Semisimple Banach algebras generated by strongly continuous representations of locally compact abelian groups
type: Item
instance of: scholarly article
title: Semisimple Banach algebras generated by strongly continuous representations of locally compact abelian groups
zbMATH Open document ID: 0837.22002
published in: Journal of Functional Analysis
publication date: 1994-12-20T00:00:00Z
review text: Let \(T\) be a strongly continuous representation of a locally compact Abelian group \(G\) in a Banach space. It is shown that the weak operator closure of the algebra generated by \(\widehat f(T)= \int_G f(g) T(g) dg\), \(f\in L^1(G)\), is a semisimple Banach algebra if the spectrum of \(T\) is scattered (each nontrivial subset contains an isolated point). The spectrum of the representation \(T\) is defined as \(\{\chi\in G^*: \widehat f(\chi)= 0\) whenever \(\widehat f(T)= 0\}\).
Mathematics Subject Classification ID: 22B10
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 43A32
zbMATH DE Number: 704515
zbMATH Keywords: strongly continuous representation
zbMATH Keywords: locally compact Abelian group
zbMATH Keywords: Banach space
zbMATH Keywords: weak operator closure
zbMATH Keywords: semisimple Banach algebra
zbMATH Keywords: scattered
zbMATH Keywords: spectrum
author: Gilbert L. Muraz
author: Vu Quoc Phong
MaRDI profile type: Publication
full work available at URL: jfan.1994.1138
OpenAlex ID: W1976582141
DOI: 10.1006/JFAN.1994.1138
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description: scientific article; zbMATH DE number 704515