A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

Abstract

Let $X$ and $Y$ be compact Hausdorff spaces, and $E$, $F$ be Banach lattices. Let $C(X,E)$ denote the Banach lattice of all continuous $E$-valued functions on $X$ equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism $\mathnormal{\Phi}: C(X,E)\to C(Y,F)$ such that $\mathnormal{\Phi}f$ is non-vanishing on $Y$ if and only if $f$ is non-vanishing on $X$, then $X$ is homeomorphic to $Y$, and $E$ is Riesz isomorphic to $F$. In this case, $\mathnormal{\Phi}$ can be written as a weighted composition operator: $\mathnormal{\Phi} f(y)=\mathnormal{\Pi}(y)(f(\varphi(y)))$, where $\varphi$ is a homeomorphism from $Y$ onto $X$, and $\mathnormal{\Pi}(y)$ is a Riesz isomorphism from $E$ onto $F$ for every $y$ in $Y$. This generalizes some known results obtained recently.