Subprojectivity of projective tensor products of Banach spaces of continuous functions

TL;DR

This paper extends the characterization of subprojectivity and $c_0$-saturation in projective tensor products of continuous function spaces from metrizable to general compact Hausdorff spaces, for any tensor power.

Contribution

It removes the metrizability hypothesis and generalizes the result to n-fold tensor products of $C(K)$ spaces.

Findings

$c_0$-saturation and subprojectivity are equivalent in these tensor products

The properties hold if and only if all spaces $K_i$ are scattered

Results extend previous work to non-metrizable spaces and higher tensor powers

Abstract

Galego and Samuel showed that if $K,L$ are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is $c_0$-saturated if and only if it is subprojective if and only if $K$ and $L$ are both scattered. We remove the hypothesis of metrizability from their result, and extend it from the case of the two-fold projective tensor product to the general $n$-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces $K_1, \ldots, K_n$, $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is $c_0$-saturated if and only if it is subprojective if and only if each $K_i$ is scattered.