\(K\)-theory for \(C^*\)-algebras of one-relator groups
type: Item
instance of: scholarly article
title: \(K\)-theory for \(C^*\)-algebras of one-relator groups
zbMATH Open document ID: 0932.46063
DOI: 10.1023/A:1007755408585
author: Cédric Béguin
author: Hela Bettaieb
author: Alain Valette
published in: \(K\)-Theory
publication date: 2000-02-28T00:00:00Z
review text: The authors compute the K-theory groups of the reduced \(C^*\)-algebra \(C^*_r({\Gamma})\) of a one-relator group \({\Gamma}\) and they prove that every such group is \(K\)-amenable in the sense of Cuntz. For a torsion-free one-relator group \({\Gamma}=\langle X|r\rangle\) such that \(r\) is not a product of commutators a direct proof is given of the fact that the Baum-Connes analytical assembly map \({\mu}_i^{\Gamma}:K_i(B{\Gamma})\to K_i(C^*_r({\Gamma}))\) for \(i=0, 1\) is an isomorphism. From recent results of Oyono and Tu, they deduce that the Baum-Connes conjecture with coefficients holds for any one-relator group, as well as for fundamental groups of Haken 3-manifolds. In particular, if \({\Gamma}\) is a torsion-free group in one of these classes, then \(C^*_r({\Gamma})\) has no nontrivial idempotent.
reviewed by: Zvonko Čerin
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K99
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 1275467
zbMATH Keywords: torsion-free one-relator group
zbMATH Keywords: reduced \(C^*\)-algebra
zbMATH Keywords: Baum-Connes conjecture
zbMATH Keywords: \(K\)-amenability
zbMATH Keywords: fundamental groups of Haken 3-manifolds
MaRDI profile type: Publication
full work available at URL: a:1007755408585
OpenAlex ID: W2011512505
description: scientific article; zbMATH DE number 1275467