An elementary approach to a lattice-valued Banach-Stone Theorem

Abstract

Let $ X $ and $ Y $ be compact Hausdorff spaces, and $ E $ be a nonzero real Banach lattice. In this note, we give an elementary proof of a lattice-valued Banach-Stone theorem by Cao, Reilly and Xiong [3] which asserts that if there exists a Riesz isomorphism $ \Phi: C(X,E)\rightarrow C(Y,\mathbb{R}) $ such that $ \Phi(f) $ has no zeros if $ f $ has none, then $ X $ is homeomorphic to $ Y $ and $ E $ is Riesz isomorphic to $ \mathbb{R} $.