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.References
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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