The relation between the Baum-Connes Conjecture and the Trace Conjecture, by Wolfgang Lueck

This paper has been revised on March 22, 2001. The change consists of adding a new item (c) to Lemma 3.4 and its proof. This fills a gap in the proof of Theorem 3.5. The rest is unchanged.

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 subring R of the rationals which is 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 from the zero-th K-group of the reduced group C^*-algebra K_0(C^*_r(G)) to the reals takes values in R. (We actually show that the image of the composite of the trace map with the Baum-Connes assembly map lies in R.) The original Trace Conjecture predicted that its image lies in the additive subgroup of the rationals generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy in 1999.

Wolfgang Lueck <>