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A Banach-Stone theorem for Riesz isomorphisms of Banach lattices. (English) Zbl 1160.46026
Summary: Let $X$ and $Y$ be compact Hausdorff spaces and $E,F$ be Banach lattices. Let $C\left(X,E\right)$ 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 ${\Phi }:C\left(X,E\right)\to C\left(Y,F\right)$ such that ${\Phi }f$ 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, ${\Phi }$ can be written as a weighted composition operator: ${\Phi }f\left(y\right)={\Pi }\left(y\right)\left(f\left(\varphi \left(y\right)\right)\right)$, where $\varphi$ is a homeomorphism from $Y$ onto $X$, and ${\Pi }\left(y\right)$ is a Riesz isomorphism from $E$ onto $F$ for every $y$ in $Y$. This generalizes some known results obtained recently.

##### MSC:
 46E40 Spaces of vector- and operator-valued functions 46B42 Banach lattices 47B65 Positive and order bounded operators