Banach-Stone theorem for Banach lattice valued continuous functions

Abstract

Let X X and Y Y be compact Hausdorff spaces, E E be a Banach lattice and F F be an AM space with unit. Let π : C ( X , E ) → C ( Y , F ) {\pi }:C(X,E)\rightarrow C(Y,F) be a Riesz isomorphism such that 0 ∉ f ( X ) 0\not \in f(X) if and only if 0 ∉ π ( f ) ( Y ) 0\not \in {\pi }(f)(Y) for each f ∈ C ( X , E ) f\in C(X,E) . We prove that X X is homeomorphic to Y Y and E E is Riesz isomorphic to F F . This generalizes some known results.