On isomorphisms of Banach spaces of continuous functions

TL;DR

This paper investigates the structure of Banach spaces of continuous functions on compact spaces, establishing conditions under which two such spaces are isomorphic and exploring implications for specific classes like Corson compact spaces.

Contribution

It provides a new criterion involving $ ext{pi}$-bases for when $C(K)$ and $C(L)$ are isomorphic, linking topological properties of $K$ and $L$.

Findings

If $C(K)$ and $C(L)$ are isomorphic, then $K$ has a $ ext{pi}$-base with certain image properties.

The results give insights into the structure of spaces like $C([0,1]^ ext{cardinality})$ and $C(K)$ for Corson compact $K$.

The paper advances understanding of the relationship between the topology of compact spaces and the Banach space isomorphisms of their continuous functions.

Abstract

We prove that if $K$ and $L$ are compact spaces and $C(K)$ and $C(L)$ are isomorphic as Banach spaces then $K$ has a $\pi$-base consisting of open sets $U$ such that $\bar{U}$ is a continuous image of some compact subspace of $L$. This gives some information on isomorphic classes of the spaces of the form $C([0,1]^\kappa)$ and $C(K)$ where $K$ is Corson compact.