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

DOI:
https://doi.org/10.1090/S0002-9939-1992-1097335-4

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 -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 -algebra is a uniform Banach algebra. Relevant examples, showing that the respective assumptions regarding regularity, -algebra norm, and uniform property of topology cannot be omitted, have been discussed.

**[1]**F. F. Bonsall and J. Duncan,*Complete normed algebras*, Springer-Verlag, Berlin, Heidelberg, and New York, 1973. MR**0423029 (54:11013)****[2]**S. J. Bhatt,*On spectra and numerical ranges in locally**-convex algebras*, Indian J. Pure Appl. Math.**14**(1983), 596-603. MR**709314 (85g:46057)****[3]**H. Goldmann,*Uniform Fréchet algebras*, North-Holland, Amsterdam, 1990. MR**1049384 (91f:46073)****[4]**B. Kramm,*A duality theorem for nuclear function algebras*, Aspects of Mathematics and its Applications (J. A. Barroso, ed.), Elsevier, Amsterdam, 1986, pp. 495-532. MR**849575 (88d:46083)****[5]**R. Larsen,*Banach algebras*, Marcel-Dekker, New York, 1973. MR**0487369 (58:7010)****[6]**E. Michael,*Locally multiplicatively convex topological algebras*, Mem. Amer. Math. Soc., vol. 11, Amer. Math. Soc., Providence, RI, 1952. MR**0051444 (14:482a)****[7]**M. Schottenloher,*Michael problem and the algebras of holomorphic functions*, Ark. Mat.**37**(1981), 241-247. MR**637767 (83b:46061)****[8]**W. Żelazko,*Selected topics in topological algebras*, Univ. Lecture Notes in Math., vol. 31, Aarhus, 1971.**[9]**-,*On maximal ideals in commutative**-convex algebras*, Studia Math.**58**(1976), 291-298. MR**0435852 (55:8803)**

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1097335-4

Keywords:
Uniform Banach algebra,
regular Banach algebra,
topological algebra,
-algebra

Article copyright:
© Copyright 1992
American Mathematical Society