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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Gel′fand subalgebra of real or non-Archimedean valued continuous functions
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by Jesús M. Domínguez PDF
Proc. Amer. Math. Soc. 90 (1984), 145-148 Request permission

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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Additional Information
  • © Copyright 1984 American Mathematical Society
  • 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