When does $C(K,X)$ contain a complemented copy of $c_0(\Gamma)$ iff $X$ does?

Abstract

Let $K$ be a compact Hausdorff space with weight w$(K)$, $\tau$ an infinite cardinal with cofinality cf$(\tau)$ 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 $c_{0}(\tau)$ 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.