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

TL;DR

This paper demonstrates that under certain set-theoretic assumptions, there exist Banach spaces containing $c_0$ with quotients isomorphic to $C(L)$, where $L$ is an Eberlein compact space, but the space itself is not a $C(K)$-space.

Contribution

It establishes the existence of non-$C(K)$-spaces with specific quotient structures under the assumption $rak p=rak c$.

Findings

Existence of a short exact sequence with $c_0$ and $C(L)$

Construction of a Banach space not isomorphic to any $C(K)$-space

Dependence on set-theoretic assumption $rak p=rak c$

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.