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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 $$C(X,E)$$ 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 $$\Phi: C(X,E)\rightarrow C(Y,F)$$ 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: $$\Phi f(y)=\Pi(y)(f(\varphi(y)))$$, where $$\varphi$$ is a homeomorphism from $$Y$$ onto $$X$$, and $$\Pi(y)$$ 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 linear operators and order-bounded operators
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