On invariant subalgebras of the Fourier-Stieltjes algebra of a locally compact group
type: Item
instance of: scholarly article
title: On invariant subalgebras of the Fourier-Stieltjes algebra of a locally compact group
zbMATH Open document ID: 0787.22005
DOI: 10.1007/BF01934339
author: A. T.-M. Lau
author: G. Schlichting
author: Bachir Bekka
published in: Mathematische Annalen
publication date: 1993-01-12T00:00:00Z
full work available at URL: https://eudml.org/doc/165015
review text: One of the most instructive Banach algebras attached to a locally compact group \(G\) is the commutative Fourier-Stieltjes algebra \(B(G)\), the dual space of the \(C^*\)-algebra \(C^*(G)\) of \(G\); it is spanned by the set \(P(G)\) of all continuous positive definite functions on \(G\). In [J. Funct. Anal. 11, 17-38 (1972; Zbl 0242.22010)], \textit{M. E. Walter} proved the locally compact groups \(G_ 1\) and \(G_ 2\) to be isomorphic if and only if the algebras \(B(G_ 1)\) and \(B(G_ 2)\) are isometrically isomorphic. The carefully written paper under review describes the structure of a [translation-] invariant *-subalgebra \(A\) of \(B(G)\) in terms of a subgroup in \(G\). A direct proof yields that in case \(A\) separates points of \(G\) and it is also conjugation-invariant with respect to the enveloping von Neumann algebra of \(C^*(G)\), it coincides with the Fourier algebra \(A(G)\) of \(G\). As a consequence, the locally compact group \(G\) is amenable if and only if there exists a bijection between the set of all closed normal subgroups of \(G\) and the set of all weak-*-closed invariant *-subalgebras \(A\) of \(G\) with \(A\neq\{0\}\); the proof uses the fact that in the amenable case, \(f\in P(G)\) is a limit on compacta of functions \(\varphi*\widetilde{\varphi}\), \(\varphi\in L^ 2(G)\), with \(\widetilde{\varphi}(x)= \overline {\varphi(x^{-1})}\), \(x\in G\). Also, if the discrete group \(G_ d\) corresponding to \(G\) is amenable, then \(B(G)\) is weak-*-dense in \(B(G_ d)\). The authors prove that not being the case for the nonamenable group \(SL(3,\mathbb{C})\), as it violates Reiter's condition \((P_ 2^*)\). Other general properties are established: If \(A\) is a norm-closed invariant *-subalgebra of \(A(G)\) and \(A\neq \{0\}\), then \(A=A(G/K)\) for some compact normal subgroup \(K\) of \(G\). The weak-*-closed \({\mathcal C}_ 0(G)\)-invariant *-subalgebra \(A\) of \(M^ 1(G)={\mathcal C}_ 0(G)^*\) [resp. the closed \(L^ \infty(G)\)-invariant *-suba…
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 43A07
zbMATH DE Number: 86202
zbMATH Keywords: amenable groups
zbMATH Keywords: Fourier-Stieltjes algebra
zbMATH Keywords: positive definite functions
reviewed by: Jean-Paul Pier
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 86202