The Kadison-Kaplansky conjecture for word-hyperbolic groups

Abstract

In this paper we prove the Kadison-Kaplansky idempotent conjecture for torsion-free word-hyperbolic groups. The conjecture asserts that the following equivalent statements hold for a torsion-free discrete group Γ: • The reduced group C∗-algebra C∗ r (Γ) contains no idempotents except 0 and 1. • The spectrum of every element of the reduced group C∗-algebra is connected. • The canonical trace on C∗ r (Γ) takes integer values on idempotents. The last assertion can be viewed as a statement about the pairing between the K-theory and the (local) cyclic cohomology of the group C∗-algebra. It is in this setting that we will prove the conjecture. Our proof is based on a partial analysis of the assembly maps in K-theory and local cyclic homology. We compare these assembly maps by means of an equivariant bivariant Chern-Connes character. Before going into details, we recall some previous work on the conjecture. The first progress was achieved by Pimsner and Voiculescu [PV] who proved the Kadison-Kaplansky conjecture for free groups as a consequence of their computation of the K-theory of the group C∗-algebra. Subsequently it was realized that more generally the Kadison-Kaplansky conjecture was a consequence of the Baum-Connes conjecture which gives a geometric description of K∗(C∗ r (Γ)) for any torsion-free discrete group. In fact the Baum-Connes conjecture states that the K-theoretic assembly map