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An Amir-Cambern theorem for subspaces of Banach lattice-valued continuous functions (2006.07195v1)

Published 11 Jun 2020 in math.FA

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 $\mathcal{Ch}{\mathcal{H}i} K_i$ of $\mathcal{H}_i$ is a weak peak point. We show that if there exists an isomorphism $T\colon \mathcal{H}_1 \to \mathcal{H}_2$ with $\Vert T \Vert \cdot \Vert T{-1} \Vert<\min \lbrace \lambda+(E_1), \lambda+(E_2) \rbrace$ such that $T$ and $T{-1}$ preserve positivity, then $\mathcal{Ch}{\mathcal{H}1} K_1$ is homeomorphic to $\mathcal{Ch}{\mathcal{H}_2} K_2$.

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