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 and be compact Hausdorff spaces, and , be Banach lattices. Let denote the Banach lattice of all continuous -valued functions on equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism such that is non-vanishing on if and only if is non-vanishing on , then is homeomorphic to , and is Riesz isomorphic to . In this case, can be written as a weighted composition operator: , where is a homeomorphism from onto , and is a Riesz isomorphism from onto for every in . 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

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.