Isomorphic classification of projective tensor products of spaces of continuous functions

TL;DR

This paper classifies the isomorphic types of projective tensor products of continuous function spaces over certain compact spaces, showing they correspond uniquely to specific ordinal-indexed spaces and are not isomorphic to any single continuous function space.

Contribution

It provides a complete isomorphic classification of these tensor products for countable compact spaces, linking them to ordinal-indexed spaces and ruling out isomorphisms with single continuous function spaces.

Findings

Each tensor product is isomorphic to a unique ordinal-indexed space tensor product.

Such tensor products are not isomorphic to any single space of continuous functions.

The classification is complete for countable, compact, Hausdorff spaces.

Abstract

We prove that for infinite, countable, compact, Hausdorff spaces $K,L$, $C(K)\widehat{\otimes}_\pi C(L)$ is isomorphic to exactly one of the spaces $C(\omega^{\omega^\xi})\widehat{\otimes}_\pi C(\omega^{\omega^\zeta})$, $0\leqslant \zeta\leqslant \xi<\omega_1$. We also prove that $C(K)\widehat{\otimes}_\pi C(L)$ is not isomorphic to $C(M)$ for any compact, Hausdorff space $M$.