Module maps and invariant subsets of Banach modules of locally compact groups
type: Item
instance of: scholarly article
title: Module maps and invariant subsets of Banach modules of locally compact groups
zbMATH Open document ID: 1295.43005
author: Hawa Alsanousi Hamouda
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2014-06-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1400592623
review text: Here \(G\) denotes a locally compact group, \(C_0(G)\) and its Banach dual \(M(G)\), the complex regular Borel measures \(\mu\) on \(G\) with convolution product; also \(L^p(G), 0 \leq p < \infty\), while \(L^1(G)\) denotes the group convolution algebra, a closed ideal of \(M(G)\). The \(\mu_f, f \in L^1\), act on \(M(G)\)-modules via test functions in \(C_0(G)\). The article is based on results of \textit{A. T. M. Lau} [Trans. Am. Math. Soc. 232, 131--142 (1977; Zbl 0363.43002)] and \textit{A. Ghaffari} [Houston J. Math. 34, No. 4, 1225--1232 (2008; Zbl 1162.43004)]; they had been concerned with the action, for \(s \in G, f \in L^p, 0 \leq p <\infty\), in particular \(s \cdot f = \delta_s * f(t) = \delta s*f\) and \(s \cdot f(t) = \delta_s *f(t) =\Delta(s)^{1/p} f(s^{-1}ts)\). The author extends some of their results to include arbitrary left Banach \(G\)-modules, \(M(G)\) modules and \(L^1(G)\) modules; his results concern equivalence of statements relating the module mappings, usually needing changes of topologies. By \(T\) being a \textit{odular mapping} is meant \(T(s \cdot x) = s \cdot Tx\) for \(s \in G\) and \(x\) in some module. Given a bounded linear mapping \(T:X \to Y\) where \(X\) and \(Y\) are left \(G\)-modules the statements for \(T\) being a \(G\)-modular mapping, an \(M(G)\)-modular mapping or an \(L^1(G)\)-modular mapping are shown to be equivalent. Providing the duals \(X^*\) and \(Y^*\) with the \(\omega\)* topology, for \(T\) a continuous mapping \(Y^* \to X^*\) the statements that \(T\) is a right \(G\)-module mapping, a right \(M(G)\)-module mapping and a right \(L^1(G)\) mapping are equivalent. For a more involved result the author provides the dual Banach spaces with the uniform norm, and denotes by \(UC(X^*)\) the closed linear subspace of \(X^*\), i.e., those \(\phi \in X^*\) for which \(s \to \phi \cdot s\) is continuous. Restriction to \(UC(X^*)\) removes an ambiguity between the \(M(G)\)-module action and the dual \(M(G)\) action. Then…
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 43A20
zbMATH DE Number: 6305494
zbMATH Keywords: locally compact groups
zbMATH Keywords: group algebras
zbMATH Keywords: measure algebras
zbMATH Keywords: Banach modules
zbMATH Keywords: module mappings
zbMATH Keywords: invariance of closed convex sets
reviewed by: Aubrey Wulfsohn
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6305494