Isomorphisms of subspaces of vector-valued continuous functions

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 ${\mathcal{C}h}_{{\mathcal{H}}_i}{K}_i$ of ${\mathcal{H}}_i$ is a weak peak point. We show that if there exists an isomorphism $T:{\mathcal{H}}_1\to {\mathcal{H}}_2$ with $\Vert T\Vert \cdot \Vert T^{-1}\Vert <\min \{\lambda (E_1),\lambda (E_2)\}$, then ${\mathcal{C}h}_{{\mathcal{H}}_1}{K}_1$ is homeomorphic to ${\mathcal{C}h}_{{\mathcal{H}}_2}{K}_2$. 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 $c_0$.