On the three space property for $C(K)$-spaces

TL;DR

The paper proves, under the assumption $\mathfrak p=\mathfrak c$, that for every Eberlein compact $L$ of weight $\mathfrak c$ there exists a nontrivial twisted sum $0\to c_0\to X\to C(L)\to 0$ where $X$ is not isomorphic to a $C(K)$-space, providing a negative answer to whether all twisted sums of $c_0$ and $C(K)$-spaces must be $C(K)$-spaces. The construction relies on an almost disjoint family to create a countable discrete extension $L$ of $M_1(K)$ and a pull-back that yields a $C(K)$-space non-equivalent twisted sum; key to the argument is embedding a copy of $\mathbb A(\mathfrak c)$ inside $M_1(K)$ and ensuring any potential free $c$-norming subset is destroyed. The results establish that the 3-space property fails for certain $K$, including $eta\omega\setminus\omega$ and Eberlein compacta of weight $\mathfrak c$, and they analyze when twisted sums preserve Lindenstrauss structure and related properties under MA-type hypotheses, highlighting open questions about broader generalizations.

Abstract

Assuming $\mathfrak p=\mathfrak c$, we show that for every Eberlein compact space $L$ of weight $\mathfrak c$ there exists a short exact sequence $0\to c_0\to X\to C(L)\to 0$, where the Banach space $X$ is not isomorphic to a $C(K)$-space.

Theorems & Definitions (27)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Lemma 2.7
• proof