Weakly pseudocompact subsets of nuclear groups
\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.
scientific article; zbMATH DE number 1308362