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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Local power series quotients of commutative Banach and Fréchet algebras
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by Marc P. Thomas PDF
Trans. Amer. Math. Soc. 355 (2003), 2139-2160 Request permission

Abstract:

We consider the relationship between derivations and local power series quotients for a locally multiplicatively convex Fréchet algebra. In §2 we derive necessary conditions for a commutative Fréchet algebra to have a local power series quotient. Our main result here is Proposition 2.6, which shows that if the generating element has finite closed descent, the algebra cannot be simply a radical algebra with identity adjoined—it must have nontrivial representation theory; if the generating element does not have finite closed descent, then the algebra cannot be a Banach algebra, and the generating element must be locally nilpotent (but non-nilpotent) in an associated quotient algebra. In §3 we consider a fundamental situation which leads to local power series quotients. Let $D$ be a derivation on a commutative radical Fréchet algebra ${\mathcal {R}}^{\sharp }$ with identity adjoined. We show in Theorem 3.10 that if the discontinuity of $D$ is not concentrated in the (Jacobson) radical, then ${\mathcal {R}}^{\sharp }$ has a local power series quotient. The question of whether such a derivation can have a separating ideal so large it actually contains the identity element has been recently settled in the affirmative by C. J. Read.
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Additional Information
  • Marc P. Thomas
  • Affiliation: Department of Mathematics, California State University at Bakersfield, Bakersfield, California 93311
  • Email: marc@cs.csubak.edu
  • Received by editor(s): August 27, 2001
  • Received by editor(s) in revised form: October 18, 2002
  • Published electronically: January 14, 2003
  • Additional Notes: The author thanks Pomona College for support as a Visiting Scholar during the summer of the Banach Algebras 1999 conference and the Centre for Mathematics and its Applications for support during the Banach Spaces, Operators, and Algebras Symposium in January 2001 at the Australian National University.
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 2139-2160
  • MSC (2000): Primary 46H05
  • DOI: https://doi.org/10.1090/S0002-9947-03-03251-3
  • MathSciNet review: 1953541