Isomorphic classification of projective tensor products of spaces of continuous functions

Abstract

We prove that for infinite, countable, compact, Hausdorff spaces $K,L$, $C(K){\hat{\otimes }}_{\pi }C(L)$ is isomorphic to exactly one of the spaces $C({\omega }^{{\omega }^{\xi }}){\hat{\otimes }}_{\pi }C({\omega }^{{\omega }^{\zeta }})$, $0⩽\zeta ⩽\xi <{\omega }_1$. We also prove that $C(K){\hat{\otimes }}_{\pi }C(L)$ is not isomorphic to $C(M)$ for any compact, Hausdorff space $M$.