The Kadison-Kaplansky conjecture for word-hyperbolic groups
type: Item
instance of: scholarly article
title: The Kadison-Kaplansky conjecture for word-hyperbolic groups
zbMATH Open document ID: 1019.22002
author: Michael Puschnigg
published in: Inventiones Mathematicae
publication date: 2003-08-17T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 22D15
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F67
zbMATH DE Number: 1965423
zbMATH Keywords: Kadison-Kaplansky conjecture
zbMATH Keywords: word-hyperbolic group
zbMATH Keywords: cyclic homology
zbMATH Keywords: Chern-Connes character
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: s002220200216
OpenAlex ID: W2124387428
Wikidata QID: Q123199684
DOI: 10.1007/S002220200216
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description: scientific article; zbMATH DE number 1965423