Some remarks on a paper by Liu and van Rooij
type: Item
instance of: scholarly article
title: Some remarks on a paper by Liu and van Rooij
zbMATH Open document ID: 1152.22005
DOI: 10.1016/S0019-3577(07)80066-8
published in: Indagationes Mathematicae. New Series
publication date: 2008-08-12T00:00:00Z
review text: In the paper [\textit{T.-S. Liu} and \textit{A. C. M. van Rooij}, Nederl. Akad. Wet., Proc., Ser. A 77, 306--316 (1974; Zbl 0305.43005)], the authors proved that if a discrete group \(G\) is amenable, then there exist non-zero left invariant operators on \(L^\infty(G)\) with values either in \(L^p(G)\), \(2\leq p<\infty\), or in \(C^0(G)\). This result raises the following question: Does the existence of a non-zero left invariant operator from \(L^\infty(G)\) into \(L^p(G)\), \(2\leq p<\infty\), or in \(C^0(G)\) imply the amenability of \(G\)? In the paper under review, the author gives several affirmative answers for locally compact groups. For instance, the author obtains the following results: Let \(G\) be locally compact and \(2\leq p<\infty\). (a) If there is a non-zero left invariant operator from \(L^\infty(G)\) into \(L^p(G)\), then \(G\) is amenable. (b) \(G\) is amenable if and only if there is a non-zero weakly compact left invariant operator from \(L^\infty(G)\) into itself. (c) \(G\) is inner amenable if and only if there is a non-zero weakly compact conjugation invariant operator from \(L^\infty(G)\) into itself. The author also obtains several characterizations for compact groups.
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 43A22
zbMATH DE Number: 5311053
zbMATH Keywords: translation invariant operators
zbMATH Keywords: amenable groups
zbMATH Keywords: weak compactness
author: Gerhard Racher
reviewed by: Dashan Fan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5311053