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Proceedings of the American Mathematical Society

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

 
 

 

The Gel′fand subalgebra of real or non-Archimedean valued continuous functions


Author: Jesús M. Domínguez
Journal: Proc. Amer. Math. Soc. 90 (1984), 145-148
MSC: Primary 54C40; Secondary 46J10, 46P05
DOI: https://doi.org/10.1090/S0002-9939-1984-0722433-7
MathSciNet review: 722433
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Abstract: Let $L$ be either the field of real numbers or a nonarchimedean rank-one valued field. For topological space $T$ we study the Gelfand subalgebra ${C_0}(T,L)$ of the algebra of all $L$-valued continuous functions $C(T,L)$. The main result is that if $T$ is a paracompact locally compact Hausdorff space, which is ultraregular if $L$ is nonarchimedean, then for $f \in C(T,L)$ the following statements are equivalent: (1) There exists a compact set $K \subset T$ such that $f(T - K)$ is finite, (2) $f$ has finite range on every discrete closed subset of $T$, and (3) $f \in {C_0}(T,L)$.


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Keywords: Valued field, continuous function algebras, Gelfand subalgebra
Article copyright: © Copyright 1984 American Mathematical Society