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Uniform spaces, spanier quasitopologies, and a duality for locally convex algebras

Part of: Commutative Banach algebras and commutative topological algebras Topological algebras, normed rings and algebras, Banach algebras Categories with structure Spaces with richer structures Special categories

Published online by Cambridge University Press:  09 April 2009

Eduardo J. Dubuc  and
Horacio Porta
Eduardo J. Dubuc
Affiliation:
University of Illinois at Urbana-Champaign UrbanaIllinois 61801, U.S.A.
Horacio Porta
Affiliation:
Facultad de Ciencias Exactas Universidad de Buenos Aires Buenos Aires, Argentina
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Abstract

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Gelfand-type duality results can be obtained for locally convex algebras using a quasitopological structure on the spectrum of an algebra (as opposed to the topologies traditionally considered). In this way, the duality between (commutative, with identity) C*-algebras and compact spaces can be extended to pro-C*-algebras and separated quasitopologies. The extension is provided by a functional representation of an algebra A as the algebra of all continuous numerical functions on a quasitopological space. The first half of the paper deals with uniform spaces and quasitopologies, and has independent interest.

MSC classification

Secondary: 46H05: General theory of topological algebras 46J10: Banach algebras of continuous functions, function algebras 18B30: Categories of topological spaces and continuous mappings 18D20: Enriched categories (over closed or monoidal categories) 54E15: Uniform structures and generalizations

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 1 , February 1980 , pp. 99 - 128
Copyright
Copyright © Australian Mathematical Society 1980

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