A Banach-stone Theorem for Riesz Isomorphisms of Banach Lattices

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.