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$\mathbb{N}$-compactness and automatic continuity
in ultrametric spaces
of bounded continuous functions


Author: Jesús Araujo
Journal: Proc. Amer. Math. Soc. 127 (1999), 2489-2496
MSC (1991): Primary 54C40, 46S10
DOI: https://doi.org/10.1090/S0002-9939-99-04781-4
Published electronically: April 15, 1999
MathSciNet review: 1487354
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Abstract: In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field $\mathbb{K}$ are studied. It is proven that the behaviour of these maps when $\mathbb{K}$ is not locally compact is very different from the case of real- or complex-valued functions: in general, for $\mathbb{N}$-compact spaces $X$ and $Y$, the existence of a (weakly) separating additive map $T: C^* (X)\rightarrow C^* (Y)$ implies that $X$ and $Y$ are homeomorphic, whereas when dealing with real-valued functions, this result is in general false, and we can just deduce the existence of a homeomorphism between the Stone-Cech compactifications of $X$ and $Y$. Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results.


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Additional Information

Jesús Araujo
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria Facultad de Ciencias, Avda. de los Castros, s. n., E-39071 Santander, Spain
Email: araujo@matesco.unican.es

DOI: https://doi.org/10.1090/S0002-9939-99-04781-4
Keywords: $\mathbb{N}$-compact, weakly separating map, nonarchimedean field
Received by editor(s): July 20, 1997
Received by editor(s) in revised form: November 6, 1997
Published electronically: April 15, 1999
Additional Notes: Research supported in part by the Spanish Dirección General de Investigación Científica y Técnica (DGICYT, PB95-0582).
Communicated by: Alan Dow
Article copyright: © Copyright 1999 American Mathematical Society

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