An Amir-Cambern theorem for subspaces of Banach lattice-valued continuous functions

Abstract

For $i=1,2$, let $E_i$ be a reflexive Banach lattice over $\mathbb{R}$ with a certain parameter ${\lambda }^{+}(E_i)>1$, 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 $\Vert T\Vert \cdot \Vert T^{-1}\Vert <\min \{{\lambda }^{+}(E_1),{\lambda }^{+}(E_2)\}$ such that $T$ and $T^{-1}$ preserve positivity, then ${\mathcal{C}h}_{{\mathcal{H}}_1}{K}_1$ is homeomorphic to ${\mathcal{C}h}_{{\mathcal{H}}_2}{K}_2$.

arXiv admin note: text overlap with arXiv:1908.09680