Tensor products with bounded continuous functions

We study that natural inclusions $C^b(X)\text{tensor}A$ 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 $\beta X \times \beta Y$ifandonlyif$X$and$Y$ are pseudocompact.