# Extension operators and twisted sums of $c_0$ and $C(K)$ spaces

## TL;DR

This paper explores the existence of nontrivial twisted sums of $c_0$ and $C(K)$ spaces, providing set-theoretic conditions under which all such sums are trivial or nontrivial, and constructing examples accordingly.

## Contribution

It offers the first examples of compact spaces where all twisted sums are trivial under certain set-theoretic assumptions and analyzes extension operators between $C(K)$ spaces.

## Key findings

- Under Martin's axiom and negation of CH, certain $K$ yield only trivial twisted sums.
- Constructs examples of nontrivial twisted sums for specific classes of compacta.
- Identifies conditions on pairs of compact spaces that prevent extension operators.

## Abstract

We investigate the following problem posed by Cabello Sanch\'ez, Castillo, Kalton, and Yost: Let $K$ be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space $X$ containing a non-complemented copy $Z$ of $c_0$ such that the quotient space $X/Z$ is isomorphic to $C(K)$? Using additional set-theoretic assumptions we give the first examples of compact spaces $K$ providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either $K$ is the Cantor cube $2^{\omega_1}$ or $K$ is a separable scattered compact space of height $3$ and weight $\omega_1$, then every twisted sum of $c_0$ and $C(K)$ is trivial. We also construct nontrivial twisted sums of $c_0$ and $C(K)$ for $K$ belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces $K\subseteq L$ which do not admit an extension operator $C(K)\to C(L)$.