On the maximal G-compactification of products of two G-spaces

Let G be any Hausdorff topological group and let βGX denote the maximal G-compactification of a G-Tychonoff space X (i.e., a Tychonoff G-space possessing a G-compactification). Recall that a completely regular Hausdorff topological space is called pseudocompact if every continuous function f : X →R is bounded. In this paper, we prove that if X and Y are two G-Tychonoff spaces such that the product X ×Y is pseudocompact, then βG(X ×Y)= βGX ×βGX (see Theorem 2.2). This is a G-equivariant version of the well-known result of Glicksberg [16], which for G a locally compact group was proved earlier by de Vries in [10]. Note that even in the case of a locally compact acting group G, our proof is shorter than that of [10, Theorem 4.1]. It follows from Proposition 2.7 that the equality βG(X ×Y)= βGX ×βGX does not imply, in general, the pseudocompactness of X ×Y even if X and Y both are infinite (cf. [16, Theorem 1]). Theorem 2.10 says that if a pseudocompact group G acts continuously on a pseudocompact space X , then βGX = βX . Let us introduce some terminology we will use in the paper. Throughout the paper, all topological spaces are assumed to be Tychonoff (i.e., completely regular and Hausdorff). The letter “G” will always denote a Hausdorff (and hence, completely regular) topological group unless otherwise stated. For the basic ideas and facts of the theory of G-spaces or topological transformation groups, we refer the reader to [5, 7, 11]. However, we recall below some more special notions and facts we need in the paper.