# Twisted sums of $c_0$ and $C(K)$-spaces: A solution to the CCKY problem

## TL;DR

This paper characterizes Banach spaces Y that admit nontrivial twisted sums with c_0, providing a solution to a longstanding problem by linking it to properties of the weak* topology and using the continuum hypothesis.

## Contribution

It offers a characterization of spaces Y allowing nontrivial twisted sums with c_0 and proves their existence with C(K) spaces under the continuum hypothesis.

## Key findings

- Characterization of Banach spaces Y for twisted sums with c_0
- Existence of nontrivial twisted sums with C(K) spaces under continuum hypothesis
- Solution to the CCKY problem in Banach space theory

## Abstract

We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the continuum hypothesis $c_0$ has a nontrivial twisted sum with every space of the form $Y=C(K)$, where $K$ is compact and not metrizable. This gives a consistent positive solution to a problem posed by Cabello, Castillo, Kalton and Yost.