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arXiv · 1709.01114

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

Elói Medina Galego·Vinícius Morelli Cortes
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Abstract

Let $K$ be a compact Hausdorff space with weight w$(K)$, $τ$ an infinite cardinal with cofinality cf$(τ)$ and $X$ a Banach space. In contrast with a classical theorem of Cembranos and Freniche it is shown that if cf$(τ)>$ w$(K)$ then the space $C(K, X)$ contains a complemented copy of $c_{0}(τ)$ if and only if $X$ does. This result is optimal for every infinite cardinal $τ$, in the sense that it can not be improved by replacing the inequality cf$(τ)>$ w$(K)$ by another weaker than it.

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Elói Medina Galego, Vinícius Morelli Cortes. 2017-09-04. When does $C(K,X)$ contain a complemented copy of $c_0(Γ)$ iff $X$ does?. https://arxiv.org/abs/1709.01114

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