This is an announcement for the paper “When does $C(K, X)$ contain a complemented copy of $c_0(\Gamma)$ iff $X$ does?” by Elói Medina Galegohttps://arxiv.org/find/math/1/au:+Galego_E/0/1/0/all/0/1, Vinícius Morelli Corteshttps://arxiv.org/find/math/1/au:+Cortes_V/0/1/0/all/0/1.
Abstract: Let $K$ be a compact Hausdorff space with weight $w(K)$, $\tau$ an infinite cardinal with cofinality $cf(\tau)>w(K)$ and $X$ a Banach space. In contrast with a classical theorem of Cembranos and Freniche it is shown that if $cf(\tau)>w(K)$ then the space $C(K, X)$ contains a complemented copy of $c_0(\Gamma)$ if and only if $X$ does. This result is optimal for every infinite cardinal $\tau$, in the sense that it can not be improved by replacing the inequality $cf(\tau)>w(K)$ by another weaker than it.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1709.01114