Uniqueness of the uniform norm with an application to topological algebras
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- by S. J. Bhatt and D. J. Karia PDF
- Proc. Amer. Math. Soc. 116 (1992), 499-503 Request permission
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.References
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
- Subhash J. Bhatt, On spectra and numerical ranges in locally $m$-convex algebras, Indian J. Pure Appl. Math. 14 (1983), no. 5, 596–603. MR 709314
- Helmut Goldmann, Uniform Fréchet algebras, North-Holland Mathematics Studies, vol. 162, North-Holland Publishing Co., Amsterdam, 1990. MR 1049384
- Bruno Kramm, A duality theorem for nuclear function algebras, Aspects of mathematics and its applications, North-Holland Math. Library, vol. 34, North-Holland, Amsterdam, 1986, pp. 495–532. MR 849575, DOI 10.1016/S0924-6509(09)70278-8
- Ronald Larsen, Banach algebras, Pure and Applied Mathematics, No. 24, Marcel Dekker, Inc., New York, 1973. An introduction. MR 0487369
- Ernest A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11 (1952), 79. MR 51444
- Martin Schottenloher, Michael problem and algebras of holomorphic functions, Arch. Math. (Basel) 37 (1981), no. 3, 241–247. MR 637767, DOI 10.1007/BF01234351 W. Żelazko, Selected topics in topological algebras, Univ. Lecture Notes in Math., vol. 31, Aarhus, 1971.
- W. Żelazko, On maximal ideals in commutative $m$-convex algebras, Studia Math. 58 (1976), no. 3, 291–298. MR 435852, DOI 10.4064/sm-58-3-291-298
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 499-503
- MSC: Primary 46H05; Secondary 46J05
- DOI: https://doi.org/10.1090/S0002-9939-1992-1097335-4
- MathSciNet review: 1097335