Let be compact Hausdorff spaces and a Banach lattice. Let be a Riesz isomorphism such that for all . The authors show that the above conditions imply that is Riesz homeomorphic to and is homeomorphic to . When is the real scalars, this result was proved by J. Cao, I. Reilly and H. Xiong [Acta Math. Hung. 98, 103–110 (2003; Zbl 1027.46025)].
A key ingredient in the proof is a lemma about Riesz isomorphisms on spaces of continuous functions on products of compact spaces. Let be a unit preserving Riesz isomorphism. If for all for all , then are homeomorphic to , respectively.