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), 499503
MSC:
Primary 46H05; Secondary 46J05
MathSciNet review:
1097335
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Abstract: Any squarepreserving 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]
Frank
F. Bonsall and John
Duncan, Complete normed algebras, SpringerVerlag, New
YorkHeidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete,
Band 80. MR
0423029 (54 #11013)
 [2]
Subhash
J. Bhatt, On spectra and numerical ranges in locally
𝑚convex algebras, Indian J. Pure Appl. Math.
14 (1983), no. 5, 596–603. MR 709314
(85g:46057)
 [3]
Helmut
Goldmann, Uniform Fréchet algebras, NorthHolland
Mathematics Studies, vol. 162, NorthHolland Publishing Co.,
Amsterdam, 1990. MR 1049384
(91f:46073)
 [4]
Bruno
Kramm, A duality theorem for nuclear function algebras,
Aspects of mathematics and its applications, NorthHolland Math. Library,
vol. 34, NorthHolland, Amsterdam, 1986, pp. 495–532. MR 849575
(88d:46083), http://dx.doi.org/10.1016/S09246509(09)702788
 [5]
Ronald
Larsen, Banach algebras, Marcel Dekker, Inc., New York, 1973.
An introduction; Pure and Applied Mathematics, No. 24. MR 0487369
(58 #7010)
 [6]
Ernest
A. Michael, Locally multiplicativelyconvex topological
algebras, Mem. Amer. Math. Soc., 1952 (1952),
no. 11, 79. MR 0051444
(14,482a)
 [7]
Martin
Schottenloher, Michael problem and algebras of holomorphic
functions, Arch. Math. (Basel) 37 (1981), no. 3,
241–247. MR
637767 (83b:46061), http://dx.doi.org/10.1007/BF01234351
 [8]
W. Żelazko, Selected topics in topological algebras, Univ. Lecture Notes in Math., vol. 31, Aarhus, 1971.
 [9]
W.
Żelazko, On maximal ideals in commutative 𝑚convex
algebras, Studia Math. 58 (1976), no. 3,
291–298. MR 0435852
(55 #8803)
 [1]
 F. F. Bonsall and J. Duncan, Complete normed algebras, SpringerVerlag, 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), 596603. MR 709314 (85g:46057)
 [3]
 H. Goldmann, Uniform Fréchet algebras, NorthHolland, 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. 495532. MR 849575 (88d:46083)
 [5]
 R. Larsen, Banach algebras, MarcelDekker, 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), 241247. 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), 291298. MR 0435852 (55:8803)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199210973354
PII:
S 00029939(1992)10973354
Keywords:
Uniform Banach algebra,
regular Banach algebra,
topological algebra,
algebra
Article copyright:
© Copyright 1992
American Mathematical Society
