the topological K-theory Ktop ∗ (G/N, B or N ) with respect to the twisted action of (G, N ) on B or N , and a twisted version of the assembly map µBor N : Ktop ∗ (G/N, B or N ) → K∗ ((B or N ) or (G, N )). In [9] we constructed a partial assembly map top top µG N,B : K∗ (G, B) → K∗ (G/N, B or N ) such that the following diagram commutes µG N,B Ktop −−−→ ∗ (G, B) −   µB y Ktop ∗ (G/N, B or N )  µBo N y r ∼ = K∗ (B or G) −−−−→ K∗ ((B or N ) or (G, N )). Using this, the first two authors were able to prove the following extension results: Theorem 2.1. Assume that B is a G-algebra and let N be a closed normal subgroup of G. Let q : G → G/N denote the quotient map and assume that one of the following conditions is satisfied (i) G/N has a compact open subgroup K̇ and for any compact subgroup Ċ of G/N , the group C = q −1 (Ċ) satisfies BC for B. (ii) G has a γ-element γ ∈ KKG 0 (C, C) (which is automatically true if G is almost connected), G/N is almost connected and K = q −1 (K̇) satisfies BC for B, where K̇ is a maximal compact subgroup of G/N . top top Then the partial assembly map µG N,B : K∗ (G, B) → K∗ (G/N, B or N ) is an isomorphism. In particular, G satisfies BC for B if and only if G/N satisfies BC for B or N . Proof. See [10, Theorem 3.3 and Theorem 3.7].  In order to avoid the use of twisted actions we may use the version of the Packer-Raeburn stabilization trick as given in [41, 17]: CONNES-KASPAROV CONJECTURE 5 Proposition 2.2 (cf [41, Theorem 3.4] and [17, Corollary 1]). Assume that G is a second countable group and let N be a closed normal subgroup of G. Let (α, τ ) be a twisted action of (G, N ) on the separable C ∗ -algebra A. Then there exists an ordinary action β : G/N → Aut(A ⊗ K), K = K(l2 (N)), such that β is stably exterior equivalent (and hence Morita equivalent) to (α, τ ). Note that BC is invariant under passing to Morita equivalent actions. Thus, in order to conclude that (G, N ) satisfies BC for B or N , it is enough to show that G/N satisfies BC for (B or N ) ⊗ K with respect to an appropriate action of G/N on (B or N ) ⊗ K. In particular, if G/N is amenable, it follows that µBor N : Ktop ∗ (G/N, B or N ) → K∗ ((B or N ) or (G, N )) is always an isomorphism. In what follows we need to study the following special situation: Assume that α : G → Aut(K) is an action of G on K = K(H) for some separable Hilbert space H. Since Aut(K) ∼ = P U (H) = U (H)/T1, we can choose a Borel map V : G → U (H) such that αs = Ad Vs for all s ∈ G. Since α is a homomorphism, we see that there exists a Borel cocycle ω ∈ Z 2 (G, T) such that Vs Vt = ω(s, t)Vst for all s, t ∈ G. The class [ω] ∈ H 2 (G, T) is called the Mackey obstruction for α being unitary. Let 1 7→ T → Gω → G → 1 be the central extension of G by T corresponding to ω, i.e., we have Gω = G × T with multiplication given by (g, z)(g 0 , z 0 ) = (gg 0 , ω(g, g 0 )zz 0 ), and the unique locally compact group topology which generates the product Borel structure on G × T (see [39]). Then the following is true Lemma 2.3. For each n ∈ Z let χn : T → T; χn (z) = z n . Let α : G → Aut(K) and Gω be as above. Then α is Morita equivalent to the twisted action (id, χ1 ) of (Gω , T) on C. Proof. Let V : G → U (H) be as in the discussion above, i.e., αs = Ad Vs and Vs Vt = ω(s, t)Vst for all s, t ∈ G. Then it is easy to check that Ṽ : Gω → U (H) defined by Ṽ(s,z) = zVs is a homomorphism which implements the desired equivalence on the K − C bimodule H (we refer to [17] for an extensive discussion of Morita equivalence for twisted actions).  Another important result is the continuity of the Baum-Connes conjecture with respect to inductive limits of the coefficients, at least if G is exact. For this we need Lemma 2.4. Assume that (Bi )i∈I is an inductive system of G-algebras and let B = limi Bi be the C ∗ -algebraic inductive limit. Assume further that one of the following conditions is satisfied: (i) All connecting maps Bi → Bj , i ≤ j ∈ I are injective, or (ii) G is exact. Then B or G = limi (Bi or G) with respect to the obvious connecting homomorphisms. Proof. If all connecting maps are injective, we may regard each Bi as a subalgebra of B. But this implies that we also have Bi or G as subalgebras of B or G, and hence the inductive 6 CHABERT, ECHTERHOFF, AND NEST limit limi (Bi or G) = ∪{Bi or G : i ∈ I} sits inside B or G. But it is easy to check that ∪{Cc (G, Bi ) : i ∈ I} ⊆ lim(Bi or G) is dense in B or G. Suppose now that G is exact. In this situation we want to reduce the proof to situation (i). Consider the canonical homomorphisms φi : Bi → B. Let Ii = ker φi and let Iij = ker φij , where the φij : Bi → Bj denote the connecting homomorphisms for j ≥ i. Of course, if i ≤ j ≤ j 0 then Iij ⊆ Iij 0 , so for each i ∈ I the system (Iij )j≥i is an inductive system with injective connecting maps. It follows directly from the definition of the inductive limit that Ii = ∪{Iij : j ≥ i} = limj≥i Iij , and hence it follows from (i) that Ii or G = limj≥i (Iij or G). By exactness of G it follows that Ii or G is the kernel of φi or G : Bi or G → B or G. By the previous discussion it follows that Ii or G = limj≥i (Iij or G) is also the kernel of the canonical homomorphism Bi or G → lim(Bj or G). Thus, dividing out the kernels, i.e., by considering the system (Bi0 )i∈I with Bi0 = Bi /Ii we conclude from another use of (i) that B or G = lim(Bi0 or G) = lim(Bi or G).  As a direct consequence we obtain Proposition 2.5. Assume that the G-algebra B is an inductive limit of the G-algebras Bi , i ∈ I, such that G satisfies BC for all Bi . Assume further that G is exact or that all connecting homomorphisms Bi → Bj are injective. Then G satisfies BC for B. Proof. It follows from Lemma 2.4 and the continuity of K-theory that K∗ (B or G) = limi K∗ (Bi or G). On the other side, it is shown in [10, Proposition 7.1] that Ktop ∗ (G, B) = top top ∼ limi K∗ (G, Bi ). Since by assumption K∗ (G, Bi ) = K∗ (Bi or G) via the assembly map, and since the assembly map commutes with the K-theory maps induced by the G-equivariant homomorphism Bi → Bj , the result follows.  As a first application we get Proposition 2.6. Let G be a separable locally compact group such that G/G0 satisfies BC for arbitrary coefficients. Then the following are equivalent: (1) For every central extension 1 → T → Ḡ → G → 1 the group Ḡ satisfies BC for C. (2) G satisfies BC with coefficients in the compact operators K ∼ = K(H) for all separable Hilbert spaces H and with respect to all possible actions of G on K. Proof. Assume that (1) holds. Let α : G → Aut(K) be any action of G on K and let [ω] ∈ H 2 (G, T) denote the Mackey obstruction for this action. Let 1 → T → Gω → G → 1 denote the central extension determined by ω. It follows from Lemma 2.3 that α is Morita equivalent to the twisted action (id, χ1 ) of (Gω , T) on C. By assumption, we know that Gω satisfies BC for C. It follows from Theorem 2.1 that (Gω , T) satisfies BC for C ∗ (T) ∼ = C0 (Z), or, equivalently, that G satisfies BC for C0 (Z, K) with respect to the appropriate action of G (use Proposition 2.2). Since our group G does not satisfy directly the assumptions of Theorem 2.1, let us briefly explain how it is used: first apply part (i) of Theorem 2.1 to N = G0 , which implies that G satisfies BC for C0 (Z, K) if and only if every compact extension C of G0 in CONNES-KASPAROV CONJECTURE 7 G satisfies BC for C0 (Z, K), and then apply part (ii) of Theorem 2.1 to the subgroup T of Cω ⊆ Gω . L Writing C0 (Z) = n∈Z C, the twisted action of (Gω , T) is given by the twisted action (id, χn ) of (Gω , T) on the n’th summand. Let q1 : C0 (Z) → C be the projection on the summand corresponding to 1 ∈ Z. Consider the diagram µC (Z) Ktop −−0−→ K∗ (C0 (Z) or (Gω , T)) ∗ (G, C0 (Z)) −   q1,∗ q1,∗  y y Ktop ∗ (G, C) −−−−→ µC K∗ (C or (Gω , T)). (Here the topological K-theory Ktop ∗ (G, C) is computed with respected to the twisted action ∼ (id, χ1 ) of G = Gω /T and µC denotes the twisted assembly map!) Since the vertical arrows are split-surjective and the upper horizontal arrow is bijective, it follows that the lower horizontal arrow is also bijective. Thus we see that (Gω , T) satisfies BC for C with respect to the twisted action (id, χ1 ). By Morita equivalence this implies that G satisfies BC for K with respect to α. For the opposite direction assume that (2) holds. Let 1 → T → Ḡ → G → 1 be as in (1). As explained above it follows from Theorem 2.1 that Ḡ satisfies BC for C if (Ḡ, T) satisfies BC for C ∗ (T) = C0 (Z). Using the stabilization trick, the latter is true if G satisfies BC for C0 (Z, K) with respect to an appropriate action of G on C0 (Z, K) which fixes the base Z. Using continuity of BC, this follows easily from the fact that G satisfies BC for arbitrary actions on K.  We also need a result on induced algebras as obtained in [10]. For this recall that if H is a closed subgroup of G and A is an H-algebra, then the induced algebra IndG H A is defined as
−1 IndG H A = {f ∈ Cb (G, A) : f (sh) = h (f (s)) and sH 7→ kf (s)k ∈ C0 (G/H)}. ∗ Equipped with the pointwise operations and the supremum-norm, IndG H A becomes a C algebra with G-action defined by s · f (t) = f (s−1 t). The following result follows from [10, Theorem 2.2]: G Theorem 2.7. Let G, H, A and IndG H A be as above. Then G satisfies BC for IndH A if and only if H satisfies BC for A. The result becomes most valuable for us when combined with the following result of [16]: Proposition 2.8. Suppose that H is a closed subgroup of G and B is a G-algebra. Let b denote the set of equivalence classes of irreducible representations of B equipped with the B usual G-action defined by s · π(b) = π(s−1 · b). Then B is isomorphic (as a G-algebra) to IndG H A for some H-algebra A if and only if there exists a G-equivariant continuous map b → G/H. Moreover, if ϕ : B b → G/H is such a map, then A can be chosen to be B/I, ϕ:B with I = ∩{ker π : ϕ(π) = eH} equipped with the obvious H-action. As a corollary of Theorem 2.7 and Proposition 2.8 we get in particular: 8 CHABERT, ECHTERHOFF, AND NEST Corollary 2.9. Suppose that G is a locally compact group and B is a G-algebra which is b Let π ∈ B b and let Gπ denote the stabilizer of type I and such that G acts transitively on B. b π for the action of G on B. Then G satisfies BC for B if and only if Gπ satisfies BC for B/ ker π ∼ = K(Hπ ), where Hπ denotes the Hilbert space of π. b it follows from results of Glimm Proof. Since there is only one orbit for the G-action on B, b b is [20], that B is homeomorphic to G/Gπ via sGπ 7→ s · π. In particular, it follows that B Hausdorff, which implies that B/ ker π ∼ = π(B) = K(Hπ ). The inverse of the above map is b to G/Gπ , and Proposition 2.8 then implies that clearly a continuous G-equivariant map of B G  B∼ = IndGπ (B/ ker π). The result then follows from Theorem 2.7. 2.1. Reduction of the proof of Theorem 1.1. The main work is required for proving the following proposition: Proposition 2.10. Assume that G is a Lie group with finitely many components and let α : G → Aut(K) be an action of G on the compact operators on some separable Hilbert space H. Then G satisfies BC for K. The body of this paper is devoted to give a proof of this result by using induction on the dimension of G. It is fairly easy to see that the above proposition implies Theorem 1. Indeed, using the first part of Theorem 2.1 we can directly reduce to the case where G is almost connected. Hence Theorem 1 follows from Proposition 2.11 (Corollary of 2.10). Let G be any almost connected group and let α : G → Aut(K) be any action of G on the compact operators on some separable Hilbert space H. Then G satisfies BC with coefficients in K. Proof. By the structure theory of almost connected groups (e.g. see [40]) we can find a compact normal subgroup C ⊆ G such that G/C is a Lie group with finitely many components. Using Theorem 2.1 we see that G satisfies BC for K if and only if G/C satisfies BC for K o C (with respect to an appropriate twisted action). Since C is compact, it follows that X := (K o C)b is discrete, and (after stabilizing if necessary), K o C ∼ = C0 (X, K). Let G̃ := G/C and let X/G̃ denote the space of G̃-orbits in X. Since X is discrete, the same is L true for X/G̃, and we get a decomposition C0 (X, K) ∼ = G̃(x)∈X/G̃ C0 (G̃(x), K). By continuity of BC (see Proposition 2.5), we conclude that G̃ satisfies BC for C0 (X, K) if and only if G̃ satisfies BC for C0 (G̃(x), K) for all x ∈ X. Using Corollary 2.9, this will follow if all stabilizers G̃x ⊆ G̃ satisfy BC for K. But since X is discrete, it follows that each stabilizer G̃x contains the connected component G̃0 of G̃. Thus, each stabilizer is a Lie group with finitely many components and the result will follow from Proposition 2.10.  As mentioned above, the main idea for the proof of Proposition 2.10 is to use induction on the dimension dim(G) of the Lie group G. For this we were very much influenced by Pukánszky’s proof of the fact that locally algebraic groups (i.e., Lie groups having the same Lie algebra as some real algebraic group) have type I group C ∗ -algebras as presented in his recent book [43]. We split the induction argument into two main parts, which deal with the cases whether G is semi-simple or not. Note that even in the semi-simple case the result CONNES-KASPAROV CONJECTURE 9 does not follow directly from the existent results, since all known results only work for trivial coefficients and require that the groups have finite centers. 3. Baum-Connes for continuous fields of C ∗ -algebras Let G be a separable locally compact group. Then G is called K-exact, if the functor A 7→ K∗ (A or G) is half-exact, that is: whenever 0 → I → A → A/I → 0 is a short exact sequence of G-algebras, then the sequence K∗ (I or G) → K∗ (A or G) → K∗ (A/I or G) is exact in the middle term. Clearly, every exact group is K-exact. Note that every almost connected group is exact by [30, Corollary 6.9]. Also, if k is a finite extension of the p-adic numbers Qp , then every linear algebraic group over k is exact. To see this first notice that the upper triangle matrices form an amenable cocompact subgroup of GL(n, k), which implies that GL(n, k) has a cocompact closed subgroup which is exact. It follows then from [30, §7] that GL(n, k) is exact. Since exactness passes to closed subgroups by [30, Theorem 4.1], the result follows.1 Recall also that an element γ ∈ KKG 0 (C, C) is called a γ-element for G if there exists a locally compact proper G-space Y , a C ∗ -algebra D equipped with a nondegenerate and G-equivariant ∗-homomorphism φ : C0 (Y ) → ZM (D), the center of the multiplier algebra M (D) of D, and (Dirac and dual-Dirac) elements α ∈ KKG 0 (D, C) β ∈ KKG 0 (C, D) such that γ = β ⊗D α and p∗Z (γ) = 1 ∈ RKKG 0 (Z; C, C) for all locally compact proper G-spaces Z, where pZ : Z → {pt}. It is a basic result of Kasparov [27, Theorem 5.7] that every almost connected group has a γ-element and it follows also from the work of Kasparov (but see also [46, §5]) that a γ-element of G is unique and that it is an idempotent with the remarkable property that for every G-algebra B the image
µB Ktop (G; B) of the assembly map is equal to the γ-part ∗ γ · K∗ (B or G) := {x ⊗Bor G jG (σB (γ)) : x ∈ K∗ (B or G)}. Here and below, we denote by jG : KKG ∗ (A, B) → KK∗ (Aor G, B or G) the (reduced) descent G homomorphism of Kasparov and we denote by σB : KKG ∗ (A, D) → KK∗ (B ⊗ A, D ⊗ B) the external tensor product homomorphism (see [27, Definition 2.5]). Note that it follows from the above discussion that a group G with γ-element satisfies BC for a given G-algebra B if and only if γ (i.e., jG (σB (γ))) acts as the identity on K∗ (B or G). We want to exploit these facts to prove the following basic result: Proposition 3.1. Suppose that X is a separable locally compact space which satisfies one of the following conditions: (a) X can be realized as the geometric realization of a (probably infinite) finite dimensional simplicial complex. 1We should point out that the argument given in [10, Remark 4.4] for exactness of GL(n, Q ) contains a p mistake! 10 CHABERT, ECHTERHOFF, AND NEST (b) X is totally disconnected. Let A be the algebra of C0 -sections of a continuous field of C ∗ -algebras {Ax : x ∈ X}, and let α : G → Aut(A) be a C0 (X)-linear action of G on A. Assume further that G is exact and has a γ-element γ ∈ KKG 0 (C, C). Then, if G satisfies BC with coefficients in each fibre Ax , G satisfies BC for A. For the general notion of continuous fields of C ∗ -algebras and their basic properties we refer to [19, 18, 5, 29]. The idea of the proof is to show first that it holds for any closed interval I ⊆ R. Then a short induction argument will show that it holds for any cube in Rn . Then the result will follow from a Mayer-Vietoris argument. For the proof we first need the following lemma. Lemma 3.2. Assume that G is a K-exact group with a γ-element γ ∈ KKG 0 (C, C). Let A be a G-algebra and let I ⊆ A be a G-invariant closed ideal of A. Then there is a natural six-term exact sequence (1 − γ) · K0 (I or G) x   −−−−→ (1 − γ) · K0 (A or G) −−−−→ (1 − γ) · K0 (A/I or G)   y (1 − γ) · K1 (A/I or G) ←−−−− (1 − γ) · K1 (A or G) ←−−−− (1 − γ) · K1 (I or G). Proof. Since G is K-exact, it follows that A 7→ K∗ (A or G) is a homotopy invariant and halfexact functor on the category of G-C ∗ -algebras which also satisfies Bott-periodicity (with respect to the trivial G-action on C0 (R2 )). Then it follows from some general arguments (e.g., see [4, Chapter IX]) that there exists a six-term exact sequence K0 (I or G) x   −−−−→ K0 (A or G) −−−−→ K0 (A/I or G)   y K1 (A/I or G) ←−−−− K1 (A or G) ←−−−− K1 (I or G). We want to show that all maps in the sequence commute with multiplication with the γelement. By the construction of the connecting maps in the above sequence as given in [4, Chapters VIII and IX], it is enough to show that for any pair of G-algebras A and B and any y ∈ KKG ∗ (A, B) K∗ (A or G) → K∗ (B or G); x 7→ x ⊗Aor G jG (y) commutes with multiplication with γ. But for this it is enough to show that jG (y) ⊗Bor G jG (σB (γ)) = jG (σA (γ)) ⊗Aor G jG (y). This follows from the fact that the descent homomorphism jG is compatible with Kasparov products and the fact that y ⊗B σB (γ) = y ⊗C γ = γ ⊗C y = σA (γ) ⊗A y, which follows from [27, Theorem 2.14]. It follows now that multiplication with 1 − γ also commutes with all maps in the above commutative diagram. Since 1 − γ is an idempotent, it is now easy to see that the full sixterm exact sequence restricts to a six-term exact sequence on the 1 − γ-parts of the respective K-theory groups of the crossed products.  CONNES-KASPAROV CONJECTURE 11 Remark 3.3. It is now a direct consequence of the above proposition that if G is a K-exact group possessing a γ-element, and if 0 → I → A → A/I → 0 is a short exact sequence of Galgebras, then G satisfying BC for two of the algebras in this sequence implies that G satisfies BC for all three algebras in the sequence. The same result holds without the assumption on the γ-element (see [10, Proposition 4.2] – which was actually deduced as an easy consequence of a result of Kasparov and Skandalis in [28]). We also need the following easy lemma. Lemma 3.4. Assume that X is a locally compact space and that A is the algebra of C0 sections of the continuous field {Ax : x ∈ X} of C ∗ -algebras. Assume further that z ∈ Ki (A), i = 0, 1, such that qx,∗ (z) = 0 for some evaluation map qx : A → Ax . Then there exists a compact neighborhood C of x such that qC,∗ (z) = 0 in K0 (A|C ), where A|C denotes the restriction of A to C and qC : A → A|C denotes the quotient map. Proof. We may assume without loss of generality that X is compact. Using suspension, it is enough to give a proof for the case i = 0. In what follows, if B is any C ∗ -algebra, we denote by B 1 the algebra obtained from B by adjoining a unit (even if B is already unital). Then {A1x : x ∈ X} is a continuous field of C ∗ -algebras in a canonical way. The algebra à of sections can be written as the set of pairs {(a, f ) : a ∈ A, f ∈ C0 (X)} with multiplication given pointwise by the multiplication rule of the fibres A1x . Moreover, we have an obvious unital embedding A1 → Ã. Assume now that z ∈ K0 (A) and x ∈ X are as in the lemma. We represent z as a formal difference [p − p0 ] for some projections p, p0 ∈ Ml (A1 ). Since qx,∗ (z) = 0 we may assume (after increasing dimension if necessary) that there exists a unitary ux ∈ Ml (A1x ) such that
ux px u∗x = p0x . After passing to u0 u0∗ if necessary, we may further assume that ux lies in the connected component of the identity of U (Ml (A1x )). Thus, there exists a unitary u ∈ Ml (A1 ) such that qx (u) = ux . Since u is a continuous section in Ã, it follows that there exists a compact neighborhood C of x such that kuy py u∗y − p0y k < 1 for all y ∈ C, which implies that [pC ] = [uC pC u∗C ] = [p0C ] ∈ K0 (A|1C ), where pC , uC , and p0C denote the restrictions of p, u, p0 to C, respectively. But this shows that qC,∗ (z) = [pC − p0C ] = 0 in K0 (A|C ).  Proof of Proposition 3.1. Since G is exact, it follows from [29, Theorem] that the crossed products {Ax or G : x ∈ X} form a continuous bundle such that A or G is the algebra of continuous sections of this bundle. Assume first that we are in situation (a), i.e. that X is a geometric realization of a finite dimensional simplicial complex. Indeed, we first consider the special case where X = [0, 1] ⊆ R. Recall from the above discussions that G satisfies BC for a given G-algebra B if and only if (1 − γ) · K∗ (B or G) = {0}. In particular, it follows from our assumptions that (1 − γ) · K∗ (Ax or G) = {0} for all x ∈ I. Assume now that z ∈ (1 − γ) · Ki (A or G), i = 0, 1, and let qx : A or G → Ax or G denote the evaluation maps for each x ∈ X. Then qx,∗ (z) ∈ (1 − γ) · Ki (Ax or G) = {0} for all x ∈ I. Thus, using Lemma 3.4, we see that there exists a partition 0 = x0 < x1 < · · · < xl = 1 such that q[xj−1 ,xj ],∗ (z) = 0 in Ki (A|[xj−1 ,xj ] or G). Now Ll let O = [0, 1] \ {x0 , . . . , xl } and let A|O = C0 (O) · A ∼ = j=1 A|(xj−1 ,xj ) . It follows from the 12 CHABERT, ECHTERHOFF, AND NEST exact sequence (1 − γ) · Ki (AO or G) → (1 − γ) · Ki (A or G) → l M (1 − γ) · Ki (Axj or G) = {0} j=0 that there exists a z 0 ∈ (1−γ)·Ki (AO or G) such that z is the image of z 0 under the inclusion. Since l M
(1 − γ) · Ki (AO or G) = (1 − γ) · Ki A|(xj−1 ,xj ) or G , j=1 Pl
we may write z 0 = j=1 zj0 with zj0 ∈ (1 − γ) · Ki A|(xj1 ,xj ) or G for each 1 ≤ j ≤ l. Thus it is enough to show that zj0 = 0 for each 1 ≤ j ≤ l. In what follows, we write Aj = A|(xj−1 ,xj ) and Āj = A|[xj−1 ,xj ] . Since (1 − γ) · Ki (Axk or G) = {0} for all 0 ≤ k ≤ l we obtain a six-term exact sequence (1 − γ) · K0 (Aj or G) −−−−→ (1 − γ) · K0 (Āj or G) −−−−→ x   0 0   y ←−−−− (1 − γ) · K1 (Āj or G) ←−−−− (1 − γ) · K1 (Aj or G). Since the image of zj0 in Ki (Āj or G) coincides with the image of z in Ki (Āj or G), we see that zj0 maps to 0 under the isomorphism (1 − γ) · K0 (Aj or G) → (1 − γ) · K0 (Āj or G), so zj0 = 0. We now show by induction on n that the result is true for [0, 1]n ⊆ Rn . For this assume that {Ax : x ∈ [0, 1]n } is a continuous field over the cube and A is the algebra of continuous sections of this field. We write [0, 1]n = ∪y∈[0,1] {y} × [0, 1]n−1 and put Ay = A|{y}×[0,1]n−1 . Then {Ay : y ∈ [0, 1]} is a continuous field over [0, 1] and A is also the section algebra of this bundle. If α is a C([0, 1]n )-linear action on A, it is also C([0, 1])-linear with respect to the bundle structure of A over [0, 1] coming from the above decomposition of the cube. Moreover, the actions on the fibres Ay are clearly C([0, 1]n−1 )-linear, so by the induction assumption we know that G satisfies BC with coefficients in Ay for all y ∈ [0, 1]. We now apply the above result to the bundle {Ay : y ∈ [0, 1]} to conclude that G satisfies BC with coefficients in A. In a next step we show that the result holds for the open cubes (0, 1)n ⊆ Rn . By similar arguments as given above it suffices to show that the result holds for open intervals. So assume that {Ax : x ∈ (0, 1)} is a continuous field with section algebra A. Let x1 < x2 ∈ (0, 1). Then it follows from the first part of the proof that G satisfies BC with coefficients in A[x1 ,x2 ] . Since, by assumption, G also satisfies BC for the fibres, a six-term sequence argument shows that it also satisfies BC with coefficients in (x1 , x2 ). Writing A = limn→∞ A|( 1 ,1− 1 ) and n n using continuity of the BC conjecture, it follows that G satisfies BC for A. Since the result of the proposition is clearly invariant under replacing the space X by a homeomorphic space Y , we now see that the result holds for all open or closed simplices. We now proof the general result for simplicial complexes via induction on the dimension of the complex. By continuity of the conjecture, the result is clear for zero-dimensional complexes. If X has dimension n, let Wn denote the interiors of all n-dimensional simplices in X. Then Wn is homeomorphic to a disjoint union of open n-dimensional cubes, so the result holds for Wn . Since X r Wn is a simplicial complex of dimension n − 1, the result is true for X r Wn CONNES-KASPAROV CONJECTURE 13 by the induction assumption. The result then follows from another easy application of the six-term sequence (see Remark 3.3). We now come to the situation where X is totally disconnected. Let z ∈ (1 − γ) · K∗ (A or G) and let qx : A or G → Ax or G denote evaluation at x. Then qx,∗ (z) ∈ (1 − γ) · K∗ (Ax or G) = {0} for all x ∈ X. By Lemma 3.4, each x ∈ X has a compact neighborhood C such that qC,∗ (z) = 0. Since X is totally disconnected, we therefore find a disjoint covering (Ci )i∈I of compact open subsets of X such that qCi ,∗ (z) = 0 for all i ∈ I. But then we have
L L A or G = i∈I (A or G)|Ci from which it follows that K∗ (A or G) ∼ = i∈I K∗ (A or G)|Ci . Since the projection (qCi ,∗ (z))i∈I is zero for all i ∈ I, it follows that z = 0.  Remark 3.5. Let G be a second countable locally compact group and let X be a second countable locally compact almost Hausdorff G-space (a topological space X is called almost Hausdorff if every closed subset C ⊆ X contains a relatively open dense Hausdorff subset O ⊆ C). Following Glimm we say that the quotient space X/G is countably separated, if all orbits G(x) are locally closed, i.e. G(x) is open in its closure G(x). Glimm showed in [20, Theorem] that X/G being countably separated is equivalent to each of the following conditions: (1) The canonical map G/Gx → G(x), gGx 7→ g · x is a homeomorphism for each x ∈ X. (2) X/G is almost Hausdorff. (3) There exists a sequence of G-invariant open subsets {Uν }ν of X, where ν runs through the ordinal numbers such that (a) Uν ⊆ Uν+1 for each ν and Uν+1 r Uν )/G is Hausdorff. (b) If ν is a limit ordinal, then Uν = ∪µ<ν Uµ . (c) There exits an ordinal number ν0 such that X = Uν0 . b of all equivalence classes of A classical example of an almost Hausdorff space is the space A ∗ irreducible representations of a type I C -algebra A equipped with the Jacobson topology (see [14]). Recall that A is type I if and only if π(A) contains the compact operators K(Hπ ) whenever π : A → B(Hπ ) is an irreducible representation of A. If A is a type I G-algebra, b under the action of G, then the restriction of the G-action and if Gπ is the stabilizer of π ∈ A to Gπ factorizes to a canonical action of Gπ on Aπ := K(Hπ ) and we get: Theorem 3.6. Suppose that G is an exact group with γ-element γ ∈ KKG 0 (C, C) and let A b is countably separated. Suppose that be a type I G-algebra such that the action of G on A b as in item (3) of the above remark such that there exists an ascending sequence {Uν }ν of A the difference sets Uν+1 r Uν satisfy the following conditions: (i) There exists a locally compact Hausdorff space Xν and a contionuous and open surjection qν : Uν+1 r Uν → Xν such that, for all x ∈ Xν , qν−1 ({x}) is a finite union of G-orbits in Uν+1 . (ii) The space Xν of (i) is (the geometric realization of ) a finite dimensional simplicial complex or Xν is totally disconnected. b The G satisfies BC for A, if the stabilizer Gπ satisfies BC for Aπ = K(Hπ ) for all π ∈ A. b (see Proof. For each ordinal ν let Aν the ideal of A corresponding to the open subset Uν of A [14, Chapter 3]). We show by transfinite induction that G satisfies BC with coefficients in Aν for each ν. Since A = Aν0 for some ν0 , the result will follow. 14 CHABERT, ECHTERHOFF, AND NEST We start by showing that G satisfies BC with coefficients in A1 . Since the quotient map q1 : U1 → X1 is open, we can regard A1 as a section algebra of a continuous bundle over X1 with fibres isomorphic to A|q−1 (x) (see [36]). By condition (ii) it follows from Proposition 3.1 1 that it suffices to prove that G satisfies BC for A|q−1 (x) for all x ∈ X1 . Fix x ∈ X1 and put 1 Z := q1−1 (x). Since Z is a finite union of G-orbits, we find a finite sequence Z = Z0 ⊇ Z1 ⊇ · · · ⊇ Zl = ∅
of open invariant subsets of Z such that Zi−1 r Zi /G is a discrete finite set. To see this let C1 be the union of all closed G-orbits in Z (such orbits must exist by the finiteness of Z/G). Then C1 is closed in Z and C1 /G is discrete. Put Z1 = Z r C1 and then define the Zi ’s, i > 1, inductively by the same procedure. Using six-term sequences (e.g., see Remark 3.3), G satisfies BC for A|Z if G satisfies BC for all A|Zi−1 rZi , which in turn follows if G satisfies b (where A|G(π) denotes the subquotient of A BC for A|G(π) for any G-orbit G(π) ⊆ Z ⊆ A b But this follows from the assumption corresponding to the locally closed subset G(π) of A). that Gπ satisfies BC for Aπ and Corollary 2.9. This completes the proof for A1 . Assume now that ν is an ordinal number and that we have already shown that G satisfies BC for Aµ for all µ < ν. If ν = µ + 1 for some ordinal µ, it follows from the same reasoning as for the case ν = 1 that G satisfies BC for Aν /Aµ . Since G satisfies BC for Aµ by the induction assumption, it follows from Remark 3.3 that G satisfies BC for Aν . Assume now that ν is a limit ordinal and G satisfies BC for Aµ for each µ < ν. Then Uν = ∪µ<ν Uµ which implies that Aν = limµ<ν Aµ is the inductive limit of the Aµ . Thus it follows from Proposition 2.5 that G satisfies BC for Aν .  4. The semi-simple case In this section we want to show that Proposition 2.10 is true if G is semi-simple. For this we first have to obtain a slight extension of Lafforgue’s results on the Baum-Connes conjecture for semi-simple groups with finite center. Let us first recall the basic idea of Lafforgue’s proof of the Baum-Connes conjecture for such groups. If G is a locally compact group we let Cc (G) denote the convolution algebra of G consisting of continuous functions with compact supports. A norm k · k on Cc (G) is called good if convolution is continuous with respect to this norm and if kf k only depends on the absolute value of f for all f ∈ Cc (G) (i.e., kf k = k|f |k for all f ∈ Cc (G)). A good completion A(G) of Cc (G) is a completion with respect to a good norm on Cc (G). Note that L1 (G) is always a good completion of Cc (G), but C ∗ (G) and Cr∗ (G) are in general not good completions of Cc (G). If A(G) is a good completion of Cc (G), then Lafforgue constructed an assembly map µA(G) : Ktop ∗ (G, C) → K∗ (A(G)). Moreover, if the identity on Cc (G) extends to a continuous embedding ι : A(G) → Cr∗ (G), he ∗ also shows that the assembly map µ : Ktop ∗ (G, C) → K∗ (Cr (G)) factors through K∗ (A(G)), i.e., µ = ι∗ ◦ µA(G) CONNES-KASPAROV CONJECTURE 15 (see [32, Proposition 1.7.6]). Thus, if we know that µA(G) is an isomorphism for all good completions of Cc (G), and if we further know that there exists a good completion A(G) ⊆ Cr∗ (G) such that the inclusion ι∗ : K∗ (A(G)) → K∗ (Cr∗ (G)) is an isomorphism, it follows that ∗ ∗ µ : Ktop ∗ (G, C) → K∗ (Cr (G)) is an isomorphism. Note that ι∗ : K∗ (A(G)) → K∗ (Cr (G)) is an isomorphism whenever A(G) is closed under holomorphic functional calculus in Cr∗ (G). Now Lafforgue was able to prove the following deep results: Theorem 4.1 (cf [32, Théorème 17.13 and Chapitre 3]). Assume that G is a second countable locally compact group such that G satisfies one of the following conditions: (a) G acts isometrically and properly on a Riemannian manifold with nonpositive sectional curvature which is bounded below. (b) G acts properly and isometrically on a “bolic” space (X, d) in the sense of [32, Definition 2.2.1]. Then µA(G) : Ktop ∗ (G, C) → K∗ (A(G)) is an isomorphism for every good completion A(G) of Cc (G). In fact, Lafforgue was even able to show that the above result holds with arbitrary C ∗ algebra coefficients, but we do not need this more general result here. Note that if G is semi-simple with finite center, then the Riemannian manifold of the theorem can be chosen to be the symmetric space G/K, where K is the maximal compact subgroup of G. In the second step for the proof of BC for semi-simple groups, Lafforgue constructed a Schwartz-algebra S(G) ⊆ Cr∗ (G) which is a good completion of Cc (G) which is closed under holomorphic functional calculus in Cr∗ (G). In fact, this construction followed a more general principle, which we are now going to describe in more detail. Assume that G is a unimodular group and K is a compact subgroup of G. Then, following Lafforgue (see [32, Chapitre 4]), we say that the pair (G, K) satisfies property (HC) if the following conditions are satisfied (HC1) There exists a continuous function d : G → [0, ∞) such that d(e) = 0, d(kgk 0 ) = d(g) for all k, k 0 ∈ K, g ∈ G and d(gg 0 ) ≤ d(g) + d(g 0 ) for all g, g 0 ∈ G. (HC2) There exists a continuous function φ : G → (0, 1] such that φ(e) = 1, φ(g −1 ) = φ(g), φ(kgk 0 ) = φ(g) for all g ∈ G and k, k 0 ∈ K, and Z φ(gkg 0 ) dk = φ(g)φ(g 0 ) for all g, g 0 ∈ G, K with respect to the normalized Haar measure on K.
(HC3) There exists a t0 ∈ R such that t 7→ φ(g)(1 + d(g))−t ∈ L2 (G) for all t > t0 . If G is a connected semi-simple group with finite center, and if K is the maximal compact subgroup of G, then Lafforgue was able to show that (G, K) satisfies (HC). The following theorem then completes the proof of BC for connected semi-simple groups with finite center. Theorem 4.2 (cf [32, Proposition 4.2.1]). Assume that G is a unimodular group and K is a compact subgroup of G such that the pair (G, K) satisfies (HC). Then there exists a good completion S(G) ⊆ Cr∗ (G) of Cc (G) such that S(G) is closed under holomorphic functional calculus in Cr∗ (G). 16 CHABERT, ECHTERHOFF, AND NEST In order to extend Lafforgue’s methods to extensions of semi-simple groups, we first show that property (HC) is closed under compact extensions. Lemma 4.3. Assume that (G, K) satisfies property (HC). Assume further that q 1 → C → G̃ → G → 1 is a group extension with C compact. Let K̃ = q −1 (K) ⊆ G̃. Then (G̃, K̃) satisfies (HC). Proof. Since compact extensions of unimodular groups are unimodular, G̃ is unimodular. Let (d, φ) be a pair of functions which satisfy HC1, HC2, and HC3 with respect to (G, K). Define ˜ ˜ φ̃) d(g) = d(q(g)) and φ̃(g) = φ(q(g)). Then a straightforward computation shows that (d, satisfies HC1, HC2, and HC3 with respect to (G̃, K̃).  The second result is slightly more technical. Lemma 4.4. Assume that G is a unimodular Lie group with finitely many components and let K be a maximal compact subgroup of G. Let G0 denote the connected component of G and let K0 be a maximal compact subgroup of G0 such that (G0 , K0 ) satisfies (HC). Then (G, K) satisfies (HC), too. Proof. First note that we may assume that K0 = K ∩ G0 . To see this observe first that, since K0 is a compact subgroup of G, we may assume without loss of generality that K0 ⊆ K ∩ G0 . But the maximality of K0 then implies equality. It follows from this that K0 is a normal subgroup of K and since G/K is connected, it follows that the inclusion K → G induces a group isomorphism K/K0 ∼ = G/G0 . Let (d0 , φ0 ) be a pair of functions satisfying conditions HC1–HC3 for (G0 , K0 ). It follows from the above remarks that we can write every element of G as a product kg with k ∈ K, g ∈ G0 . We then define Z Z −1 d(kg) = d0 (lgl ) dl and φ(kg) = φ0 (lgl−1 ) dl. K K To see that d and φ are well defined assume that we have two factorizations kg = k 0 g 0 with k, k 0 ∈ K, g, g 0 ∈ G0 . Then g = k −1 k 0 g 0 with h := k −1 k 0 ∈ K0 . Since K normalizes K0 , it follows from the left and right K0 -invariance of d0 that Z Z Z
d(kg) = d0 (lgl−1 ) dl = d0 (lhg 0 l−1 ) dl = d0 (lhl−1 )(lg 0 l−1 ) dl K K ZK = d0 (lg 0 l−1 ) dl = d(k 0 g 0 ). K A similar computation shows that φ is well defined. We are now going to check properties HC1–HC3 for (d, φ). It follows directly from the definition of d and φ that they are left invariant under the action of K. To see right invariance, we compute for h ∈ K: Z Z l7→lh −1 −1 −1 −1 d(kgh) = d(khh gh) = d(h gh) = d0 (lh ghl ) dl = d0 (lhl−1 ) dl = d(kg). K K So d is also right invariant and a similar computation show that the same is true for φ. Since Haar measure on K is normalized, it follows that d(e) = 0 and φ(e) = 1. Moreover, if CONNES-KASPAROV CONJECTURE 17 kg, hg 0 ∈ G with k, h ∈ K, g, g 0 ∈ G0 we get Z Z −1 0 −1 0 −1 0 d0 (lh−1 ghl−1 lg 0 l−1 ) dl d0 (lh ghg l ) dl = d(kghg ) = d(khh ghg ) = K K Z Z d0 (lg 0 l−1 ) dl = d(kg) + d(hg 0 ). d0 (lh−1 ghl−1 ) dl + ≤ K K In order to prove the multiplication rule for φ we use Weil’s formula  Z  Z Z Z Z ˙ ϕ(lm) dm dl, ϕ(lm) dm dl = ϕ(l) dl = K K/K0 K K0 K0 with respect to normalized Haar measures on K, K0 , and K/K0 to compute Z Z Z Z 0 −1 0 φ(glmhg 0 h−1 ) dm dl φ(glhg h ) dl = φ(kglhg ) dl= K K0 K K Z Z (∗) = φ(l−1 glmhg 0 h−1 ) dm dl K K Z Z 0Z = φ0 (nl−1 glmhg 0 h−1 n−1 ) dn dm dl K K0 K Z Z Z (∗∗) = φ0 (nl−1 gln−1 mnhg 0 h−1 n−1 ) dm dl dn K K K0 Z Z (∗∗∗) = φ0 (nl−1 gln−1 )φ0 (nhg 0 h−1 n−1 ) dl dn K K Z  Z  l7→l−1 n −1 0 −1 = φ0 (lgl ) dl φ0 (ng n ) dn K K 0 = φ(kg)φ(hg ). Here the equation (*) follows from the K-invariance of φ, (**) follows from Fubini together with the transformation m 7→ n−1 mn, and (***) follows from property HC2 for φ0 . This completes the proof of HC1 and HC2. For the proof of HC3 we write Fs (g) := φ(g)(1 + d(g))−s , s > 0. Since G = KG0 , it follows from the K-invariance of φ and d that the integrals of Fs2 over the G0 -cosets coincide. Thus, since G/G0 is finite, it is enough to show that Fs |G0 ∈ L2 (G0 ) for some s > 0. For this we first choose a set of representatives t1 , . . . , tn ∈ K for K/K0 with t1 = e. Then, for g ∈ G0 , we obtain the inequality n 1 + d(g) = 1 + 1X 1 d0 (ti gt−1 i ) ≥ 1 + d0 (g), n n i=1 d(g))n from which it follows that ≥ 1 + d0 (g) for all g ∈ G0 . Thus, if t ∈
R such that
(1 + g 7→ φ0 (g)(1 + d0 (g))−t ∈ L2 (G0 ), then we also have g 7→ φ0 (g)(1 + d(g))−nt ∈ L2 (G0 ). So let s = nt with t as above. Then we get n −s Fs (g) = φ(g)(1 + d(g)) 1X −s φ0 (ti gt−1 = i )(1 + d(g)) n i=1 n 1X −1 −s = φ0 (ti gt−1 i )(1 + d(ti gti )) . n i=1 18 CHABERT, ECHTERHOFF, AND NEST
−1 −s Since G is unimodular, it follows that each summand g 7→ φ0 (ti gt−1 ∈ i )(1 + d(ti gti )) 2 2  L (G0 ), and hence Fs |G0 ∈ L (G0 ). We are now ready to combine the above results to get Proposition 4.5. Let G be a locally compact group with finitely many components. Assume further that G has a compact normal subgroup C ⊆ G0 such that G0 /C is a real semi-simple Lie group with finite center. Then G satisfies BC with trivial coefficients. Proof. Let K0 denote the maximal compact subgroup of G0 . Then G0 /K0 ∼ = (G0 /C)/(K0 /C) is a symmetric space and therefore has nonpositive sectional curvature. Moreover, if K is a maximal compact subgroup of G such that K ∩ G0 = K0 , we see that G/K ∼ = G0 /K0 as a Riemannian manifold. Since G acts isometrically and properly on G/K, G satisfies the assumptions of Theorem 4.1. By Lafforgue’s results we also know that (G0 /C, K0 /C) satisfies (HC). Lemmas 4.3 and 4.4 then imply that (G, K) also satisfies property (HC). Thus, it follows from the combination of Theorem 4.1 with Theorem 4.2 that G satisfies BC with coefficients in C.  Using the results on continuous fields of actions as presented in the previous section, we are now able to prove Proposition 4.6. Assume that G is a real Lie group with finitely many components such L that G0 is reductive, i.e., the Lie algebra g of G is a direct sum of two ideals g = s z with s semi-simple and z abelian. Then G satisfies BC for C. Proof. Let Z = Z(G0 ) denote the center of G0 . Using Theorem 2.1 it is enough to show that b K), where the action of G/Z on the G/Z satisfies BC with coefficients in C ∗ (Z) ⊗ K ∼ = C0 (Z, dual space Zb of Z is given via conjugation. Since Z is central in G0 , it follows that this action b is a manifold (since Z factors through an action of the finite group G/G0 . Moreover, since Z b is a compactly generated abelian group), it follows that the quotients of the orbit-types in Z are manifolds. From this we easily obtain a finite decomposition sequence b ∅ = U0 ⊆ U1 ⊆ · · · ⊆ Ul = Z b such that the quotients of the differences Uj r Uj−1 are of open G-invariant subsets of Z homeomorphic to geometric realizations of finite dimensional simplicial complexes. Moreover, b contain G0 /Z, which is semi-simple with trivial since all stabilizers for the action of G/Z on Z center, it follows from a combination of Proposition 4.5 with Proposition 2.6 that all stabilizers satisfy BC for K. The result then follows from Theorem 3.6.  Since any central extension of a semi-simple group is reductive, we now get the desired result for general semi-simple groups. Corollary 4.7. Let G be a semi-simple Lie group with finitely many components and let 1 → T → Ḡ → G → 1 be a central extension of G by T. Then Ḡ satisfies BC for C. As a consequence (using Proposition 2.6), G satisfies BC for K with respect to arbitrary actions of G on K. CONNES-KASPAROV CONJECTURE 19 Remark 4.8. When we first wrote down this article, we had a copy of the original thesis of Lafforgue, where he shows that connected semi-simple Lie groups with finite center satisfy BC with trivial coefficients. In the meantime Lafforgue has extended this result to all reductive groups G such that the semi-simple part of G has finite center (i.e. the semi-simple subgroup S ⊆ G corresponding to the Levi summand s ⊆ g has finite center). Note that this result ^ does not cover all reductive real groups. For instance, the universal covering group SL 2 (R) of SL2 (R) is a semi-simple group with infinite center. The statements about real reductive groups as given in the surveys [34, 35] are somewhat misleading in this respect. We close this section with a result on p-adic groups: Proposition 4.9. Let k be a finite extension of some Qp and let G be the group of k-rational points of a reductive linear algebraic group over k. Then G satisfies BC for K with respect to any action of G on K. Proof. Using Proposition 2.6, it suffices to show that every central extension G̃ of G by T satisfies BC with trivial coefficients. By the results of [28, 32] we know that G acts properly and isometrically on a “bolic” space (X, d) in the sense of [32, Definition 2.2.1] and that G has a compact subgroup K such that (G, K) satisfies property (HC) (see [32, Théorème 2.2.2, Proposition 4.1.2 and §4.3]). Lemma 4.3 then implies that (G̃, K̃), where K̃ denotes the inverse image of K in G̃ has the same properties. Hence the result follows from Theorems 4.1 and 4.2.  5. The general case of almost connected groups We now want to give a proof of Proposition 2.10. As outlined in §2, this will imply the proof of Theorem 1 and therefore finishes the case of almost connected groups in Theorem 1.1. As indicated before, we are going to use an induction argument on the dimension n = dim(G). Since any one-dimensional Lie group with finitely many components is amenable, and since amenable groups satisfy BC for arbitrary coefficients, the case n = 1 is clear. Assume now that G is an arbitrary Lie group with finitely many components. Let G0 denote the connected component of G. Let N denote the nilradical of G and let n and g denote the Lie algebras of N and G, respectively. If n = {0}, then G is semi-simple and the result follows from the previous section. So we may assume that n 6= {0}. It is shown in [43, Lemma 4 on p. 24] that the subgroup H of G0 corresponding to the subalgebra h = [g, g] + n of g is closed in G0 . Further, if s is a Levi section in g, i.e., s is a maximal semi-simple subalgebra of g, then h = s + n (e.g., see the discussion in the proof of [43, Sublemma on p. 24]). In particular, H/N is semi-simple. Clearly, H is a normal subgroup of G and G/H is a finite extension of a connected abelian Lie group. Let M denote the inverse image of the maximal compact subgroup of G/H in G. It follows then from Theorem 2.1 that G satisfies BC for K if and only if M satisfies BC for K. Note that the connected component of M/H is a compact connected abelian Lie group, hence a torus group. Thus, replacing G by M , we may from now on assume that G has the following structure: There exist closed normal subgroups (5.1) N ⊆ H ⊆ G0 ⊆ G 20 CHABERT, ECHTERHOFF, AND NEST such that N is a non-trivial connected nilpotent Lie group, H/N is semi-simple, G0 /H is a torus group and G/G0 is finite. Moreover, by induction we may assume that every almost connected Lie group with smaller dimension satisfies BC for K with respect to arbitrary actions on K, or, equivalently (by Proposition 2.6), every central extension by T satisfies BC for C. It is now useful to recall the following result of Chevalley (see [12, Proposition 5, p.324]): Proposition 5.1. Let g ⊆ gl(V ) be a Lie-algebra of endomorphisms of the finite dimensional real vector space V . Then g is algebraic (i.e. it corresponds to a real algebraic subgroup G ⊆ GL(V )) if and only if there exist subalgebras s, a, n of g with g = s + a + n, s is semisimple, n is the largest ideal of g consisting of nilpotent endomorphisms, and a is an algebraic abelian subalgebra of gl(V ) consisting of semi-simple endomorphisms such that [s, a] ⊆ a. Using Ado’s theorem (see [12, Théorème 5 on p. 333]) and Proposition 5.1, it follows that the group H considered above is locally algebraic, i.e., the Lie algebra h has a faithful representation as an algebraic Lie subalgebra into some gl(V ). Using this structure, the main idea is to apply the Mackey machine to a suitable abelian subgroup S of N which is normal in G. The fact that G is very close to an algebraic group implies that the action of G on the dual Sb of S has very good topological properties, which is precisely what we need to make everything work. As a first hint that this approach is feasible we prove: Lemma 5.2. Assume that G is a Lie group with finitely many components. Let H ⊆ G0 be a connected closed normal subgroup of G such that H is locally algebraic, and such that G/H is compact. Let N denote the nilpotent radical of H, and let S ⊆ N be a connected abelian normal subgroup of G. Let Sb denote the character group of S and let G act on Sb via conjugation. Then the following assertions are true: b (i) The orbit space S/G is countably separated, i.e., all G-orbits in Sb are locally closed. b then Gχ /(Gχ )0 is (ii) If Gχ is the stabilizer of some χ ∈ Sb for the action of G on S, amenable. Proof. We first show that it is sufficient to prove the result for the case G = H. Indeed, if b b we already know that S/H is countably separated, then we observe that S/H is a topolog∼ b b ical G/H-space such that S/G = (S/H)/(G/H). But it is an easy exercise to prove that the quotient space of a countably separated space by a compact group action is countably separated. Assume now that Gχ is the stabilizer of some χ ∈ Sb in G. Then Hχ = Gχ ∩ H is the stabilizer in H. It follows that Hχ is a normal subgroup of Gχ such that Gχ /Hχ is compact. If Hχ /(Hχ )0 is amenable, it also follows that Gχ /(Hχ )0 , and hence also Gχ /(Gχ )0 are amenable. So, for the rest of the proof we assume that G = H. In the next step we reduce to the case where H is simply connected. For this let H̃ denote the universal covering group of H. Then H̃ has the same Lie algebra as H, and therefore it is locally algebraic. Let q : H̃ → H denote the quotient map and let C = ker q. Then C is a discrete central subgroup of H̃. Let r ⊆ h denote the Lie algebra of S and let S̃ denote the connected closed normal subgroup of H̃ corresponding to r. Then S̃ is a vector subgroup of the nilpotent radical Ñ of H̃ and the quotient map H̃ → H maps S̃ surjectively onto S, i.e., we have S ∼ = S̃/(S̃ ∩ C). In CONNES-KASPAROV CONJECTURE 21 b̃ and we have particular, it follows that we may view Sb as a closed H̃-invariant subspace of S, b̃ H̃ is countably b b H̃ (since the central subgroup C acts trivially on S). b Thus, if S/ S/H = S/ b separated, the same is true for S/H. We now consider the stabilizers. It follows from the above considerations that if Hχ is the b then q −1 (Hχ ) ⊆ H̃ is the stabilizer of χ in H̃. Thus it follows that stabilizer of some χ ∈ S, Hχ = H̃χ /(C ∩ H̃χ ). Since the connected component of H̃χ is mapped onto the connected component of Hχ under the quotient map, it follows that Hχ /(Hχ )0 is a quotient of H̃χ /(H̃χ )0 . Thus, if the latter is amenable, the same is true for Hχ /(Hχ )0 . Thus, in what follows we may assume without loss of generality that H is simply connected. We are then in precisely the same situation as in the proof of Case (A) of the proof of the Theorem on page 2 of [43], and from now on we can follow the line of arguments as given b on pages 2 and 3 of [43] to see that S/H is countably separated. Moreover, the arguments presented in steps c) and d) on page 3 of Punkánzsky’s book imply that for each stabilizer Hχ the quotient Hχ /(Hχ )0 is a finite extension of an abelian group, and hence is amenable.  Remark 5.3. We should point out that the result on the stabilizers in Lemma 5.2 is most satisfying: Indeed if we know that every almost connected Lie group with dimension dim(G) < n satisfies BC for K, say, then, by an easy application of Theorem 2.1 the same is true for all Lie groups H with dim(H) < n and H/H0 amenable! b Unfortunately, the result on the orbit space S/G is not sufficient for a direct application of Theorem 3.6. So we have to do some extra work to obtain more information on the structure b of S/G. To do this we have to do two steps: (i) Reduce to cases where the action of G on Sb factors through an algebraic action of some real algebraic group G0 (or a subgroup of finite index in G0 ). (ii) Show that that the topological orbit-spaces of algebraic group actions on real affine varieties have nice stratifications as required by Theorem 3.6. Note that Pukánszky does the first reduction for the cocompact subgroup H of G, which allowed us to draw the conclusions of the previous lemma. However, with a bit more work we obtain similar conclusion for G. The following result is certainly well-known to the experts, but since we didn’t find a direct reference we included the easy proof. Lemma 5.4. Assume that G is a Lie group with finitely many components such that G has a connected closed normal subgroup H with H semi-simple and G0 /H a torus group. Let V be a finite dimensional real vector space and let ρ : G → GL(V ) be any continuous homomorphism. Then the Zariski closure G0 of ρ(G) is a (reductive) real algebraic group which contains ρ(G) as a subgroup of finite index. Proof. Let R : g → gl(V ) denote the differential of ρ and let h denote the ideal of g corresponding to H. Then R(h) is semi-simple (or trivial). Since ρ(H) is a semi-simple subgroup of GL(V ) it is closed in GL(V ). This follows from the fact that every semi-simple subalgebra of gl(V ) is algebraic (by Proposition 5.1), which implies that ρ(H) is the connected component of some algebraic linear subgroup of GL(V ). Since G/H is compact, it follows that ρ(G) is a closed subgroup of GL(V ), too. To simplify notation we assume from now on that G itself is a closed subgroup of GL(V ) and that ρ is the identity map. Let g = s + z be a Levi decomposition of g. Since G0 /H is 22 CHABERT, ECHTERHOFF, AND NEST abelian and H is semi-simple, it follows that s = [g, g] = h and [h, z] = [z, z] = {0}. Thus L g = h z is reductive. We now show that g is an algebraic subalgebra of gl(V ). By Proposition 5.1 it suffices to show that z is algebraic and consists of semi-simple elements. But this will follow if we can show that Z = exp(z) ⊆ gl(V ) is compact, and hence a torus group. Since Z ∩ H is finite (since every linear semi-simple group has finite center), the restriction to Z of the quotient map q : G → G/H has finite kernel. Since q(Z) = G0 /H, q(Z) is compact by assumption, and hence Z is compact, too. It follows that the algebraic closure G̃ of G0 is a reductive algebraic group which contains G0 as a subgroup of finite index. Since every element of G fixes the Lie algebra g via the adjoint action, it also normalizes G̃. Therefore, G0 = GG̃ is a reductive algebraic group which contains G as a subgroup of finite index.  We now show that quotients of linear algebraic group actions on affine varieties have nice stratifications in the sense of Theorem 3.6. We are very grateful to Jörg Schürmann and Peter Slodowy for some valuable comments, which helped us to replace a previous version of the following result (which, as was pointed out to us by Jörg Schürmann, contained a gap) by Proposition 5.5. Suppose that G is a closed subgroup of finite index of a Zariski closed subgroup G0 of GL(n, R) and that V ⊆ Rn is a G0 -invariant Zariski closed subset of Rn . Then there exists a stratification ∅ = V0 ⊆ V1 ⊆ · · · ⊆ Vl = V of open G-invariant subsets Vi of V such that (Vi \ Vi−1 )/G admits a continuous and open finite-to-one map onto a differentiable manifold. Since every manifold has a triangulation, the above result really gives what we need to apply Theorem 3.6. For the proof we need the following lemma about certain decompositions of continuous semi-algebraic maps. Lemma 5.6. Let X, Y be semi-algebraic sets and let f : X → Y be a continuous semialgebraic map (see [6] for the notations). Then there exists a stratification ∅ = Z0 ⊆ Z1 · · · ⊆ Zl = f (X), with each Zi open in f (X), Zi \ Zi−1 is a differentiable manifold and f : f −1 (Zi \ Zi−1 ) → Zi \ Zi−1 is open (in the euclidean topology) for all 1 ≤ i ≤ l. Proof. Since the image of a semi-algebraic set under a semi-algebraic map is semi-algebraic (see [6, Proposition 2.2.7]), we may assume without loss of generality that Y = f (X). By [6, Corollary 9.3.3] there exists a closed semi-algebraic subset Y1 ⊆ Y with dim(Y1 ) < dim(Y ), such that Y \ Y1 is a finite disjoint union of connected components (combine with [6, Theorem 2.4.5]) and such that the restriction of f to the inverse image of each component is a projection, hence open. Thus f : f −1 (Y \ Y1 ) → Y \ Y1 is open, too. Indeed, the construction (using [6, Proposition 9.18]) implies that Y \ Y1 is homeomorphic to a submanifold of some Rm . Put Z0 = Y \ Y1 . Since dim(Y1 ) < dim(Y ), the result follows by induction.  CONNES-KASPAROV CONJECTURE 23 Remark 5.7. Let G ⊆ GL(n, R) be a real linear algebraic group, and let GC ⊆ GL(n, C) be its complexification. Then it follows from [7, Proposition 2.3] that each GC -orbit in Cn contains at most finitely many G-orbits in Rn ⊆ Cn . Proof of Proposition 5.5. We first note that we may assume without loss of generality that G = G0 . Indeed, since G has finite index in G0 , every G0 -orbit decomposes into finitely many G-orbits. Thus, if ∅ = V0 ⊆ V1 ⊆ · · · ⊆ Vl = V is a stratification of V for the G0 -action with the required properties, it is also a stratification for the G-action with the same properties. Thus we assume from now on that G is a Zariski closed subgroup of GL(V ). Let VC ⊆ Cn denote the complexification of V . Consider the diagram V −−−−→ VC     y y V /G −−−−→ VC /GC . By the theorem of Rosenlicht ([45], but see also [31, Satz 2.2 on p. 23]), there exists a sequence VC = W0 ⊇ W1 ⊇ W2 ⊇ · · · ⊇ Wr = ∅, of Zariski-closed GC -invariant subsets of strictly decreasing dimension such that Wi \ Wi+1 has closed GC -orbits and the geometric quotient by GC of Wi \ Wi+1 exists. This means that the quotient (Wi \ Wi+1 )/GC can be realized as an algebraic set and the quotient map is also algebraic. Let O be the first of the sets Wi \ Wi+1 which has nonempty intersection with V . Restricting the maps in the above diagram gives V ∩O −−−−→ O     y y (V ∩ O)/G −−−−→ O/GC . The resulting map f from V ∩ O to O/GC is an algebraic map, and hence it is a continuous semi-algebraic map. Thus it follows from Lemma 5.6 that, if Y denotes the image of X := V ∩ O in O/GC , then Y has a stratification ∅ = Z0 ⊆ Z1 ⊆ · · · ⊆ Zs = Y such that f : f −1 (Zi \ Zi−1 ) → Zi \ Zi−1 is open for all 1 ≤ i ≤ s, each Zi is open in Y , and the difference sets Zi \ Zi−1 are submanifolds of some Rm . Put Vi = f −1 (Zi ) for 0 ≤ i ≤ s. Then Vs = V ∩ O. By Remark 5.7, if we pass through the lower left corner of the diagram, the corresponding maps (Vi \ Vi−1 )/G → Zi \ Zi−1 are open, finite-to-one, onto the manifolds Zi \ Zi−1 . Now replace V by the invariant Zariski-closed subset V \O. Repeating the above arguments finitely many times gives the desired stratification (the procedure stops after finitely many steps, since any increasing sequence of Zariski open sets eventually stabilizes).  Using the above results, we are now able to prove Proposition 5.8. Suppose that G is a Lie group with finitely many components and with connected closed normal subgroups N ⊆ H ⊆ G0 ⊆ G as in (5.1), i.e., N is the nilradical of H, H/N is semi-simple and G0 /H is a torus group. Let S ⊆ Z(N ) be a connected closed 24 CHABERT, ECHTERHOFF, AND NEST subgroup which is normal in G, where Z(N ) denotes the center of N . Then Sb decomposes into a countable disjoint union of open G-invariant sets Vn such that each Vn has a stratification ∅ = U0 ⊆ U1 ⊆ · · · ⊆ Ul = Vn (where l may depend on n) of open G-invariant subsets of Vn , and continuous open surjections qi : Ui r Ui−1 → Yi , 1 ≤ i ≤ l, such that each Yi is a differentiable manifold and inverse images of points in Yi are finite unions of G-orbits in Vn for all 1 ≤ i ≤ l. Proof. Let s denote the ideal of g corresponding to S. Then we may identify Sb with a closed G-stable subset of s∗ of the form R × Z with R being a vector subgroup of s∗ and Z a finitely generated free abelian group. Note that Z can be identified with the dual of the maximal compact subgroup in S, and therefore we can decompose Z into a disjoint union of G-orbits, which are all finite since G0 acts trivially on Z. It then follows that Sb can be decomposed into a disjoint union of G-invariant sets of the form R × F with F ⊆ Z finite. The action of G on Sb is given via the coadjoint representation Ad∗s : G → GL(s∗ ). Since S ⊆ Z(N ), it follows that this representation factors through a representation of G/N . Thus it follows from the general assumptions on G and Lemma 5.4 that the algebraic closure G0 of Ad∗s (G) in GL(s∗ ) is a reductive algebraic group which contains the image of Ad∗s (G) as a subgroup of finite index. Since the G-stable sets of the form R × F of the previous paragraph are closed algebraic subvarieties of s∗ , it follows that these sets are also invariant under the action of the Zariski closure G0 of Ad∗s (G). Thus it follows from Proposition 5.5 that for each such set we obtain a stratification ∅ = U0 ⊆ U1 ⊆ · · · ⊆ Ul = R × F with the required properties.  We are now ready for the final step: Proof of Proposition 2.10. By the discussion at the beginning of this section we may assume without loss of generality that G is as in (5.1), i.e., we have connected closed normal subgroups N ⊆ H ⊆ G0 ⊆ G such that N is a nontrivial nilpotent group H/N is semi-simple and G0 /H is a torus group. For the induction step we have to show that every central extension 1 → T → Ḡ → G → 1 satisfies BC for C. Let N̄ , H̄ and Ḡ0 denote the inverse images of N , H and G0 in Ḡ. Then the sequence of normal subgroups N̄ ⊆ H̄ ⊆ Ḡ0 ⊆ Ḡ has the same general properties as the sequence N ⊆ H ⊆ G0 ⊆ G, in particular, N̄ is the nilradical of H̄ and H̄ is locally algebraic. Let T denote the central copy of T in Ḡ coming from the given central extension. We now divide the proof into the following cases: C(1) The center S = Z(N̄ ) of N̄ has dimension greater or equal to two. C(2) Z(N̄ ) = T . CONNES-KASPAROV CONJECTURE 25 We start with Case C(1): By Theorem 2.1 (and the discussion following that theorem) b K) where the action of it suffices to show that Ḡ/S satisfies BC with coefficients in C0 (S, b Ḡ/S on S is given by conjugation. By Theorem 3.6 it suffices to show that all stabilizers (Ḡ/S)χ = Ḡχ /S satisfy BC for K and that Sb has a nice stratification. While the latter follows from Proposition 5.8, the requirement on the stabilizers follows from Lemma 5.2, Remark 5.3, and the induction assumption since dim(Ḡχ /S) ≤ dim(Ḡ) − 2 < dim(G). This finishes the proof in Case C(1). For the proof of Case C(2) we have to do some more reduction steps in order to use the same line of arguments as in C(1). For this it is useful to consider the following two subcases: (2)a If Z̄(N ) denotes
the inverse image of the center Z(N ) of N in Ḡ, then Z(Z̄(N )) = T . (2)b dim Z(Z̄(N )) ≥ 2. In Case (2)a we consider the normal subgroup S = Z̄(N ) of G. Then S is a connected twostep nilpotent Lie group with one-dimensional center T , and therefore a Heisenberg group. It follows that Cr∗ (S) = C ∗ (S) can be written as the direct sum M C ∗ (S) = Aχ χ∈Tb with d ). Aχ ∼ = K, if χ 6= 1, and A1 = C0 (S/T Since Ḡ acts trivially on Tb, it follows that the decomposition action of Ḡ/S on C ∗ (S) ⊗ K induces an action on each fibre Aχ , and, by Theorem 2.1 together with Proposition 2.5, it follows that Ḡ satisfies BC with coefficients in C if Ḡ/S satisfies BC with coefficients in Aχ ⊗ K for each χ ∈ Tb. If χ 6= 1, we get Aχ ⊗ K ∼ = K, and the desired result follows from the induction assumption and the fact that dim(Ḡ/S) < dim(G). So we only have to deal with the case χ = 1, where we have to deal with the fibre d , K) = C0 (Z(N \), K). But here we are exactly in the same situation as in the proof of C0 (S/T \) factors through an action of G/N and all stabilizers Case C(1), since the action of Ḡ on Z(N of the characters of Z(N ) have dimension strictly smaller than dim(G). We have to work a bit more for the Proof of Case (2)b. Here we put S = Z(Z̄(N )). Then S is a connected abelian subgroup of N̄ and it follows from Lemma 5.2, Remark 5.3, the fact that dim(Ḡ/S) < dim(G) and the induction assumption that all stabilizers for the action of Ḡ/S on Sb satisfy BC for K. b Ḡ. For each χ ∈ Tb we define Again we study the structure of the orbit space S/ Sbχ = {µ ∈ Sb : µ|T = χ}. Since T is central in Ḡ, it follows that Ḡ acts trivially on Tb, and hence that Sbχ is Ḡ-invariant for all χ ∈ Tb. Since Tb is discrete, we may write M b K) ∼ C0 (S, C0 (Sbχ , K) = χ∈Tb with fiberwise action of Ḡ/S. Thus by continuity of BC it suffices to deal with the single d , and since fibers. For χ = 1 we are looking at the action of Ḡ/S ∼ = G/(S/T ) on Sb1 ∼ = S/T 26 CHABERT, ECHTERHOFF, AND NEST S/T is a central subgroup of N we may again argue precisely as in the proof of Case C(1) to see that Ḡ/S satisfies BC for C0 (Sb1 , K). In order to deal with the other fibers we are now going to show that Ḡ acts transitively on b Sχ for each nontrivial character χ ∈ Tb. It follows then directly from Corollary 2.9 that Ḡ/S satisfies BC for C0 (Sbχ , K). In fact, Lemma 5.9 below shows that N̄ already acts transitively on Sbχ for χ 6= 1 and the result will follow from that lemma.  The following lemma is certainly well known to the experts on the representation theory of nilpotent groups. For the readers convenience we give the elementary proof. Lemma 5.9. Assume that N is a connected nilpotent Lie group with one-dimensional center Z(N ) = T . Let S be a closed connected abelian normal subgroup of N such that T ⊆ S and S/T ⊆ Z(N/T ). Let 1 6= χ ∈ Tb and let Sbχ = {µ ∈ Sb : µ|T = χ}. Then N acts transitively on Sbχ by conjugation. Proof. We may assume without loss of generality that N is simply connected. In fact, if this is not the case, we pass to the universal covering group Ñ of N and the universal covering S̃ ⊆ Ñ of S and observe that there exists a discrete subgroup D ⊆ T̃ = Z(Ñ ) such that b̃ for all N = Ñ /D, S = S̃/D, T = T̃ /D and Sbχ can then be (equivariantly) identified with S χ b̃ b χ ∈ T ⊆ T. Let n, s and t denote the Lie algebras of N , S and T , respectively. Since N is simply connected, we can write N = {exp(X) : X ∈ n} with multiplication given by the Campbell-Hausdorff formula. In particular, if Y ∈ s, then exp(X) exp(Y ) = exp(X + Y + [X, Y ]) for all X ∈ n, since it follows from the assumption that S/T ⊆ Z(N/T ) that [X, Y ] ∈ t = z(n) and all commutators with [X, Y ] vanish. In particular, if we conjugate exp(Y ) by exp(X) we get the formula (5.2) exp(X) exp(Y ) exp(−X) = exp(Y + [X, Y ]) for all Y ∈ s. Assume now that dim(s) = n + 1 and let 0 6= Z ∈ t. There exists a basis {Y1 , . . . , Yn , Z} of s and elements X1 , . . . , Xn ∈ n such that (5.3) [Xi , Yi ] = Z and [Xi , Yj ] = 0 for all 1 ≤ i, j ≤ n, i 6= j. Indeed this follows from an easy Schmidt-orthogonalization procedure applied to the bilinear form (·, ·) : n × s → R; (X, Y ) = λ ⇔ [X, Y ] = λZ. We identify S with s (via exp) and Sb with s∗ . The conjugation action of N on Sb is then transferred to the coadjoint action Ad∗ . If {f1 , . . . , fn , g} is a dual basis for the basis {Y1 , . . . , Yn , Z} of s, the result will follow if we can show that Ad∗ (N )(λg) = span{f1 , . . . , fn } + λg CONNES-KASPAROV CONJECTURE 27 for all 0 6= λ ∈ R. By rescaling we may assume that λ = 1. But for λ1 , . . . , λn ∈ R we can compute   Ad∗ (exp(λ1 X1 + · · · + λn Xn ))(g) (Yi ) = g(Yi + λi [Xi , Yi ]) = λi g(Z) = λi .   Since Z is central in n it follows that Ad∗ (exp(X))(g) (Z) = g(Z) = 1 for all X ∈ n. Thus Ad∗ (exp(λ1 X1 + · · · + λn Xn ))(g) = λ1 f1 + · · · + λn fn + g.  6. The p-adic case In this section k denotes a finite algebraic extension of a field of p-adic numbers Qp , p prime. In what follows, by a k-group G we shall always understand a Zariski closed subgroup of GL(n, k). As pointed out in §3, all k-groups are exact. It is shown in [28] that all k-groups have a γ-element. Hence all k-groups G satisfy the general assumptions made in §3. Since k has characteristic zero, it follows from general structure theory (e.g., see [23, VIII, Theorem 4.3]) that G is a semidirect product N o R of the unipotent radical of G (which is a k-group) by some reductive k-subgroup R of G. We want to show by induction on the dimension of G (i.e., the dimension of the Lie-algebra g of G), that G satisfies the Baum-Connes conjecture with trivial coefficients. In fact, as in the real case, to perform the argument it is necessary to generalize the result a little bit, since we need to include certain actions of G on the algebra of compact operators K(H). We first need some information on the unitary representation theory of unipotent k-groups. For this let n denote the Lie-algebra of N , and let exp : n → N and log : N → n denote the exponential map and its inverse. Let n∗ denote the dual space of the underlying vector space of n and let Ad∗ : N → GL(n∗ ) denote the coadjoint representation of N on n∗ . By Kirillov’s theory, established for p-adic unipotent groups by Moore in [38], there exists a b of N as follows: bijection between the quotient space n∗ / Ad∗ (N ) and the unitary dual N Fix any character ε ∈ b k of order zero in the sense of [48, II, Definition 4]. For f ∈ n∗ , let mf be a maximal subalgebra of n such that f ([mf , mf ]) = {0}. Let Mf = exp(mf ) ⊆ N . Then m 7→ χf (m) := ε(f (log(m)) is a character of Mf and the induced representation πf := indN Mf χf is an irreducible representation of N , whose equivalence class does not depend on the choice of mf . The resulting map b ; f → πf n∗ → N b . By [24, is constant on Ad∗ (N )-orbits and induces a bijection between n∗ / Ad∗ (N ) and N ∗ ∗ Theorem II], this bijection is a homeomorphism. Since the Ad (N )-orbits in n are closed, it ∗ (N ) are closed, which implies that C ∗ (N ) is type I (cf [38, b = C\ follows that the points of N Theorem 4]). We are now going to specify the actions of G on K(H) which we want to include into our picture: For this suppose that G and G0 are k-groups such that G0 is an (algebraic) semi-direct product G0 = M o G with M a normal unipotent k-subgroup of G0 . Then Cr∗ (G0 ) = C ∗ (M ) o G and the unipotent radical N 0 of G0 equals M o N , where N denotes the unipotent radical of G. 28 CHABERT, ECHTERHOFF, AND NEST ∗ (M ) such that the unitary equivalence class of π is G-invariant. Since c∼ Let π ∈ M = C\ is type I, the action of G on C ∗ (M ) factors through an action of G on K(Hπ ), and we may define C ∗ (M ) Definition 6.1. Suppose that G is a k-group and that H is a Hilbert-space. An action of G on K(H) is called unipotent, if there exists an extension G0 = M o G as above, and a c such that H = Hπ and the action of G on K(Hπ ) is induced from G-invariant element π ∈ M ∗ the action of G on C (M ) as above. Now, the p-adic case of Theorem 1.1 will follow from Proposition 6.2. Let k be a finite extension of Qp and let G be a k-group acting unipotently on K(H). Then G satsfies BC for K(H). Proof. Let G be any k-group. If dim G = 1, then G is (almost) abelian, and the result is true by the general fact that all amenable groups satisfy BC for arbitrary coefficients ([22]). Assume now that dim(G) > 1. If G is reductive, then the result is true by Proposition 4.9. If G is not reductive, then G = N o R, where N is the unipotent radical of G and R is a reductive k-subgroup of G with dim(R) strictly smaller than the dimension of G. Let α : G → K(H) be any unipotent action of G. Since R is totally disconnected and since N is an amenable closed subgroup of G, it follows from Theorem 2.1 that it suffices to show that R satisfies BC for A := K(H) o N with respect to the canonical action of R on this algebra (since the extension of R by N is topologically split, there is no need to consider twisted actions at this point). Let G0 = M o G be a semidirect product of G by a unipotent k-group M as in Definition ∗ (M ) be G-invariant such that H = H and such that α is induced from 6.1, and let π ∈ C\ π ∗ the action of G on C (M ). Since π is also N -invariant, A = K(H) o N is the quotient of C ∗ (M o N ) ∼ oM : = C ∗ (M ) o N corresponding to the closed G-invariant subset L := {ρ ∈ N\ ∗ b ker ρ|M = ker π} (where the kernels are taken in C (M )). In particular, we have L = A. ∗ Moreover, since C (M ) is type I, the same is true for A. Now write N 0 = M o N and let n0 denote the Lie algebra of N 0 . Let Ad∗ : G0 → GL((n0 )∗ ) denote the adjoint action of G0 on (n0 )∗ . Since this is an algebraic action, all Ad∗ (G0 )-orbits in (n0 )∗ are locally closed in the Zariski topology, and therefore also in the Hausdorff topology. c0 let f ∈ (n0 )∗ such that ρ = πf . Then the R-orbit R(ρ) ⊆ N c0 corresponds to the For ρ ∈ N c0 (e.g., see [37, §4]). Since the Ad∗ (G0 )-orbit of f under the Kirillov correspondence for N 0 ∗ c0 , it follows that Kirillov correspondence is a homeomorphism between (n ) / Ad∗ (N 0 ) and N ∗ 0 ∗ 0 0 c c0 N /R is homeomorphic to (n ) / Ad (G ). In particular, it follows that every R-orbit in N c0 and on A c0 is countable separated b⊆N is locally closed. It follows that the action of R on N (see Remark 3.5). By Glimm’s Theorem ([20, Theorem]) we obtain an ascending family {Uν }ν of open subsets c0 satisfying the conditions as described in item (3) of Remark 3.5. Moreover, since all of N difference sets Uν+1 rUν are orbit spaces of locally compact subsets of the totally disconnected space (n0 )∗ , it follows that all these difference sets are totally disconnected, and hence the Hausdorff quotients (Uν+1 r Uν )/R are totally disconnected. Thus, taking the intersections b ν , we obtain an ascending family {UνA }ν of A b satisfying the conditions of Theorem UνA := A∩U 3.6. CONNES-KASPAROV CONJECTURE 29 Using Theorem 3.6 the proposition will follow as soon as we have checked that for all b ⊆ C ∗ (N 0 )), the stabilizer Rρ of ρ in R satisfies BC for ρ ∈ C ∗ (N 0 ) (and hence for all ρ ∈ A K(Hρ ). By [15, Lemme 12], if ρ = πf for some f ∈ (n0 )∗ , then Rρ = G0f ∩R, where G0f denotes the stabilizer of f ∈ n∗ in G0 . Since the action of G0 on n∗ is algebraic, G0f and hence Rρ is a k-group with dimension strictly smaller than dim(G). Since the action of Rρ on K(Hρ ) is clearly unipotent, the result follows from our induction assumption.  7. Relations to the K-theory of the maximal compact subgroup In this section we want to describe the relations between the K-theory of Cr∗ (G) and the K-theory of C ∗ (L), where L denotes the maximal compact subgroup of the almost connected group G (we chose the letter L to avoid confusion). We should mention that all results presented here (except the conclusions drawn out of our main theorem) are well known, but since they have important impact on our results, we found it useful to give at least a brief report. The main references for these results are [13, 25], and we refer especially to [13, §4] for a more geometric discussion of some the results presented in this section. If G and L are as above, it follows from work of Abels (see [1]) that G/L is a universal proper G-space. Thus we have Ktop ∗ (G, A) resG L ∼ ∼L = KKG ∗ (C0 (G/L), A) = KK∗ (C0 (G/L), A), where the second isomorphism follows from [27, Corollary to Theorem 5.7]. Also by the work of Abels [1], G/L is a Riemannian manifold which is L-equivariantly diffeomorphic to the tangent space V := TeL equipped with the adjoint action of L on V . It follows then from Kasparov’s work in [25] (see [9, Lemma 7.7] for a more extensive discussion) that tensoring with C0 (V ) gives a natural isomorphism L σC0 (V ) : KKL ∗ (C0 (V ), A) → KK∗ (C0 (V ) ⊗ C0 (V ), A ⊗ C0 (V )), and by Kasparov’s Bott-periodicity theorem (see [25, Theorem 7]) we know that C0 (V ) ⊗ C0 (V ), equipped with the diagonal action, is KKL -equivalent to C (but see also the discussion below). Thus we obtain the following chain of isomorphisms σC (V ) 0 L ∼ ∼ KKL = KKL ∗ (C0 (G/L), A) = KK∗ (C0 (V ), A) ∗ (C0 (V )⊗C0 (V ), A⊗C0 (V ))
L ∼ = KK (C, A⊗C0 (V )) = K∗ (A⊗C0 (V )) o L , ∗ where the last isomorphism follows from the Green-Julg theorem. Hence, as a direct consequence of Theorem 1 we can deduce Theorem 7.1. Assume that G is an almost connected (second countable) group with maximal compact subgroup L. Let K = K(H) be the algebra of compact operators on the separable Hilbert space H equipped with any action of G. Then K∗ (K or G) is naturally isomorphic to
K∗ (K⊗C0 (V )) o L . By Kasparov’s Bott-periodicity theorem (see [25, Theorem 7]) it follows that C0 (V ) is KKL -equivalent to the graded complex Clifford algebra Cl(V ) (with respect to a compatible inner product on V ), equipped with the action of L induced by the given action on V . So we can replace C0 (V ) by the graded C ∗ -algebra Cl(V ), but then we have to use graded K-theory! 30 CHABERT, ECHTERHOFF, AND NEST Let us look a bit closer to the implications of this Bott-periodicity theorem. Assume for the moment that V is even dimensional and that the action of L on V preserves a given orientation of V , i.e., the action factors through a homomorphism ϕ : L → SO(V ). We have a central extension 0 → T → Spinc (V ) → SO(V ) → 0 of SO(V ), where Spinc (V ) ⊆ CL(V ) denotes the group of complex spinors (e.g. see [2]). The corresponding action of L on Cl(V ) is given by the homomorphism L → SO(V ) ∼ = Spinc (V )/T = Ad(Spinc (V )). Now choose a fixed orthonormal base {e1 , . . . , en } of V . Then the grading of Cl(V ) is given by conjugation with the symmetry J = e1 ·e2 · · · en ∈ Cl(V ). One can show that, up to a sign, J does not depend on the choice of this basis, and the sign only depends on the orientation of the basis. In particular, J is invariant under conjugation with elements in Spinc (V ). From this it follows that the graded L-algebra Cl(V ) is L-equivariantly Morita equivalent to the trivially graded L-algebra Cl(V ) – a Morita equivalence is given by the module Cl(V ) with given L-action and grading automorphism given by left multiplication with J. Moreover, since n = dim(V ) is even, Cl(V ) is isomorphic to the simple matrix algebra M2n (C). Assume now that dim(G/L) is odd. Then, replacing G by G × R (with trivial action of R on K) we get K∗ (K or G) = K∗+1 (K or (G × R)). Moreover, if the action of L on V = TeL is orientation preserving, the same is true for the resulting action of L on V × R, which we identify with the tangent space at eL in the group G × R. Hence, modulo a dimension shift, we can use the above considerations also for this case. Thus, as a consequence of Theorem 7.1 we obtain Theorem 7.2. Assume that G is an almost connected group with maximal compact subgroup L such that the adjoint action of L on V = TeL is orientation preserving. Then there are natural isomorphisms
K∗ (K or G) ∼ = K∗ (K ⊗ Cl(V )) o L if dim(G/L) is even and
K∗+1 (K or G) ∼ = K∗ (K ⊗ Cl(V × R)) o L if dim(G/L) is odd. Here all algebras are trivially graded! Perhaps, the above result has its most satisfying formulation if translated into the language of twisted group algebras. For this let ω ∈ Z 2 (G, T) denote a representative of the Mackey obstruction for the action of G on K (see the discussion preceding Lemma 2.3). Then K or G is isomorphic to Cr∗ (G, ω) ⊗ K, where Cr∗ (G, ω) denotes the reduced twisted group algebra Cr∗ (G, ω) (e.g., see [21, Theorem 18]). Recall that Cr∗ (G, ω) can be defined either as the reduced twisted crossed product C or (Gω , T) with respect to the twisted action (id, χ1 ) (which, by Lemma 2.3, is Morita equivalent to the given action on K), or as the completion of L1 (G) ⊆ B(L2 (G)), where L1 (G) acts on L2 (G) by the twisted convolution Z f ∗ ξ(s) = f (t)ω(t, t−1 s)ξ(t−1 s) dt, f ∈ L1 (G), ξ ∈ L2 (G). G CONNES-KASPAROV CONJECTURE 31 Up to isomorphism, Cr∗ (G, ω) only depends on the class [ω] ∈ H 2 (G, T). Conversely, given any cocycle, the representation λω : G → U (L2 (G)) given by
λω (t)ξ (s) = ω(t, t−1 s)ξ(t−1 s) determines an action of G on K(L2 (G)) with Mackey obstruction represented by ω. Note that the Mackey obstruction for the action of L on K is given by the restriction of ω to L and the obstruction for the action of L on Cl(V ) ∼ = M2n (C) (if dim(V ) is even) is given by the pull-back, say µL , to L of a cocycle representing the central extension 1 → T → Spinc (V ) → SO(V ) → 1. Since Spinc (V ) ∼ = (T × Spin(V ))/Z2 (diagonal action), where 1 → Z2 → Spin(V ) → SO(V ) → 1 is the real group of spinors, the cocycle µL can be chosen to take values in the subgroup Z2 ⊆ T, and therefore µ2L = 1. Note that µL is trivial if and only if the homomorphism ϕ : L → SO(V ) factorizes through Spinc (V ) (i.e., if and only if G/L carries a G-invariant Spinc -structure). If dim(V ) is odd, we may define µL in the same way as above, noticing that this cocycle is equivalent to the pull back of (a cocycle representing) the extension 1 → T → Spinc (V × R) → SO(V × R) → 1, which follows from the fact that L acts trivially on R! Since the Mackey obstruction of a tensor product of actions is the product of the Mackey obstructions of the factors, we obtain Theorem 7.3. Assume that G is an almost connected group with maximal compact subgroup L such that the adjoint action of L on V = TeL is orientation preserving. Let n = dim(G/L) and let ω ∈ Z 2 (G, T) be any cocycle on G. Then

K∗ C ∗ (G, ω) ∼ = K∗+n C ∗ (L, ω · µL ) . r In particular, in the special case where ω is trivial, we obtain an isomorphism

K∗ Cr∗ (G) ∼ = K∗+n C ∗ (L, µL ) . Again, µL is trivial if and only if G/L carries a G-invariant Spinc -structure. In general, since C ∗ (L, ω · µL ) is the quotient of the central extension Lω·µL of L by T corresponding to the character χ1 of T, it follows that C ∗ (L, ω · µL ) is a direct sum of (possibly infinitely many) matrix algebras. Thus as a direct corollary of the above result we obtain:
Corollary 7.4. Assume that G, L and ω are as in Theorem 7.3. Then K0+n Cr∗ (G, ω) is isomorphic
to a free abelian group in at most countably many generators and K1+n Cr∗ (G, ω) = {0}. This result has interesting consequence towards the question of existence of square integrable representations of connected unimodular Lie groups. In fact, combining the above corollary with [44, Theorem 4.6] gives: Corollary 7.5 (cf [44, Corollary 4.7]). Let G be a connected unimodular Lie group. Then all square-integrable factor representations of G are type I. Moreover, G has no square-integrable factor representations if dim(G/L) is odd, where L denotes the maximal compact subgroup of G. 32 CHABERT, ECHTERHOFF, AND NEST We refer to [44] for more detailed discussions on this kind of applications of the positive solution of the Connes-Kasparov conjecture. Note that Theorem 7.3 and Corollary 7.4 do not hold in general without the assumption that the action of L on V = TeL is orientation preserving. In fact an easy six-term-sequence argument shows that it cannot hold for the group G = R o Z2 , where Z2 acts on R by reflection through 0. References [1] H. Abels. Parallelizability of proper actions, global K-slices and maximal compact subgroups, Math. Ann. 212, 1–19 (1974). [2] M. Atiyah, R. Bott and A. Shapiro. Clifford Modules, Topology 3, Suppl. 1, 3-38 (1964). [3] P. Baum, A. Connes and N. Higson. Classifying space for proper actions and K-theory of group C ∗ algebras, Contemp. Math. 167, 241-291 (1994). [4] B. Blackadar. K-theory for operator algebras, MSRI pub., 5, (1986), Springer Verlag. [5] E. Blanchard. Deformations de C ∗ -algebres de Hopf, Bull. Soc. Math. Fr. 124, 141-215 (1996). [6] J. Bochnak, M. Coste, and M.-F. Roy. Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete Vol. 36, Springer 1998. [7] A. Borel and Harish-Chandra. Arithmetic subgroups of algebraic groups. Annals of Math. 75 (1962), 485–535. [8] A. Borel. Linear Algebraic Groups. Springer Verlag GTM 126 (1991). [9] J. Chabert and S. Echterhoff. Twisted equivariant KK-theory and the Baum-Connes conjecture for group extensions, K-Theory 23 (2001), 157–200. [10] J. Chabert and S. Echterhoff. Permanence properties of the Baum-Connes conjecture, Doc. Math. 6, 127–183 (2001). [11] J. Chabert, S. Echterhoff, and Ralf Meyer. Deux remarques sur la conjecture de Baum-Connes, C. R. Acad. Sci., Paris, Ser. I 332, Série I, 607–610 (2001). [12] C. Chevalley. Theorie des groupes de Lie. Groupes algebriques. Theoremes generaux sur les algebres de Lie. 2ieme ed. Hermann & Cie. IX, Paris 1968. [13] A. Connes and H. Moscovici. The L2 -index theorem for homogeneous spaces of Lie groups, Ann. Math., II. Ser. 115, 291-330 (1982). [14] J. Dixmier. C ∗ -algebras (English Edition). North Holland Publishing Company 1977. [15] M. Duflo. Théorie De Mackey pour les groupes de Lie algébriques. Acta Math 149 (1983), 153–213. [16] S. Echterhoff. On induced covariant systems. Proc. Amer. Math. Soc. 108 (1990), 703–708. [17] S. Echterhoff. Morita equivalent actions and a new version of the Packer-Raeburn stabilization trick. J. London Math. Soc. (2), 50 (1994), 170–186. [18] G. Elliott, T. Natsume, R. Nest. The Heisenberg group and K-theory, K-Theory 7, 409-428 (1993). [19] J. Fell. The structure of algebras of operator fields, Acta Math. 106, 233-280 (1961). [20] J. Glimm. Locally compact transformation groups, Trans. Am. Math. Soc. 101, 124-138 (1961). [21] P. Green. The local structure of twisted covariance algebras. Acta. Math., 140 (1978), 191–250. [22] N. Higson and G. Kasparov, E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144, 23–74, (2001). [23] G.P. Hochschild Basic theory of algebraic groups and Lie algebras. Springer Verlag GTM 75 (1981). [24] R. Howe. The Fourier transform for nilpotent locally compact groups: I. Pac. J. Math. 73 (1977), 307–327. [25] G. Kasparov. The operator K-functor and extensions of C ∗ -algebras, Math. USSR Izvestija, Vol. 16, No. 3 (1981) 513–572. [26] G. Kasparov. K-theory, group C ∗ -algebras, higher signatures (Conspectus). in: Novikov conjectures, index theorems and rigidity. London Math. Soc, lecture notes series 226 (1995), 101–146. [27] G. Kasparov. Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91, (1988) 147-201. [28] G. Kasparov and G. Skandalis, Groups acting properly on “bolic” spaces and the Novikov conjecture. To appear in Ann. Math. CONNES-KASPAROV CONJECTURE 33 [29] E. Kirchberg and S. Wassermann. Exact groups and continuous bundles of C ∗ -algebras, Math. Ann. 315, 169-203 (1999). [30] E. Kirchberg and S. Wassermann. Permanence properties of C ∗ -exact groups. Doc. Math. 5 (2000), 513– 558. [31] H. Kraft, P. Slodowy, and T.A. Springer. Algebraische Transformationsgruppen und Invariantentheorie. DMV-Seminar Band 13, Birkhäuser 1989. [32] V. Lafforgue, K-théory bivariante pour les algèbres de Banach et conjecture de Baum-Connes, PhD Dissertation, Universite Paris Sud, (1999). [33] V. Lafforgue, K-théory bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. math. 149 (2002), 1–95. [34] V. Lafforgue, Banach KK-Theory and the Baum-Connes conjecture, Progress in Mathematics 202 (2001), 31–46. [35] V. Lafforgue, ICM-lecture. [36] R.Y. Lee, On the C span ∗-algebras of operator fields. Indiana Univ. Math. J. 25 (1976), 303–314. [37] G. Lion and P. Perrin Extension des Representations de groupe unipotents p-adiques. Calculs d’obstructions. Springer Lecture Notes in Math. 880 (1981), 337–356. [38] C. Moore. Decomposition of unitary representations defined by discrete subgroups of nilpotent groups. Ann. of Math., 82 (1965), 146–182. [39] G. Mackey. Borel structure in groups and their duals, Trans. Am. Math. Soc. 85, 134-165 (1957). [40] D. Montgomery and L. Zippin. Topological transformation groups. Interscience Tracts in Pure and Applied Mathematics. New York: Interscience Publishers, Inc. XI (1955). [41] J. Packer and I. Raeburn. Twisted crossed products of C∗-algebras, Math. Proc. Camb. Philos. Soc. 106, 293-311 (1989). [42] G.K. Pedersen. C ∗ -Algebras and their Automorphism Groups. Academic Press, London, 1979. [43] Lajos Pukánszky. Characters of connected Lie groups. Mathematical surveys and Monographs Vol 71. American Mathematical Society, Rhode Island 1999. [44] J. Rosenberg. Group C ∗ -algebras and topological invariants. In: Operator algebras and group representations, Proc. int. Conf., Neptun/Rom. 1980, Vol. II, Monogr. Stud. Math. 18, 95-115 (1984). [45] M. Rosenlicht. A remark on quotient spaces. An. Acad. Brasil. Ciênc. 35 (1963), 487–489. [46] J.L. Tu. La conjecture de Novikov pour les feuilletages hyperboliques, K-theory, 16, No.2, 129-184 (1999). [47] A. Wassermann. Une demonstration de la conjecture of Connes-Kasparov pour les groupes de Lie lineaires connexes reductifs, C. R. Acad. Sci., Paris, Ser. I 304, 559-562 (1987). [48] A. Weil Basic number theory. Die Grundlehren der Mathematischen Wissenschaften, Band 144. SpringerVerlag, New York-Berlin, 1974. Jérôme Chabert: Université Blaise Pascal, Bât. de mathématiques, 63177, Aubière, France Westfälische Wilhelms-Universität Münster, Mathematisches Institut, Einsteinstr. 62 D48149 Münster, Germany Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark E-mail address: chabert@math.univ-bpclermont.fr, echters@math.uni-muenster.de, rnest@math.ku.dk View publication stats