Tensor products of maximal abelian subalgebras of C*-algebras

It is shown that if C₁ and C₂ are maximal abelian self-adjoint subalgebras (masas) of C*-algebras A₁ and A₂, respectively, then the completion C₁⊗ C₂ of the algebraic tensor product C₁ C₂ of C₁ and C₂ in any C*-tensor product A₁⊗_β A₂ is maximal abelian provided that C₁ has the extension property of Kadison and Singer and C₂ contains an approximate identity for A₂. An example is given to show that C₁⊗ C₂ can fail to be a masa in A₁⊗_β A₂ with A₁ and A₂ unital if neither C₁ nor C₂ 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.