The Baum-Connes conjecture
type: Item
instance of: scholarly article
title: The Baum-Connes conjecture
zbMATH Open document ID: 0911.46041
author: Nigel Higson
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/228190
full work available at URL: Higson.MAN.html
review text: Let \(G\) be a group which is locally compact, Hausdorff and second countable. The group \(C^*\)-algebra, denoted \(C^*(G)\), is the enveloping \(C^*\)-algebra of the convolution algebra \(L_1(G)\). The reduced \(C^*\)-algebra of \(G\), denoted \(C^*_{\text{red}}(G)\), is the completion of \(L_1(G)\) in its regular representation as bounded operators on \(L_2(G)\). (The \(C^*\)-algebra \(C^*_{\text{red}}(G)\) coincides with \(C^*(G)\) iff \(G\) is amenable.) The purpose of this report is to present a short account of past and recent work on the Baum-Connes conjecture concerning the groups \(K_*(C^*(G))\) and \(K_*(C^*_{\text{red}}(G))\). The Baum-Connes conjecture for torsion-free groups is the following: if \(G\) is a discrete and torsion-free group then the assembly map \(\mu_{\text{red}}: K_*(BG)\to K_*(C^*_{\text{red}}(G))\), obtained from the so-named assembly map \(\mu: K_*(BG)\to K_*(C^*(G))\) using the regular representation \(C^*(G)\to C^*_{\text{red}}(G)\), is an isomorphism. The advantage of the new version of this conjecture is that it is simpler and applies more generally than the earlier statement. Associated to any \(G\) there is a proper \(G\)-space \({\mathcal E}G\), which is universal in the sense that any other proper \(G\)-space maps into it in a way which is unique up to equivariant homotopy. Using Kasparov's KK-theory Baum, Connes and Higson formed the equivariant \(K\)-homology group \(K^G_*({\mathcal E}G)\). A class in \(K_*({\mathcal E}G)\) is represented by an abstract \(G\)-equivariant elliptic operator on \({\mathcal E}G\). If \(G\) is discrete and torsion-free then \({\mathcal E}G\) is universal principal space \(EG\) and \(K^G_*({\mathcal E}G)= K_*(BG)\). For general \(G\) there is an assembly map \(m_{\text{red}}: K^G_*({\mathcal E}G)\to K_*(C^*_{\text{red}}(G))\), very similar to the one already considered: a cycle for \(K^G_*({\mathcal E}G)\) is abstract elliptic operator \(D\) on a proper \(G\)-space \({\mathcal E}G\) and \(m_{\text…
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K56
zbMATH DE Number: 1184298
zbMATH Keywords: Baum-Connes conjecture
zbMATH Keywords: group \(C^*\)-algebras
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: index theory
zbMATH Keywords: enveloping \(C^*\)-algebra
zbMATH Keywords: convolution algebra
zbMATH Keywords: torsion-free group
zbMATH Keywords: Kasparov's KK-theory
zbMATH Keywords: equivariant \(K\)-homology group
zbMATH Keywords: \(G\)-equivariant elliptic operator
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184298