-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

Published electronically:
April 15, 1999

MathSciNet review:
1487354

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field are studied. It is proven that the behaviour of these maps when is not locally compact is very different from the case of real- or complex-valued functions: in general, for -compact spaces and , the existence of a (weakly) separating additive map implies that and 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 and . 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:
http://dx.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