The trace on the K-theory of group C*-algebras

Abstract

The canonical trace on the reduced C*-algebra of a discrete group gives rise to a homomorphism from the K-theory of this C^*-algebra to the real numbers. This paper addresses the range of this homomorphism. For torsion free groups, the Baum-Connes conjecture and Atiyah's L2-index theorem implies that the range consists of the integers. If the group is not torsion free, Baum and Connes conjecture that the trace takes values in the rational numbers. We give a direct and elementary proof that if G acts on a tree and admits a homomorphism αto another group H whose restriction to every stabilizer group of a vertex is injective, then the range of the trace for G, tr_G(K(C_r^*G)) is contained in the range of the trace for H, tr_H(K(C_r^*H)). This follows from a general relative Fredholm module technique. Examples are in particular HNN-extensions of H where the stable letter acts by conjugation with an element of H, or amalgamated free products G=H*_U H of two copies of the same groups along a subgroup U.