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Proceedings of the American Mathematical Society

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

 

 

A non-Archimedean Stone-Banach theorem


Authors: Edward Beckenstein and Lawrence Narici
Journal: Proc. Amer. Math. Soc. 100 (1987), 242-246
MSC: Primary 46P05
MathSciNet review: 884460
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Abstract: If the spaces $ C(T,R)$ and $ C(S,R)$ of continuous functions on $ S$ and $ T$ are linearly isometric, then $ T$ and $ S$ are homeomorphic. By the classical Stone-Banach theorem the only linear isometries of $ C(T,R)$ onto $ C(S,R)$ are of the form $ x \to a(x \circ h)$, where $ h$ is a homeomorphism of $ S$ onto $ T$ and $ a \in C(S,F)$ is of magnitude 1 for all $ s$ in $ S$. What happens if $ R$ is replaced by a field with a valuation? In brief, the result fails. We discuss "how" by way of developing a necessary and sufficient condition for the theorem to hold, along with some examples to illustrate the point.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0884460-1
Keywords: Nonarchimedean, Stone-Banach theorem
Article copyright: © Copyright 1987 American Mathematical Society