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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

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