The Banach–Stone theorem revisited

Abstract

Let X and Y be compact Hausdorff spaces, and E and F be locally solid Riesz spaces. If π : C ( X , E ) → C ( Y , F ) is a 1-biseparating Riesz isomorphism then X and Y are homeomorphic, and E and F are Riesz isomorphic. This generalizes the main results of [Z. Ercan, S. Önal, Banach–Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (9) (2007) 2827–2829] and [X. Miao, C. Xinhe, H. Jiling, Banach–Stone theorems and Riesz algebras, J. Math. Anal. Appl. 313 (1) (2006) 177–183], and answers a conjecture in [Z. Ercan, S. Önal, Banach–Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (9) (2007) 2827–2829].