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

TL;DR

This paper characterizes when the space of continuous functions $C(K,X)$ contains a complemented copy of $c_0( au)$, showing it depends precisely on whether $X$ does, under certain cofinality and weight conditions.

Contribution

It establishes a sharp criterion involving cofinality and weight for the existence of complemented copies of $c_0( au)$ in $C(K,X)$, extending classical results.

Findings

If cf$( au) >$ w$(K)$, then $C(K,X)$ contains a complemented $c_0( au)$ iff $X$ does.

The result is optimal; the inequality cannot be weakened.

Provides a precise condition linking the structure of $C(K,X)$ to properties of $X$ and the cardinal $ au$.

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.