Stone Cech compactification of cartesian product Let $X,Y$ be topological spaces. By the universal property of the Stone-Cech compactification, there is a canonical map
$$\Phi: \beta(X\times Y)\to \beta(X)\times \beta(Y).$$
It is not very hard to show that $\Phi$ is always a surjection.
My question: Is it known when $\Phi$ is injective?
I read somewhere that $\beta(X\times Y)\cong \beta(X)\times \beta(Y)$ (through an arbitrary homeomorphism?) if and only if $X,Y$ are pseudocompact. Maybe if these spaces are homeomorphic, the canonical map is automatically injective as well?
 A: Let $X, Y$ be completely regular topological spaces, consider $X \times Y$ as a subspace of $\beta X \times \beta Y$.
Hence $\beta X \times \beta Y$ is a compactification of $X \times Y$. As was mentioned by the OP, there exists a unique continuous extension $\Phi: \beta(X \times Y) \rightarrow \beta X \times \beta Y$ and $\Phi$ is surjective.
The following are equivalent:

*

*$\Phi$ is injective.

*$\Phi$ is a homeomorphism.

*$\beta X \times \beta Y$ is "the" Stone-Cech compactification of $X \times Y$, i.e., there exists a homeomorphism $\Psi: \beta(X \times Y) \rightarrow \beta X \times \beta Y$, which extends the embedding.

*$X$ is finite or $Y$ is finite or $X\times Y$ is pseudocompact.

Proof: 1 $\Leftrightarrow$ 2 and 2 $\Leftrightarrow$ 3 are obvious (for "$\Leftarrow$" note that $\Phi$ and $\Psi$ coincide on $X \times Y$, hence $\Phi = \Psi$). 
3 $\Leftrightarrow$ 4 is the hard part, and essentially due to Glicksberg, see his paper cited above by GEdgar, see also Engelking, General Topology, 3.12.21 (c). (For "$\Leftarrow$" note that, if $X$ is finite, $X = \bigoplus_{i=1}^n \{x_i \}$, then $\beta (X \times Y) \cong \beta(\bigoplus_{i=1}^n Y) \cong \bigoplus_{i=1}^n \beta Y \cong X \times \beta Y \cong \beta X \times \beta Y$.)
It should be noted that if $X \times Y$ is pseudocompact, $X,Y \neq \emptyset$, then, of course, $X$ and $Y$ are pseudocompact. But, in general, not vice versa, see Engelking, General Topology 3.10.19 and the comment after 3.10.25 for a counter-example. See also there 3.10.26 and 3.12.21 (b) for a discussion, when a product of two pesudocompact spaces is pseudocompact, or Glicksberg's paper §5.
