Extension of mappings from the product of pseudocompact spaces

Abstract

Let $X$ and $Y$ be pseudocompact spaces and let the function $\Phi: X\times Y\to \mathbb R$ be separately continuous. The following conditions are equivalent: (1) there is a dense $G_\delta$ subset of $D\subset Y$ so that $\Phi$ is continuous at every point of $X\times D$ (Namioka property); (2) $\Phi$ is quasicontinuous; (3) $\Phi$ extends to a separately continuous function on $\beta X\times \beta Y$. This theorem makes it possible to combine studies of the Namioka property and generalizations of the Eberlein-Grothendieck theorem on the precompactness of subsets of function spaces. We also obtain a characterization of separately continuous functions on the product of several pseudocompact spaces extending to separately continuous functions on products of Stone-Cech extensions of spaces. These results are used to study groups and Mal'tsev spaces with separately continuous operations.