Some groups whose reduced \(C^*\)-algebras have stable rank one
type: Item
instance of: scholarly article
title: Some groups whose reduced \(C^*\)-algebras have stable rank one
zbMATH Open document ID: 0929.22008
DOI: 10.1016/S0021-7824(99)00015-X
author: Ken Dykema
author: Pierre de la Harpe
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2000-01-05T00:00:00Z
review text: The authors prove that the reduced group \(C^\ast\)-algebra \(C^\ast_{\text{red}}(\Gamma)\) of a discrete group \(\Gamma\) has stable rank one if \(\Gamma\) has (1) the free semigroup property and (2) the \(\ell^2\)-spectral radius property. This means that for any finite subset \(F\subset\Gamma\) there is \(\gamma\in\Gamma\) such that (1) the semigroup generated by \(\gamma F\) is free over \(F\) and that (2) for any \(a\in\text{span } \gamma F\) the spectral radius can be computed from the trace norm of \(\Gamma\). In particular, this holds for amalgamated free products \(\Gamma=G_1\ast_H G_2\) with certain finite \(H\) and for hyperbolic groups which are either torsion-free and non-elementary or represented as a cocompact lattice in a real, noncompact, simple, connected Lie group of real rank 1 and trivial center. Finally, they also discuss the real rank and formulate some open problems.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 22D25
zbMATH DE Number: 1344452
zbMATH Keywords: stable rank
zbMATH Keywords: reduced group \(C^\ast\)-algebra
zbMATH Keywords: free products with amalgamation
zbMATH Keywords: hyperbolic groups
MaRDI profile type: Publication
full work available at URL: s0021-7824(99)00015-x
OpenAlex ID: W1973284092
description: scientific article; zbMATH DE number 1344452