The relation between the Baum-Connes Conjecture and the Trace Conjecture

We prove a version of the L2-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 ⊂ ΛG ⊂ 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 ΛG . 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 [15]. 0. Introduction and statements of results Throughout this paper let G be a discrete group. The Baum-Connes Conjecture for G says that the assembly map asmb : K G 0 (EG) → K0(C∗ r (G)) from the equivariant K -homology of the classifying space for proper Gactions EG to the topological K -theory of the reduced C∗-algebra C∗ r (G) is bijective [3, page 8], [5, Conjecture 3.1]. In connection with this conjecture Baum and Connes [3, page 21] also made the sometimes so called Trace Conjecture. It says that the image of the composition K0(C ∗ r (G)) i −→K0(N (G)) trN (G) −−−→ R Mathematics Subject Classification (2000): 19L47, 19K56, 55N91