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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Email: jinxichen@home.swjtu.edu.cn

Zi Li Chen
Affiliation: Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, People’s Republic of China
Email: zlchen@home.swjtu.edu.cn

Ngai-Ching Wong
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
Email: wong@math.nsysu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09582-8
PII: S 0002-9939(08)09582-8
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.