The \(C^ \ast\)-algebra generated by operators with compact support on a locally compact group
type: Item
instance of: scholarly article
title: The \(C^ \ast\)-algebra generated by operators with compact support on a locally compact group
zbMATH Open document ID: 0788.22006
author: A. T.-M. Lau
author: V. Losert
published in: Journal of Functional Analysis
publication date: 1993-05-02T00:00:00Z
review text: Let \(VN(G)\) be the von Neumann algebra of a locally compact group \(G\). The \(C^*\)-subalgebra of \(VN(G)\) generated by operators with compact support is denoted by \(UCB(\widehat G)\). Alternatively, if \(A(G)\) is the Fourier algebra of \(G\), and thus the predual of \(VN(G)\), \(UCB(\widehat G)\) can be identified with the norm-closed span of \(A(G).VN(G)\), the subspace of \(VN(G)\) obtained from the natural action of the algebra \(A(G)\) on it. \(B_ \rho(G)\) is the closure in the compact open topology of the linear span of the positive definite functions of compact support on \(G\). When \(G\) is commutative, \(UCB(\widehat G)\) can be identified with the space of bounded uniformly continuous functions on \(\widehat G\), and \(B_ \rho(G)\) with \(M(\widehat G)\). The space \(B_ \rho(G)\) contains \(A(G)\) and, since positive definite functions have a natural realization as linear functionals on \(VN(G)\), it can be regarded as a subspace of \(UCB(\widehat G)^*\). The second dual \(A(G)^{**}= VN(G)^*\) has an algebra structure induced by \(A(G)\), and this in turn provides an algebra multiplication on \(UCB(\widehat G)^*\). \(B_ \rho(G)\) turns out to be naturally embedded in the algebraic centre of \(UCB(\widehat G)^*\). The main aim of this paper is to discuss the question of whether \(B_ \rho(G)\) is equal to the centre of \(UCB(\widehat G)^*\); this is known to be the case if \(G\) is discrete or commutative. The principal result asserts that if \(G\) is second countable and the closure of its commutator subgroup is not open, then \(B_ \rho(G)\) is the centre. As a consequence, the authors find that if \(G\) is in addition amenable then the centre of \(VN(G)^*\) is \(A(G)\), and they show that this conclusion also holds for amenable discrete, and for commutative, groups. Many other interesting results are presented in the paper. For example, if \(VN(G)\) has a unique topologically invariant mean then \(G\) must be discrete, and \(UCB(\widehat G)= A(G).V…
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 43A07
zbMATH DE Number: 152033
zbMATH Keywords: group \(C^*\)-algebra
zbMATH Keywords: second dual algebra
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: locally compact group
zbMATH Keywords: Fourier algebra
zbMATH Keywords: positive definite functions of compact support
zbMATH Keywords: invariant mean
reviewed by: J. Pym
MaRDI profile type: Publication
full work available at URL: jfan.1993.1024
OpenAlex ID: W1968630586
DOI: 10.1006/JFAN.1993.1024
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description: scientific article; zbMATH DE number 152033