Some groups whose reduced \(C^*\)-algebra is simple
type: Item
instance of: scholarly article
title: Some groups whose reduced \(C^*\)-algebra is simple
zbMATH Open document ID: 0827.22001
DOI: 10.1007/BF02698898
published in: Publications Mathématiques
publication date: 1995-11-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104098
review text: Group \(C^*\)-algebras play vital roles, both in \(C^*\)-algebra theory and in harmonic analysis on locally compact groups. Therefore, the question of when the reduced \(C^*\)-algebra of a discrete group \(\Gamma\), \(C^*_r(\Gamma)\), is simple is of significance and has long been a matter of interest. The first milestone in the area was Powers' theorem that if \(\Gamma\) is a non-abelian free group, then \(C^*_r(\Gamma)\) is simple and has a unique normalized trace. Subsequently, this has been generalized by several authors. In the paper under review the authors obtain further important results. Let \(G\) be a connected real semisimple Lie group without compact factors and with trivial centre, and let \(\Gamma\) be a lattice in \(G\). The authors verify the long standing conjective that \(C^*_r (\Gamma)\) is simple (and has a unique normalized trace). Actually, they prove a considerably stronger result. Namely, let \(G\) be as above, suppose that \(\Gamma\) is a Zariski dense subgroup of \(G\) and endow \(\Gamma\) with the discrete topology. Then \(C^*_r (\Gamma)\) is simple and has a unique normalized trace (Theorem 1). Notice that by Borel's density theorem, Theorem 1 in particular applies to lattices. Moreover, Theorem 1 implies the following extension of a result of Howe and Rosenberg. Let \(G\) be a connected semisimple algebraic group defined over a field \(k\) of characteristic zero and with trivial centre. Let \(\Gamma\) be the group of \(k\)-rational points of \(G\), equipped with the discrete topology. Then \(C^*_r(\Gamma)\) is simple. In addition, there are a generalization of Theorem 1 to reduced crossed product \(C^*\)-algebras as well as some other interesting results. A crucial property for the discrete group \(\Gamma\) studied in the paper, which implies simplicity of \(C^*_r (\Gamma)\), is the following property \((P)\). A discrete group \(\Gamma\) with left regular representation \(\lambda_\Gamma\) is said to have property \((P)\) if given any fin…
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E32
zbMATH DE Number: 753835
zbMATH Keywords: locally compact groups
zbMATH Keywords: reduced \(C^*\)-algebra
zbMATH Keywords: discrete group
zbMATH Keywords: connected real semisimple Lie group
zbMATH Keywords: lattice
zbMATH Keywords: Zariski dense subgroup
zbMATH Keywords: Borel's density theorem
zbMATH Keywords: connected semisimple algebraic group
zbMATH Keywords: group of \(k\)-rational points
zbMATH Keywords: reduced crossed product \(C^*\)-algebras
zbMATH Keywords: simplicity
author: Bachir Bekka
author: Michael G. Cowling
author: Pierre de la Harpe
reviewed by: Eberhard Kaniuth
MaRDI profile type: Publication
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OpenAlex ID: W1970376392
description: scientific article; zbMATH DE number 753835