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Transactions of the American Mathematical Society

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Local power series quotients of commutative Banach and Fréchet algebras

Author: Marc P. Thomas
Journal: Trans. Amer. Math. Soc. 355 (2003), 2139-2160
MSC (2000): Primary 46H05
Published electronically: January 14, 2003
MathSciNet review: 1953541
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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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Marc P. Thomas
Affiliation: Department of Mathematics, California State University at Bakersfield, Bakersfield, California 93311

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.
Article copyright: © Copyright 2003 American Mathematical Society