The ball fixed point property in spaces of continuous functions

TL;DR

This paper investigates the conditions under which spaces of continuous functions on compact spaces have the ball fixed point property, linking topological properties of the underlying space to fixed point existence.

Contribution

It characterizes classes of compact spaces where the BFPP holds for $C(K)$ spaces and provides counterexamples, advancing understanding of fixed point properties in functional analysis.

Findings

BFPP holds for certain extremally disconnected compact spaces.

Counterexamples show BFPP fails for some $F$-spaces.

$ ext{C}(eta ext{N}^*)$ fails BFPP under CH.

Abstract

A Banach space $X$ has the ball fixed point property (BFPP) if for every closed ball $B$ and for every nonexpansive mapping $T\colon B\to B$, there is a fixed point. We study the BFPP for $C(K)$-spaces. Our goal is to determine topological properties over $K$ that may determine the failure or fulfillment of the BFPP for the space of continuous functions $C(K)$. We prove that the class of compact spaces $K$ for which the BFPP holds lies between the class of extremally disconnected compact spaces and the class of compact $F$-spaces. We give a family of examples of $F$-spaces $K$ for which the BFPP fails. As a result, we prove that for every cardinal $\kappa$, $\kappa$-order completeness or $\kappa$-hyperconvexity of $C(K)$ are not enough for the BFPP and we obtain that $\ell_\infty/c_0 = C(\mathbb{N}^*)$ fails BFPP under the Continuum Hypothesis. The space $C([0,+\infty)^*)$ is also analyzed. It is left as an open problem whether all compact spaces for which the BFPP holds are in fact the extremally disconnected compact sets.