On complemented copies of $c_0(\omega_1)$ in $C(K^n)$ spaces

TL;DR

This paper investigates the existence of complemented copies of $c_0( ext{ω}_1)$ in spaces of continuous functions on product spaces, revealing set-theoretic conditions affecting their structure.

Contribution

It constructs specific compact spaces under $ ext{club}$-like assumptions showing the nuanced behavior of complemented copies of $c_0( ext{ω}_1)$ in $C(K^n)$ spaces, clarifying set-theoretic influences.

Findings

Existence of compact spaces where $C(K_n^{n+1})$ contains a complemented $c_0( ext{ω}_1)$ but $C(K_n^n)$ does not.

Set-theoretic assumptions like $ ext{club}$ are necessary for certain structural properties.

Clarification of the relationship between set theory and the structure of Banach spaces of continuous functions.

Abstract

Given a compact Hausdorff space $K$ we consider the Banach space of real continuous functions $C(K^n)$ or equivalently the $n$-fold injective tensor product $\hat\bigotimes_{\varepsilon}C(K)$ or the Banach space of vector valued continuous functions $C(K, C(K, C(K ..., C(K)...)$. We address the question of the existence of complemented copies of $c_0(\omega_1)$ in $\hat\bigotimes_{\varepsilon}C(K)$ under the hypothesis that $C(K)$ contains an isomorphic copy of $c_0(\omega_1)$. This is related to the results of E. Saab and P. Saab that $X\hat\otimes_\varepsilon Y$ contains a complemented copy of $c_0$, if one of the infinite dimensional Banach spaces $X$ or $Y$ contains a copy of $c_0$ and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if $C(K)$ has density $\omega_1$ and contains a copy of $c_0(\omega_1)$, then $C(K\times K)$ contains a complemented copy $c_0(\omega_1)$. The main result is that under the assumption of $\clubsuit$ for every $n\in N$ there is a compact Hausdorff space $K_n$ of weight $\omega_1$ such that $C(K)$ is Lindel\"of in the weak topology, $C(K_n)$ contains a copy of $c_0(\omega_1)$, $C(K_n^n)$ does not contain a complemented copy of $c_0(\omega_1)$ while $C(K_n^{n+1})$ does contain a complemented copy of $c_0(\omega_1)$. This shows that additional set-theoretic assumptions in Galego and Hagler's nonseparable version of Cembrano and Freniche's theorem are necessary as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc.