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Dieudonné completion and \(b_{f}\)-group actions. (English) Zbl 1106.22003
The problems of existence and realization of equivariant compactifications in topological dynamics have been dealt with by many authors. Let \(G\) be a Hausdorff topological group continuously acting on a Tychonoff space \(X\). The main result in this paper (Theorem 3.6) asserts that, if \(G\) is a \(b_f\)-space, and \(X\) is pseudocompact, then there exist such compactifications for \(X\) (i. e. \(X\) can be equivariantly embedded into a compact \(G\)-space as a dense subset; this property is usually referred to as “\(X\) is \(G\)-Tychonoff”), and the topological space underlying the maximal one is the Stone-Čech’s compactification \(\beta X\) of \(X\). (A \(b_f\)-space is characterized by the following property: if restrictions to bounded subsets of a real-valued function defined on such a space are continuous, then the function is continuous.)
The authors deal also with the converse question: what can be said of a space which is \(G\)-Tychonoff and whose maximal equivariant compactification is Stone-Čech’s? They observe that the proof of an important result in this direction, proved by J. de Vries [Topology theory and applications, 5th Colloq., Eger/Hung. 1983, Colloq. Math. Soc. János Bolyai 41, 655–666 (1985; Zbl 0598.54019)], does not use this reference’s initial assumption of \(G\) being locally compact, and thus derive, among other results, that \(X\) is pseudocompact whenever \(G\) is metrizable and the set of elements in \(X\) with open isotropy group has empty interior (Corollary 3.10).
Along the way to these results the authors prove a few interesting facts concerning free topological groups and \(b_f\)-spaces, e. g. both the free topological group and the free Abelian topological group over a pseudocompact space are \(b_f\)-spaces (Theorem 2.4).

22A10 Analysis on general topological groups
54H20 Topological dynamics (MSC2010)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54H15 Transformation groups and semigroups (topological aspects)
Full Text: DOI
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