Nonseparable $C(K)$-spaces can be twisted when $K$ is a finite height compact

TL;DR

The paper demonstrates that for certain nonmetrizable compact spaces with specific properties, there exist nontrivial extensions of $C(K)$ by $c_0$, revealing new insights into the structure of these Banach spaces.

Contribution

It proves the existence of nontrivial exact sequences involving $C(K)$ spaces when $K$ is a nonmetrizable compact with empty $ ext{ω}$-derived set, addressing a longstanding open problem.

Findings

Existence of nontrivial extensions $0 o c_0 o E o C(K) o 0$ for certain compact spaces.

Partial resolution of whether $Ext(C(K), c_0) eq 0$ for nonmetrizable compact $K$.

Shows that $C(K)$-spaces can be twisted in the nonseparable case.

Abstract

We show that, given a nonmetrizable compact space $K$ having $\omega$-derived set empty, there always exist nontrivial exact sequences $0\to c_0\to E\to C(K)\to 0$. This partially solves a problem posed in several papers: Is $Ext(C(K), c_0)\neq 0$ for $K$ a nonmetrizable compact set?