Extension of locally pseudocompact group actions

Abstract

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonne complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonne completion γG on X, (2) any action of a locally pseudocompact topological group G on a bf-space (in particular, on a first countable space) X can uniquely be extended to an action of the Weil completion \(\overline{G}\) on the Dieudonne completion γX of X. As a consequence, we obtain that, for each locally pseudocompact topological group G, every G-space with the bf-property admits an equivariant embedding into a compact Hausdorff G-space. Furthermore, for each pseudocompact group G, every metrizable G-space has a G-invariant metric compatible with its topology. We also give a direct construction of such an invariant metric.