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 ${\text {Ch}}_{{\mathcal {H}}_i} K_i$ 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 $\left\| T\right\| \cdot \left\| T^{-1}\right\| <\min \lbrace \lambda ^+(E_1), \lambda ^+(E_2) \rbrace$ such that T and $T^{-1}$ preserve positivity, then ${\text {Ch}}_{{\mathcal {H}}_1} K_1$ is homeomorphic to ${\text {Ch}}_{{\mathcal {H}}_2} K_2$.