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

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

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
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

References

Arens, R. (1946), ‘A topology for spaces of transformations’, Ann. of Math. (2) 47, 480–495.CrossRefGoogle Scholar
Beckenstein, E., Narici, L. and Suffel, C. (1977), Topological algebras (Notas de Matemática No. 24, North-Holland, Amsterdam).Google Scholar
Bourdaud, G. (1975/1976), ‘Sur la dualité des algèbres localement convexes’, C. R. Acad. Sci. Paris 281, 1011–1014; 282, 313–316.Google Scholar
Bourbaki, N. (1940), Éléments de mathématiques, Topologie générale (Hermann, Paris).Google Scholar
Bourbaki, N. (1967), Éléments de mathématiques, Thèories spectrales (Hermann, Paris).Google Scholar
Clark, A. (1969), ‘Quasitopologies and compactly generated spaces’ Ph.D. Dissertation.Google Scholar
Day, B. (1972), ‘A reflexion theorem for closed categories’, J. Pure Appl. Algebra 2, 1–11.CrossRefGoogle Scholar
Dixmier, J. (1964), Les C*-algèbres et leurs réprésentations (Gauthier-Villars, Paris).Google Scholar
Dubuc, E. J. (1970), Kan extensions in enriched category theory (Lecture Notes No. 145, Springer-Verlag, Berlin).CrossRefGoogle Scholar
Dubuc, E. J. (1977), ‘Concrete quasitopoi’, Proc. Durham Conference (Springer-Verlag, Berlin) (to appear).Google Scholar
Dubuc, E. J. and Porta, H. (1971), ‘Convenient categories of topological algebras and their duality theory’, J. Pure Appl. Algebra 1, 281–316.CrossRefGoogle Scholar
Eilenberg, S. (1969), Spanier quasitopologies (Lecture notes, University of Paris).Google Scholar
Ginsburg, S. and Isbell, J. R. (1959), ‘Some operators on uniform spaces’, Trans. Amer. Math. Soc. 93, 145–168.CrossRefGoogle Scholar
Hsia, D. (1959), ‘Seminormed Banach rings with involutions’, Izv. Akad. Nauk SSSR, Ser. Mat 23, 509–528 (Russian).Google Scholar
Isbell, J. R. (1964), Uniform spaces (Math. Surveys No. 2 A.M.S., Providence).CrossRefGoogle Scholar
Kelley, J. (1955), General topology (Van Nostrand, New York).Google Scholar
Lamartin, W. F. (1977), On the foundations of k-group theory, Dissertationes Mathematicæ, CXLVI (polska. Akad. Nauk, Warszawa).Google Scholar
McCord, M. C. (1970), ‘Classifying spaces and infinite symmetric products’, Trans. Amer. Math. Soc. 146, 273–298.CrossRefGoogle Scholar
MacLane, S. (1971), Categories for the working mothematician (Springer-Verlag, Berlin).Google Scholar
Michael, E. (1952), ‘Locally multiplicatively-convex topological algebras’, Amer. Math. Soc. Memoirs 11.Google Scholar
Schubert, H. (1972), Categories (Springer-Verlag, Berlin).CrossRefGoogle Scholar
Spanier, E. (1959), ‘Infinite symmetric products, function spaces and duality’, Ann. of Math. 69, 142–198.CrossRefGoogle Scholar
Spanier, E. (1962), ‘Quasitopologies’, Duke Math. J. 30, 1–14.Google Scholar