Isomorphisms of subspaces of vector-valued continuous functions

TL;DR

This paper establishes conditions under which isomorphic subspaces of vector-valued continuous functions imply homeomorphic Choquet boundaries, extending Banach-Stone theorems to certain vector-valued function spaces.

Contribution

It proves isomorphism implies boundary homeomorphism for subspaces of vector-valued continuous functions with specific geometric conditions.

Findings

Isomorphism with controlled norm implies boundary homeomorphism.

Provides a Banach-Stone type theorem for subspaces without $c_0$ copies.

Extends classical results to vector-valued function spaces with reflexive Banach spaces.

Abstract

We deal with isomorphic Banach-Stone type theorems for closed subspaces of vector-valued continuous functions. Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. For $i=1,2$, let $E_i$ be a reflexive Banach space over $\mathbb{F}$ with a certain parameter $\lambda(E_i)>1$, which in the real case coincides with the Schaffer constant of $E_i$, let $K_i$ be a locally compact (Hausdorff) topological space and let $\mathcal{H}_i$ be a closed subspace of $\mathcal{C}_0(K_i, E_i)$ such that each point of the Choquet boundary $\mathcal{Ch}_{\mathcal{H}_i} K_i$ of $\mathcal{H}_i$ is a weak peak point. We show that if there exists an isomorphism $T:\mathcal{H}_1 \rightarrow \mathcal{H}_2$ with $\Vert T \Vert \cdot \Vert T^{-1} \Vert<\min \lbrace \lambda(E_1), \lambda(E_2) \rbrace$, then $\mathcal{Ch}_{\mathcal{H}_1} K_1$ is homeomorphic to $\mathcal{Ch}_{\mathcal{H}_2} K_2$. Next we provide an analogous version of the weak vector-valued Banach-Stone theorem for subspaces, where the target spaces do not contain an isomorphic copy of $c_0$.