Locally pseudocompact topological groups
By a result of \textit{W. Comfort} and \textit{K. Ross} [Pac. J. Math. 16, 483- 496 (1966; Zbl 0214.285)], the following conditions are equivalent for a dense subgroup \(G\) of a compact topological group \(\overline {G}\): (a) \(G\) is pseudocompact; (b) \(G\) is \(C^*\)-embedded in \(\overline {G}\); (c) \(G\) is \(G_\delta\)-dense in \(\overline {G}\), that is, \(G\) intersects every non-empty \(G_\delta\)-subset of \(\overline {G}\). -- What happens if the ``enveloping'' group \(\overline {G}\) is locally pseudo-compact? Following A. Weil, the authors call a group \(G\) locally bounded if there exists a bounded neighborhood \(U\) of the identity in \(G\), that is, \(U\) can be covered by finitely many translations of any non-empty open subset of \(G\). The main result of the paper is the following theorem which answers the question above. Theorem. If \(G\) is a locally bounded group with Weil completion \(\overline {G}\), then the following conditions are equivalent: (a) \(G\) is locally pseudocompact; (b) \(G\) is \(C^*\)-embedded in \(\overline{G}\); (c) the set \(\text{cl}_G U\) is \(C^*\)-embedded in \(\text{cl}_{\overline {G}} U\) for some non-empty open subset \(U\) of \(G\); (d) the set \(\text{cl}_G U\) is \(C^*\)-embedded in \(\text{cl}_{\overline {G}} U\) for every bounded open subset \(U\) of \(G\). -- The paper also contains helpful discussions and an extensive bibliography.
scientific article; zbMATH DE number 762125
scientific article