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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).

##### MSC:
 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)
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