-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 | 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.

**[A]**J. Araujo,*Nonarchimedean \v{S}ilov boundaries and multiplicative isometries*, Indag. Math., N. S.**8**(1997), 417-431. CMP**98:12****[ABN1]**J. Araujo, E. Beckenstein and L. Narici,*Biseparating maps and homeomorphic real-compactifications*, J. Math. Anal. App.**192**(1995), 258-265. MR**96b:46038****[ABN2]**J. Araujo, E. Beckenstein and L. Narici,*Separating maps and the nonarchimedean Hewitt theorem*, Ann. Math. Blaise Pascal**2**(1995), 19-27. MR**96d:46099****[ABN3]**J. Araujo, E. Beckenstein and L. Narici,*When is a separating map biseparating?*, Archiv Math.**67**(1996), 395-407. MR**97f:47026****[AF]**J. Araujo and J. J. Font,*Linear isometries between subspaces of continuous functions*, Trans. A.M.S.**349**(1997), 413-428. MR**97d:46026****[BNT]**E. Beckenstein, L. Narici and A. R. Todd,*Automatic continuity of linear maps on spaces of continuous functions*, Manuscripta Math.**62**(1988), 257-275. MR**89j:47020****[GJ]**L. Gillman and M. Jerison,*Rings of continuous functions*. Van Nostrand, Princeton, 1960. MR**22:6994****[H]**M. Henriksen,*On the equivalence of the ring, lattice, and semigroup of continuous functions*, Proc. Amer. Math. Soc.**7**(1956), 959-960. MR**18:559a****[J]**K. Jarosz,*Automatic continuity of separating linear isomorphisms*, Canad. Math. Bull.**33**(1990), 139-144. MR**92j:46049****[vP]**M. van der Put,*Algèbres de fonctions continues -adiques*, Indag. Math.**30**(1968), 401-420. MR**39:784****[vR]**A. C. M. van Rooij,*Nonarchimedean Functional Analysis*, Dekker, New York 1978. MR**81a:46084****[S]**R. Staum,*The algebra of bounded continuous functions into a nonarchimedean field*, Pacific J. Math.**50**(1974), 169-185. MR**49:5803**

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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