Spectral properties of continuous representations of topological groups
type: Item
instance of: scholarly article
title: Spectral properties of continuous representations of topological groups
zbMATH Open document ID: 1228.47041
author: M. Cianfarani
author: Jean-Martin Paoli
author: J.-C. Tomasi
published in: Archiv der Mathematik
publication date: 2011-05-11T00:00:00Z
review text: The investigations are motivated by investigations of spectral properties and characterizations of uniform continuity of one-parameter \(C_0\)-operator groups \(\{T_t\}\) -- hence representations of the group \(\mathbb{R}\) -- e.g., by the angular distribution of the spectrum \(\sigma^1(T_t) :=\{\lambda/|\lambda|: \lambda\in \mathrm{Spec}(T_t)\}\), cf. [\textit{K. Latrach, J. M. Paoli} and \textit{P. Simonnet}, ``A spectral characterization of the uniform continuity of strongly continuous groups'', Arch. Math. 90, No.~5, 420--428 (2008; Zbl 1154.47033)]. As shown by counterexamples, these results are not valid for locally bounded representations of arbitrary locally compact Abelian groups. This motivates investigations of strongly, but not uniformly continuous representations \(\theta\) of (second countable) locally compact Abelian groups \(G\) and of related characterizations of uniform continuity. A representation \(\theta\) of \(G\) by linear operators of a Banach space generates a commutative Banach algebra (of operators), this leads to representations of \(G\) on a commutative Banach algebra, and hence Gelfand-theoretical methods and tools are available for analyzing spectral properties of \(\theta\), cf., e.g., [\textit{J. Esterle}, ``The zero-\(\sqrt{3}\) and zero-two laws for representations of locally compact abelian groups'', J. Contemp. Math. Anal., Armen. Acad. Sci. 38, No.~5, 9--19 (2003); translation from Izv. Nats. Akad. Nauk Armen., Mat. 38, No.~5, 11--22 (2003; Zbl 1160.22301)]. A key result for characterizations of uniform continuity is Proposition 2.4, saying that strongly continuous representations \(\theta\) of \(G\) on a commutative Banach algebra \(\mathcal{A}\) are uniformly continuous iff, for all \(\chi\in\widehat{\mathcal{A}}\), \(\chi\circ\theta\) is continuous. The main result (Theorem 4.6) characterizes uniform continuity of \(\theta\) by spectral properties, similar (but not equal) to the aforementioned characterizations of uniform con…
reviewed by: Wilfried Hazod
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 43A40
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 46H35
Mathematics Subject Classification ID: 46J99
zbMATH DE Number: 5886758
zbMATH Keywords: spectra
zbMATH Keywords: linear operators
zbMATH Keywords: strongly continuous linear representations
zbMATH Keywords: uniformly continuous representations
zbMATH Keywords: locally compact Abelian groups
zbMATH Keywords: representations on Banach algebras
MaRDI profile type: Publication
full work available at URL: s00013-011-0219-4
OpenAlex ID: W2072117842
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DOI: 10.1007/S00013-011-0219-4
description: scientific article; zbMATH DE number 5886758