Pontryagin reflexive groups are not determined by their continuous characters
type: Item
instance of: scholarly article
title: Pontryagin reflexive groups are not determined by their continuous characters
zbMATH Open document ID: 0922.43004
published in: Rocky Mountain Journal of Mathematics
publication date: 1999-10-14T00:00:00Z
full work available at URL: CONT281.html
review text: Let \(G\) be an abelian topological group. Let \(\tau\) denote the topology of \(G\) and \(\tau_\omega\) the weak topology induced by all continuous characters of \(G\). The group \(G\) endowed with the topology \(\tau_\omega\) is denoted by \(G_\omega\). \textit{I. Glicksberg} [Can. J. Math. 14, 269-276 (1962; Zbl 0109.20001)] proved: (i) \(G\) and \(G_\omega\) have the same compact subsets. (ii) The topology of \(G\) is determined by the set of all its continuous characters. \textit{R. Venkataraman} [Pac. J. Math. 57, 591-595 (1975; Zbl 0308.22009)] asserted that (i) and (ii) hold for reflexive groups. The authors show that (ii) does not hold for reflexive groups. They provide sufficient conditions on reflexive groups for (ii) to hold. That (i) is false for reflexive groups is the content of \textit{D. Remus} and \textit{F. J. Trigos-Arrieta} [Proc. Am. Math. Soc. 117, 1195-1200 (1993; Zbl 0826.22002)].
Mathematics Subject Classification ID: 43A40
Mathematics Subject Classification ID: 46A20
Mathematics Subject Classification ID: 22B05
Mathematics Subject Classification ID: 22A05
zbMATH DE Number: 1286531
zbMATH Keywords: abelian topological group
zbMATH Keywords: continuous characters
zbMATH Keywords: reflexive groups
author: M. J. Chasco
author: E. Martín-Peinador
reviewed by: Harkrishan L. Vasudeva
MaRDI profile type: Publication
OpenAlex ID: W1984405036
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DOI: 10.1216/RMJM/1181071826
description: scientific article; zbMATH DE number 1286531