Banach-Stone theorem for Banach lattice valued continuous functions

Abstract:

Let $X$ and $Y$ be compact Hausdorff spaces, $E$ be a Banach lattice and $F$ be an AM space with unit. Let ${\pi }:C(X,E)\rightarrow C(Y,F)$ be a Riesz isomorphism such that $0\not \in f(X)$ if and only if $0\not \in {\pi }(f)(Y)$ for each $f\in C(X,E)$. We prove that $X$ is homeomorphic to $Y$ and $E$ is Riesz isomorphic to $F$. This generalizes some known results.