0 The trace on the K - theory of group C ∗ - algebras

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 studies the range of this homomorphism. For torsion free groups, the Baum-Connes conjecture together with Atiyah’s L-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 trG(K(C ∗ rG)) ⊂ trH(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 productsG = H∗UH of two copies of the same groups along a subgroup U . MSC: 19K (primary); 19K14, 19K35, 19K56 (secondary)