A Lattice-valued Banach-Stone Theorem

Abstract

Let $X$ and $Y$ be two compact Hausdorff spaces, and $E$ be a Banach lattice. We show that if there is a non-vanishing preserving Riesz isomorphism $Phi: C(X, E) to C(Y)$, then $X$ is homeomorphic to $Y$ and $E$ is Riesz isomorphic to $mathbb R$.