The Baum-Connes Vermutung and the Trace-Conjecture

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 $zzsubsetLambd{a}^{G}subsetqq$ 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 modifiedTrace Conjecture which says that the image of the standard trace $K_0(C_r^{\ast }(G))or$ takes values in $Lambd{a}^G$. The original Trace Conjecture due to Baum and Connespredicted 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 disproven by Ranja Roy recently.