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A weak vector-valued Banach-Stone theorem


Authors: Leandro Candido and Elói Medina Galego
Journal: Proc. Amer. Math. Soc. 141 (2013), 3529-3538
MSC (2010): Primary 46B03, 46B25; Secondary 46E27, 46E40
DOI: https://doi.org/10.1090/S0002-9939-2013-11634-5
Published electronically: June 25, 2013
MathSciNet review: 3080174
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Abstract: For a locally compact Hausdorff space $ X$ and a Banach space $ E$, we denote by $ C_0(X, E)$ the space of $ E$-valued continuous functions on $ X$ which vanish at infinity, endowed with the supremum norm. In the spirit of the classical Banach-Stone theorem, we prove that if $ C_0(X, E)$ is isomorphic to $ C_{0}(Y, E)$, where $ E$ has non-trivial cotype and such that $ E$ is separable or $ E^*$ has the Radon-Nikodým property, then either $ X$ and $ Y$ are finite or $ X$ and $ Y$ have the same cardinality. In other words, we obtain a vector-valued extension of a 1978 B. Cengiz result, the scalar case $ E=\mathbb{R}$ or $ E=\mathbb{C}.$


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Additional Information

Leandro Candido
Affiliation: Department of Mathematics, University of São Paulo, IME, Rua do Matão 1010, São Paulo, Brazil
Email: lc@ime.usp.br

Elói Medina Galego
Affiliation: Department of Mathematics, University of São Paulo, IME, Rua do Matão 1010, São Paulo, Brazil
Email: eloi@ime.usp.br

DOI: https://doi.org/10.1090/S0002-9939-2013-11634-5
Keywords: Banach-Stone theorem
Received by editor(s): October 28, 2011
Received by editor(s) in revised form: December 26, 2011
Published electronically: June 25, 2013
Additional Notes: The first author was supported by CNPq, process number 142423/2011-4
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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