### The trace on the K-theory of group C*-algebras, by Thomas Schick

By Thomas Schick, Penn State and Uni Muenster.

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 \alpha to 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.

Thomas Schick <thomas.schick@math.uni-muenster.de>