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

 

Banach-Stone theorem for Banach lattice valued continuous functions


Authors: Z. Ercan and S. Önal
Journal: Proc. Amer. Math. Soc. 135 (2007), 2827-2829
MSC (2000): Primary 46E40; Secondary 46B42
Published electronically: May 8, 2007
MathSciNet review: 2317958
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Abstract: Let $ X$ and $ Y$ be compact Hausdorff spaces, $ E$ be a Banach lattice and $ F$ be an AM space with unit. Let $ {\pi}:C(X,E)\rightarrow C(Y,F)$ be a Riesz isomorphism such that $ 0\not \in f(X)$ if and only if $ 0\not \in {\pi}(f)(Y)$ for each $ f\in C(X,E)$. We prove that $ X$ is homeomorphic to $ Y$ and $ E$ is Riesz isomorphic to $ F$. This generalizes some known results.


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Additional Information

Z. Ercan
Affiliation: Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
Email: zercan@metu.edu.tr

S. Önal
Affiliation: Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
Email: osul@metu.edu.tr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08788-6
PII: S 0002-9939(07)08788-6
Keywords: Riesz isomorphism, Banach lattices, Banach-Stone Theorem
Received by editor(s): June 16, 2005
Received by editor(s) in revised form: May 21, 2006
Published electronically: May 8, 2007
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society