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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A Banach-Stone theorem for Riesz isomorphisms of Banach lattices
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by Jin Xi Chen, Zi Li Chen and Ngai-Ching Wong PDF
Proc. Amer. Math. Soc. 136 (2008), 3869-3874 Request permission

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
  • Received by editor(s): June 1, 2007
  • Published electronically: June 24, 2008
  • Communicated by: N. Tomczak-Jaegermann
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 3869-3874
  • MSC (2000): Primary 46E40; Secondary 46B42, 47B65
  • DOI: https://doi.org/10.1090/S0002-9939-08-09582-8
  • MathSciNet review: 2425726