  It is shown that if $C_1$ and $C_2$ are maximal abelian self-adjoint
subalgebras (masas) of C*-algebras $A_1$ and $A_2$, respectively, then the
completion $C_1\otimes C_2$ of the algebraic tensor product $C_1\odot C_2$ of
$C_1$ and $C_2$ in any C*-tensor product $A_1\otimes_{\beta} A_2$ is maximal
abelian provided that $C_1$ has the extension property of Kadison and Singer
and $C_2$ contains an approximate identity for $A_2$. An example is given to
show that $C_1\otimes C_2$ can fail to be a masa in $A_1\otimes_{\beta} A_2$
with $A_1$ and $A_2$ unital if neither $C_1$ nor $C_2$ has the extension
property. This gives an answer to a long-standing question, but leaves open
some other interesting problems, one of which turns out to have a potentially
intriguing implication for the Kadison-Singer extension problem.
