On complemented copies of $c_0({\omega }_1)$ in $C(K^n)$ spaces

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 ${\widehat{⨂}}_{\epsilon }C(K)$ 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 ${\widehat{⨂}}_{\epsilon }C(K)$ 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 $X{\hat{\otimes }}_{\epsilon }Y$ 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 $♣$ for every $n\in N$ there is a compact Hausdorff space $K_n$ of weight ${\omega }_1$ such that $C(K)$ is Lindelöf 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.