The Kadison-Kaplansky conjecture for word-hyperbolic groups
The Kadison-Kaplansky conjecture states the following: Let \(\Gamma\) be a torsion-free group and let \(C^*_r(\Gamma)\) be its reduced \(C^*-\)algebra, i.e. the closure in operator norm of the group ring \({\mathbb C}\Gamma\) acting by convolution on the Hilbert space \(\ell^2(\Gamma)\). Then the following equivalent properties hold: 1) \(C^*_r(\Gamma)\) contains no idempotents except 0 and 1. 2) The spectrum of every element of \(C^*_r(\Gamma)\) is connected. 3) The canonical trace on \(C^*_r(\Gamma)\) takes integer values on idempotents. In this paper, the author proves the following theorem : Let \(\Gamma\) be a torsion-free word hyperbolic group. Then the Kadison-Kaplansky conjecture holds for \(\Gamma\). In other words, the three equivalent properties above are satisfied for such a group \(\Gamma\). The author's proof is based on analysis of the assembly maps in \(K-\)theory and local cyclic homology. He compares these assembly maps by means of an equivariant bivariant Chern-Connes character.
scientific article; zbMATH DE number 1965423
scientific article