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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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