Weakly pseudocompact subsets of nuclear groups
type: Item
instance of: scholarly article
title: Weakly pseudocompact subsets of nuclear groups
zbMATH Open document ID: 0935.22004
DOI: 10.1016/S0022-4049(98)00034-6
published in: Journal of Pure and Applied Algebra
publication date: 2000-05-04T00:00:00Z
review text: \textit{W. Banaszczyk} introduced in [Additive subgroups of topological vector spaces, Lect. Notes Math., vol. 1466 (Berlin etc. 1991; Zbl 0743.46002)] the class of nuclear groups. This class contains every locally compact Abelian (LCA) group and the additive groups of all nuclear locally convex spaces. For an Abelian topological group \(G\) the weak topology is the topology induced by all continuous characters of \(G\). Let \(G^+\) be the group \(G\) endowed with the weak topology. \(G\) respects compactness (pseudocompactness, countable compactness, functional boundedness) if \(G\) and \(G^+\) have the same compact (pseudocompact, countably compact, functionally bounded) sets. It is a well-known result of I. Glicksberg that LCA groups respect compactness. Later on, \textit{F. J. Trigos-Arrieta} proved that LCA groups respect pseudocompactness, functional boundedness and the Lindelöf property. Recently he and \textit{D. Remus} [Topology Appl. 97, 85-98 (1999)] have shown that this and Glicksberg's result still hold for Moore groups. Now the authors prove the following interesting theorem: Nuclear groups respect compactness, countable compactness and pseudocompactness, and complete nuclear groups respect functional boundedness. It remains open if the assumption of completeness can be removed.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 1308362
zbMATH Keywords: locally compact Abelian group
zbMATH Keywords: nuclear groups
zbMATH Keywords: nuclear locally convex spaces
zbMATH Keywords: compactness
zbMATH Keywords: countable compactness
zbMATH Keywords: pseudocompactness
zbMATH Keywords: functional boundedness
author: Wojciech Banaszczyk
author: E. Martín-Peinador
reviewed by: Dieter Remus
MaRDI profile type: Publication
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cites work: Summable families in nuclear groups
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cites work: Abelian Groups which Satisfy Pontryagin Duality need not Respect Compactness
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cites work: The maximal totally bounded group topology on G and the biggest minimal G-space, for Abelian groups G
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description: scientific article; zbMATH DE number 1308362