Plasticity of the unit ball of some \(C(K)\) spaces (Q6063631)

A metric space \(M\) is said to be \textit{plastic} if every non-expansive bijection \(F: M \to M\) is an isometry. There is an open question posed in 2016 (see [\textit{B.~Cascales} et al., Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 110, No.~2, 723--727 (2016; Zbl 1362.46011)]), whether the unit ball \(B_X\) of every Banach space \(X\) is plastic. The author adds new examples of Banach spaces for which the answer is positive. Namely, it is demonstrated that, for a compact metrizable space \(K\) with finitely many accumulation points, \(B_{C(K)}\) is plastic. This extends the recent result by \textit{N.~Leo} for \(K\) being the one-point compactification of \(\mathbb N\) [J. Math. Anal. Appl. 507, No.~1, Article ID 125718, 13~p. (2022, Zbl 1484.46013)]. Another result of the article says that, for every zero-dimensional compact Hausdorff space \(K\) with a dense set of isolated points, any non-expansive homeomorphism \(F : B_{C(K)} \to B_{C(K)}\) is an isometry. In particular, this implies that every non-expansive homeomorphism \(F : B_{\ell_\infty} \to B_{\ell_\infty}\) is an isometry.