Q:

Products and Stone-Čech compactification

Let $X$, $Y$ be complete metrizable spaces, $\beta X$, $\beta Y$ be their Stone-Čech compactifications. It is known that $C_b(X) \simeq C(\beta X)$. Is it possible to say something about the relation between $C_b(X \times Y)$ and $C(\beta X \times \beta Y)$?

A:

For all Tychonoff spaces $X$ and $Y$ we know that $C_b(X \times Y) = C(\beta (X \times Y))$ (isometry as Banach spaces, as well as a ring isomorphism), and $\beta(X \times Y) \cong \beta X \times \beta Y$ iff $X \times Y$ is pseudocompact (see the original paper). The latter happens, in your case, iff $X \times Y$ is compact (pseudocompactness and compactness coincide for metrizable spaces; we only need first countability and normality to see that) iff $X$,$Y$ are compact and everything trivializes, as $\beta X = X$ etc.

