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 theK-theory of this C^*-algebra to the real numbers. Thi s paper studies the range of this homomorphism. For torsion-free groups, the Baum-Connes conjecture, together with Atiyah's L²-index theorem, implies that the range consists of the integers.

We give a direct and elementary proof that if G acts on a tree and admits a homomorphism α to another group H whose restriction α|_Gv to every stabilizer group of a vertex is injective, then

tr_GG(K(C^*_r H))⊂tr_H(K(C^*_rH)).

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_*UH of two copies of the same groups along a subgroup U.