The behaviour of quasi-linear maps on $C(K)$-spaces

TL;DR

This paper investigates quasi-linear maps on $C(K)$-spaces, demonstrating that non-trivial maps exhibit non-trivial behavior on subspaces isomorphic to $c_0$, with implications for twisted sums of $ ext{l}_1$ and $c_0$.

Contribution

It establishes the non-triviality of quasi-linear maps on $C(K)$-spaces on specific subspaces, combining topological and functional analysis techniques.

Findings

Non-locally trivial quasi-linear maps are non-trivial on subspaces isomorphic to $c_0$

The result is shown to be optimal through examples

Application to existence of nontrivial twisted sums of $ ext{l}_1$ and $c_0$

Abstract

In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a $C(K)$ must be nontrivial on a subspace isomorphic to $c_0$. We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of $\ell_1$ and $c_0$.