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A class of commutative Banach algebras with unique complete norm topology and continuous derivations


Author: John A. Lindberg
Journal: Proc. Amer. Math. Soc. 29 (1971), 516-520
MSC: Primary 46.55
DOI: https://doi.org/10.1090/S0002-9939-1971-0278076-6
MathSciNet review: 0278076
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Abstract: Let A be a semisimple commutative complex algebra with identity and $ \alpha (x)$ a monic polynomial over A. Two results are proved. If $ B = A[x]/(\alpha (x))$ is a Banach algebra under some norm, then B has a unique complete norm topology. Furthermore, B has nontrivial derivations if and only if B has a nontrivial radical.


References [Enhancements On Off] (What's this?)

  • [1] R. Arens and K. Hoffman, Algebraic extension of normed algebras, Proc. Amer. Math. Soc. 7 (1956), 203-210. MR 17, 1113. MR 0077901 (17:1113e)
  • [2] D. T. Brown, A class of Banach algebras with a unique norm topology, Proc. Proc. Amer. Math. Soc. 17 (1966) 1429-1434. MR 34 #4938. MR 0205103 (34:4938)
  • [3] B. E. Johnson, Continuity of derivations on commutative algebras, Amer. J. Math. 91 (1969), 1-10. MR 39 #7433. MR 0246127 (39:7433)
  • [4] R. J. Loy, Uniqueness of the complete norm topology and continuity of derivations on Banach algebras, Tôhoku Math. J. 22 (1970), 371-378. MR 0276768 (43:2508)
  • [5] J. A. Lindberg, Algebraic extensions of commutative Banach algebras, Pacific J. Math. 14 (1964), 559-583. MR 30 #3380. MR 0173166 (30:3380)
  • [6] C. E. Rickart, General theory of Banach algebras, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #5903. MR 0115101 (22:5903)
  • [7] O. Zariski and P. Samuel, Commutative algebra. Vol. I, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1958. MR 19, 833. MR 0090581 (19:833e)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0278076-6
Keywords: Uniqueness of complete norm topology, continuous derivations, normed extensions, commutative Banach algebras
Article copyright: © Copyright 1971 American Mathematical Society

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