Weyl Character Formula in KK-Theory

We formulate a version of Baum-Connes' conjecture for a discrete quantum group, building on our earlier work ([9]). Given such a quantum group A, we construct a directed family {E F } of C * -algebras (F varying over some suitable index set), borrowing the ideas of [5], such that there is a natural action of A on each E F satisfying the assumptions of [9], which makes it possible to define the "analytical assembly map", say µ r,F i , i = 0, 1, as in [9], from the A-equivariant K-homolgy groups of E F to the K-theory groups of the "reduced" dual r (c.f.

Journal of Functional Analysis, 2005

Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an "addition formula" for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs are provided, one using topology and the other one, surprisingly involved, sticking to analysis, on the basis of the previous result. As a second application, we construct, in a purely analytical language, various homomorphisms linking the homology of a group in low degree, the Khomology of its classifying space and the analytic K-theory of its C * -algebra, in close connection with the Baum-Connes assembly map. For groups classified by a 2-complex, this allows to reformulate the Baum-Connes Conjecture.

Inventiones Mathematicae, 2002

We prove a version of the L 2-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring Z ⊂ Λ G ⊂ Q obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K0(C * r (G)) → R takes values in Λ G. The original Trace Conjecture predicted that its image lies in the additive subgroup of R generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [13].

Topology, 1967

Annales de l’institut Fourier, 2015

In this paper, we develop a quantitative K-theory for filtered C *-algebras. Particularly interesting examples of filtered C *-algebras include group C *-algebras, crossed product C *-algebras and Roe algebras. We prove a quantitative version of the six term exact sequence and a quantitative Bott periodicity. We apply the quantitative K-theory to formulate a quantitative version of the Baum-Connes conjecture and prove that the quantitative Baum-Connes conjecture holds for a large class of groups.

Publications mathématiques de l'IHÉS, 2003

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. This research has been supported by the Deutsche Forschungsgemeinschaft (SFB 478). where X runs through the G-compact subspaces of E(G) (i.e., X/G is compact) ordered by inclusion, and KK G * (C 0 (X), A) denotes Kasparov's equivariant KK-theory. If A = C, we simply write K top * (G) for K top * (G, C). The construction of Baum, Connes and Higson presented in [3, §9] determines a homomorphism , 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 µ A 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/G 0 satisfies BC for arbitrary coefficients, where G 0 denotes the connected component of G. (By the results of Higson and Kasparov [22] this is in particular true if G/G 0 is amenable or, more general, if G/G 0 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 K top * (G, A) 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 K top * (G, A) can be computed by means of the K-theory of the crossed product A L. We give a brief discussion of these relations in §7 below. As was already pointed out in [44], our results have interesting We dedicate this paper to the memory of Peter Slodowy, who lost his fight against cancer in November 2002. 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 . 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 there exists a twisted action of (G, N ) on B r N such that the twisted crossed product (B r N ) r (G, N ) is canonically isomorphic to B r G. Moreover, we can use the twisted equivariant KK-theory of [9] to define the topological K-theory K top * (G/N, B r N ) with respect to the twisted action of (G, N ) on B r N , and a twisted version of the assembly map µ B r N : K top * (G/N, B r N ) → K * ((B r N ) r (G, N )). In [9] we constructed a partial assembly map µ G N,B : K top * (G, B) → K top * (G/N, B r N ) such that the following diagram commutes K top * (G, B) µ G N,B ----→ K top * (G/N, B r N ) µ B

Glasgow Mathematical Journal, 2017

We provide a new computation of the K-theory of the group C *algebra of the solvable Baumslag-Solitar group BS(1, n) (n = 1); our computation is based on the Pimsner-Voiculescu 6-terms exact sequence, by viewing BS(1, n) as a semi-direct product Z[1/n] Z. We deduce from it a new proof of the Baum-Connes conjecture with trivial coefficients for BS(1, n).

Duke Mathematical Journal, 2001

The canonical trace on the reduced C *

American Journal of Mathematics, 2005

Rognes and Weibel used Voevodsky's work on the Milnor conjecture to deduce the strong Dwyer-Friedlander form of the Lichtenbaum-Quillen conjecture at the prime 2. In consequence (the 2-completion of) the classifying space for algebraic K-theory of the integers Z[1/2] can be expressed as a fiber product of well-understood spaces BO and BGL(F 3 ) + over BU. Similar results are now obtained for Hermitian K-theory and the classifying spaces of the integral symplectic and orthogonal groups. For the integers Z[1/2], this leads to computations of the 2-primary Hermitian K-groups and affirmation of the Lichtenbaum-Quillen conjecture in the framework of Hermitian K-theory. 0. Introduction. In [9], Bökstedt introduced the study of the commuting square

2007

Using the analytic assembly map that appears in the Baum-Connes conjecture in noncommutative geometry, we generalise the $\Spin^c$-version of the Guillemin-Sternberg conjecture that `quantisation commutes with reduction' to (discrete series representations of) semisimple groups $G$ with maximal compact subgroups $K$ acting cocompactly on symplectic manifolds. We prove this statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements, the set of elements of $\g^*$ with compact stabilisers. This assumption on the image of the momentum map is equivalent to the assumption that $M = G \times_K N$, for a compact Hamiltonian $K$-manifold $N$. The proof comes down to a reduction to the compact case. This reduction is based on a `quantisation commutes with induction'-principle, and involves a notion of induction of Hamiltonian group actions. This principle, in turn, is based on a version of the naturality of the assembly map for the inclusion of $K$ into $G$.