Amenability for twisted covariance algebras and group \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Amenability for twisted covariance algebras and group \(C^*\)-algebras
zbMATH Open document ID: 0737.43003
DOI: 10.1016/0022-1236(91)90102-B
author: A. T.-M. Lau
author: Alan L. T. Paterson
published in: Journal of Functional Analysis
publication date: 1992-06-26T00:00:00Z
review text: Three notions of amenability have turned out to be important: Amenable groups, amenable Banach algebras and more specifically amenable \(C^*\)- algebras, amenable von Neumann algebras. For \(C^*\)-algebras, amenability means nuclearity; for von Neumann algebras, amenability signifies injectivity. The locally compact group \(G\) is amenable if and only if its group algebra \(L_ 1(G)\) is an amenable Banach algebra; the \(C^*\)-algebra \(A\) is amenable if and only if its second dual space \(A''\) constitutes an amenable von Neumann algebra. Now a locally compact group \(G\) is called \(C^*\)-amenable group in case its \(C^*\)-algebra \(C^*(G)\), i.e., the completion of \(L_ 1(G)\), is amenable. Every amenable group is \(C^*\)-amenable; the converse does not hold. The article describes profound investigations on stability properties inside the class of \(C^*\)-amenable groups. They are carried out in the large framework of twisted covariance algebras [\textit{P. Green}, Acta Math. 140, 191-250 (1978; Zbl 0407.46053)]. The authors take care to produce a rather self-contained and original approach to the theories of crossed products and twisted crossed products. Let \(G\) be a locally compact group and \(A\) a \(C^*\)-algebra. One considers the Banach \(^*\)-algebra \(L_ 1(G,A) = L_ 1(G)\overset\widehat{}{\smash\otimes} A\) of Bochner integrable functions from \(G\) into \(A\). Convolution of \(L_ 1 (G,A)\) is defined via a continuous group homomorphism \(\alpha : G \rightarrow Aut A\); \((G,A,\alpha)\) is a covariant system. If \(N\) is a closed normal subgroup of \(G\), one assumes the existence of a continuous homomorphism \({\mathcal T}\) of \(N\) into the unitary group \(U({\mathcal M}(A))\) of the algebra \({\mathcal M}(A)\) of multipliers on \(A\). Then \((G,A,\alpha,N,{\mathcal T})\) forms a twisted covariant system if for all \(n \in N\), \(x \in G\), \(a \in A\), \({\mathcal T}(n)a{\mathcal T}(n^{-1}) = \alpha_ n (a)\), \({\mathcal T}(xnx^{-1}) = a_ x ({\mathc…
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 22D15
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L40
zbMATH DE Number: 21943
zbMATH Keywords: amenable Banach algebras
zbMATH Keywords: amenable C*-algebras
zbMATH Keywords: amenable von Neumann algebra
zbMATH Keywords: locally compact group
zbMATH Keywords: group algebra
zbMATH Keywords: C*-amenable groups
zbMATH Keywords: twisted covariance algebra
zbMATH Keywords: twisted crossed products
zbMATH Keywords: Bochner integrable function
zbMATH Keywords: multipliers
zbMATH Keywords: twisted covariant system
reviewed by: Jean-Paul Pier
MaRDI profile type: Publication
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full work available at URL: 0022-1236(91)90102-b
OpenAlex ID: W2067830599
description: scientific article; zbMATH DE number 21943