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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 Φ:C(X,E)C(Y,F) such that Φf 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, Φ can be written as a weighted composition operator: Φf(y)=Π(y)(f(ϕ(y))), where ϕ is a homeomorphism from Y onto X, and Π(y) is a Riesz isomorphism from E onto F for every y in Y. This generalizes some known results obtained recently.

46E40Spaces of vector- and operator-valued functions
46B42Banach lattices
47B65Positive and order bounded operators