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On the boundedness of multiplicative and positive functionals

Published online by Cambridge University Press:  09 April 2009

Taqdir Husain  and
Shu-Bun Ng
Taqdir Husain
Affiliation:
McMaster University Hamilton, Ontario, Canada
Shu-Bun Ng
Affiliation:
U.B.C., Vancouver, British Columbia
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Abstract

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Let A be a complex sequentially complete, locally convex (not necessarily commutative) topological algebra with the defining family {pα}α∈D of seminorms in which (*): for each sequence xn → 0 there exists xm ∈ {xn} such that xmk → 0 as k → ∞. Then each multiplicative linear functional on a Fréchet algebra satisfying the above condition (*) is Continuous.

These results answer open questions (1) and (2) (Mem. Amer. Math. Soc. 11, 1953) in the affirmative for Fréchet algebras in which (*) holds. It is also shown that a positive linear functional on such algebras with identity and continuous involution is continuous, thus partially generalizing Shah's result (1959).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 4 , June 1976 , pp. 498 - 503
Copyright
Copyright © Australian Mathematical Society 1976

References

Arens, R. (1965), ‘On a class of locally convex algebras’, Proc. London Math. Soc. 15, 399–421.Google Scholar
Bourbaki, N. (1953. 1955), Espaces Vectoriels Topologiques (Chap. 1–V, Hermann).Google Scholar
Dixon, P. G. and Fremlin, D. H., (1972), ‘A remark concerning multiplicative functionals on LMC algebras’, J. London, Math. Soc. (2) 5, 231–232.CrossRefGoogle Scholar
Husain, T. and Ng, S-B. (1974), ‘Boundedness of multiplicative linear functionals’, Canad. Math. Bulletin 17, 213–215.CrossRefGoogle Scholar
Husain, T. and Ng, S-B. (1974), ‘On continuity of Algebra homomorphisms and uniqueness of metric topology’, Math. Zeit. 139, 1–4.CrossRefGoogle Scholar
Michael, E. A. (1953), ‘Locally m-convex topological algebras’, Mem. Amer. Math. Soc. No. 11.Google Scholar
Rickart, C. (1960), General theory of Banach algebras (van Nostrand).Google Scholar
Shah, T-S. (1959), ‘On seminormed rings with involution’, Izv. Akad. Nauk, SSSR, ser. Mat. 23, 509–528.Google Scholar
Zelazko, W. (1973), Banach algebras, algebras (Elsevier).Google Scholar
Zelazko, W. (1960), ‘On locally bounded and m-convex topological algebras’, Studia Math. 19, 333–356.CrossRefGoogle Scholar