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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Uniqueness of the uniform norm with an application to topological algebras


Authors: S. J. Bhatt and D. J. Karia
Journal: Proc. Amer. Math. Soc. 116 (1992), 499-503
MSC: Primary 46H05; Secondary 46J05
MathSciNet review: 1097335
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Abstract: Any square-preserving linear seminorm on a unital commutative algebra is submultiplicative; and the uniform norm on a uniform Banach algebra is the only uniform $ Q$-algebra norm on it. This is proved and is used to show that (i) uniform norm on a regular uniform Banach algebra is unique among all uniform (not necessarily complete) norms and (ii) a complete uniform topological algebra that is a $ Q$-algebra is a uniform Banach algebra. Relevant examples, showing that the respective assumptions regarding regularity, $ Q$-algebra norm, and uniform property of topology cannot be omitted, have been discussed.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1097335-4
PII: S 0002-9939(1992)1097335-4
Keywords: Uniform Banach algebra, regular Banach algebra, topological algebra, $ Q$-algebra
Article copyright: © Copyright 1992 American Mathematical Society