Dieudonné completion and -group actions

If G ( X ) denotes either the free topological group or the free Abelian topological group over a topological space X, we prove that ∏ i = 1 n G ( X i ) is a hemibounded b f -group whenever each X i is a pseudocompact space (which provides a new way to generate this kind of topological groups), and we show that the equality μ ( X × ∏ i = 1 n G ( X i ) ) = μ X × ∏ i = 1 n G ( β X i ) holds whenever X is a hemibounded b f -space (where μY stands for the Dieudonné completion of Y). By means of the Dieudonné completion we prove that every pseudocompact space X is G-Tychonoff whenever G is a b f -group and that the maximal G-compactification of X coincides with βX. We apply this result to obtain a partial version for G-spaces of Glicksbergʹs theorem on pseudocompactness and we analyze when the maximal G-compactification of a G-space X coincides with the Stone–Čech compactification of X in the case when G is a metrizable group.