On complemented copies of $c_0(\omega _1)$ in $C(K^n)$ spaces
Volume 233 / 2016
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 $\hat{\otimes}^{n}_{\varepsilon}C(K)$ or the Banach space of vector valued continuous functions $C(K, C(K, C(K \dots, C(K)\dots)$. We address the question of the existence of complemented copies of $c_0(\omega_1)$ in $\hat{\otimes}^n_{\varepsilon}C(K)$ under the hypothesis that $C(K)$ contains such a copy. This is related to the results of E. Saab and P. Saab that $X\mathbin{\hat\otimes_\varepsilon} Y$ 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 of $c_0(\omega_1)$.
Our main result is that under the assumption of $\clubsuit$ for every $n\in \mathbb N$ 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.
