A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

2009/06/23 by Jin Xi Chen, Chen, Jin Xi, Zi Li Chen +3
Mathematics · #46B42 #47B65 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.0906.4196

openalex publication_date 2009/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \mathnormalΦ: C(X,E)→ C(Y,F) such that \mathnormalΦf 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, \mathnormalΦ can be written as a weighted composition operator: \mathnormalΦ f(y)=\mathnormalΠ(y)(f(φ(y))), where φ is a homeomorphism from Y onto X, and \mathnormalΠ(y) is a Riesz isomorphism from E onto F for every y in Y. This generalizes some known results obtained recently.

Related

• Banach-lattice isomorphisms of C₀(K,X) spaces which determine the locally compact spaces K
• Banach-Stone Theorems for maps preserving common zeros
• Order isomorphisms on function spaces
• Riesz representation theorems for vector lattices and Banach lattices of regular operators
• Lattice norms on the unitization of a truncated normed Riesz space