Isomorphisms of $\mathcal{C}(K, E)$ spaces and height of $K$

TL;DR

This paper investigates the geometric structure of spaces of continuous functions on compact spaces with Banach space values, establishing lower bounds on their Banach-Mazur distances based on the height of the underlying spaces, and showing non-isomorphism when these heights differ significantly.

Contribution

It introduces new lower estimates for the Banach-Mazur distance between such function spaces, linking these estimates to the ordinal height of the underlying compact spaces, a novel approach even for real-valued functions.

Findings

Lower bounds on Banach-Mazur distances depend on the height of $K$.

Spaces are not isomorphic if the heights of $K_1$ and $K_2$ differ substantially.

Results apply to spaces of continuous functions with Banach space values, not just real-valued functions.

Abstract

Let $K_1$, $K_2$ be compact Hausdorff spaces and $E_1, E_2$ be Banach spaces not containing a copy of $c_0$. We establish lower estimates of the Banach-Mazur distance between the spaces of continuous functions $\mathcal{C}(K_1, E_1)$ and $\mathcal{C}(K_2, E_2)$ based on the ordinals $ht(K_1)$, $ht(K_2)$, which are new even for the case of spaces of real valued functions on ordinal intervals. As a corollary we deduce that $\mathcal{C}(K_1, E_1)$ and $\mathcal{C}(K_2, E_2)$ are not isomorphic if $ht(K_1)$ is substantially different from $ht(K_2)$.