Plasticity of the unit ball of some $C(K)$ spaces

TL;DR

This paper investigates the geometric property called plasticity of the unit ball in certain $C(K)$ spaces, showing under specific conditions that all non-expansive bijections are actually isometries.

Contribution

It establishes new conditions under which the unit ball of $C(K)$ spaces exhibits plasticity, extending understanding of their geometric structure.

Findings

Unit ball of $C(K)$ is plastic when $K$ has finitely many accumulation points.

Non-expansive homeomorphisms are isometries for $C(K)$ with dense isolated points.

Results apply to specific classes of compact metrizable and zero-dimensional spaces.

Abstract

We show that if $K$ is a compact metrizable space with finitely many accumulation points, then the closed unit ball of $C(K)$ is a plastic metric space, which means that any non-expansive bijection from $B_{C(K)}$ onto itself is in fact an isometry. We also show that if $K$ is a zero-dimensional compact Hausdorff space with a dense set of isolated points, then any non-expansive homeomorphism of $B_{C(K)}$ is an isometry.