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The Gelfand 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
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)$.

References [Enhancements On Off] (What's this?)

  • [1] G. Bachman, E. Beckenstein, L. Narici and S. Warner, Rings of continuous functions with values in a topological field, Trans. Amer. Math. Soc. 204 (1975), 91-112. MR 0402687 (53:6503)
  • [2] E. Beckenstein, L. Narici and C. Suffel, Topological algebras, North-Holland, Amsterdam, 1977. MR 0473835 (57:13495)
  • [3] N. Bourbaki, Topologie générale, Chapitre 1, Hermann, Paris, 1971.
  • [4] J. M. Dominguez, Sobre la subálgebra de Gelfand del anillo de funciones continuas con valores en un cuerpo valuado no-arquimediano, Rev. Mat. Hisp.-Amer. (to appear).
  • [5] L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, New York, Heidelberg and Berlin, 1976. MR 0407579 (53:11352)
  • [6] N. Shilkret, Non-Archimedean Gelfand theory, Pacific J. Math. 32 (1970), 541-550. MR 0257752 (41:2401)

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

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