Tensor products with bounded continuous functions

Abstract

We study that natural inclusions $C^b(X) \tensor A$ into $C^b(X,A)$ and $C^b(X, C^b(Y))$ into $C^b(X \times Y)$. In particular, excepting trivial cases, both these maps are isomorphisms only when $X$ and $Y$ are pseudocompact. This implies a result of Glicksberg showing that the Stone-Cech compactificiation $\beta(X \times Y)$ is naturally identified with $\beta X \times \beta Y$ if and only if $X$ and $Y$ are pseudocompact.