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A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

Authors: Jin Xi Chen, Zi Li Chen and Ngai-Ching Wong
Journal: Proc. Amer. Math. Soc. 136 (2008), 3869-3874
MSC (2000): Primary 46E40; Secondary 46B42, 47B65
Published electronically: June 24, 2008
MathSciNet review: 2425726
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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 $ \Phi: C(X,E)\rightarrow C(Y,F)$ such that $ \Phi f$ 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, $ \Phi$ can be written as a weighted composition operator: $ \Phi f(y)=\Pi(y)(f(\varphi(y)))$, where $ \varphi$ is a homeomorphism from $ Y$ onto $ X$, and $ \Pi(y)$ is a Riesz isomorphism from $ E$ onto $ F$ for every $ y$ in $ Y$. This generalizes some known results obtained recently.

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Additional Information

Jin Xi Chen
Affiliation: Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, People’s Republic of China

Zi Li Chen
Affiliation: Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, People’s Republic of China

Ngai-Ching Wong
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Keywords: Banach lattice, Banach-Stone theorem, Riesz isomorphism, weighted composition operator
Received by editor(s): June 1, 2007
Published electronically: June 24, 2008
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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