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

Leandro Candido¹ ; Piotr Koszmider²

¹Instituto de Ciência e Tecnologia Universidade Federal de São Paulo Campus São José dos Campos – Parque Tecnológico Avenida Cesare Monsueto Giulio Lattes, 1211 12231-280, São José dos Campos, SP, Brazil
²Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland

Studia Mathematica, Tome 233 (2016) no. 3, pp. 209-226

Résumé

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.

Détail

DOI : 10.4064/sm8181-4-2016

Keywords: given compact hausdorff space consider banach space real continuous functions equivalently n fold injective tensor product hat otimes varepsilon banach space vector valued continuous functions dots dots address question existence complemented copies omega hat otimes varepsilon under hypothesis contains copy related results saab saab mathbin hat otimes varepsilon contains complemented copy infinite dimensional banach spaces contains copy galego hagler follows martin maximum has density omega contains copy omega times contains complemented copy omega main result under assumption clubsuit every mathbb there compact hausdorff space weight omega lindel weak topology contains copy omega n does contain complemented copy omega while does contain complemented copy omega shows additional set theoretic assumptions galego hagler nonseparable version cembrano freniche theorem necessary clarifies negative direction matter unsettled paper dow junnila pelant whether half pcc banach spaces weakly pcc