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 be a derivation on a commutative radical Fréchet algebra with identity adjoined. We show in Theorem 3.10 that if the discontinuity of is not concentrated in the (Jacobson) radical, then 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.

**[Allan1]**G. R. Allan,*Embedding the algebra of formal power series in a Banach algebra*, Proc. London Math. Soc. (3)**25**(1972), 329–340. MR**0305071****[Allan2]**Graham R. Allan,*Fréchet algebras and formal power series*, Studia Math.**119**(1996), no. 3, 271–288. MR**1397495****[Arens]**Richard Arens,*A generalization of normed rings*, Pacific J. Math.**2**(1952), 455–471. MR**0051445****[Bonsall-Duncan]**Frank F. Bonsall and John Duncan,*Complete normed algebras*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. MR**0423029****[Dales-McClure]**H. G. Dales and J. P. McClure,*Higher point derivations on commutative Banach algebras. II*, J. London Math. Soc. (2)**16**(1977), no. 2, 313–325. MR**0473839****[Esterle]**Jean Esterle,*Mittag-Leffler methods in the theory of Banach algebras and a new approach to Michael’s problem*, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983) Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 107–129. MR**769501**, 10.1090/conm/032/769501**[Johnson]**B. E. Johnson,*Continuity of derivations on commutative algebras*, Amer. J. Math.**91**(1969), 1–10. MR**0246127****[Johnson-Sinclair]**B. E. Johnson and A. M. Sinclair,*Continuity of derivations and a problem of Kaplansky*, Amer. J. Math.**90**(1968), 1067–1073. MR**0239419****[Loy]**Richard J. Loy,*Multilinear mappings and Banach algebras*, J. London Math. Soc. (2)**14**(1976), no. 3, 423–429. MR**0454641****[Michael]**Ernest A. Michael,*Locally multiplicatively-convex topological algebras*, Mem. Amer. Math. Soc.,**No. 11**(1952), 79. MR**0051444****[Read1]**C. J. Read,*All primes have closed range*, Bull. London Math. Soc.**33**(2001), no. 3, 341–346. MR**1817773**, 10.1017/S0024609301008025**[Read2]**Read, C. J., Derivations on Fréchet algebras with large separating subspaces, preprint.**[Rickart]**Charles E. Rickart,*General theory of Banach algebras*, The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0115101****[Sinclair]**Allan M. Sinclair,*Automatic continuity of linear operators*, Cambridge University Press, Cambridge-New York-Melbourne, 1976. London Mathematical Society Lecture Note Series, No. 21. MR**0487371****[Thomas1]**Marc P. Thomas,*Automatic continuity for linear functions intertwining continuous linear operators on Fréchet spaces*, Canad. J. Math.**30**(1978), no. 3, 518–530. MR**0482325****[Thomas2]**Marc P. Thomas,*The image of a derivation is contained in the radical*, Ann. of Math. (2)**128**(1988), no. 3, 435–460. MR**970607**, 10.2307/1971432**[Thomas3]**Marc P. Thomas,*Primitive ideals and derivations on noncommutative Banach algebras*, Pacific J. Math.**159**(1993), no. 1, 139–152. MR**1211389****[Thomas4]**Marc P. Thomas,*Elements in the radical of a Banach algebra obeying the unbounded Kleinecke-Shirokov conjecture*, Banach algebras ’97 (Blaubeuren), de Gruyter, Berlin, 1998, pp. 475–487. MR**1656623****[Villena]**A. R. Villena,*Derivations on Jordan-Banach algebras*, Studia Math.**118**(1996), no. 3, 205–229. MR**1388029****[Zariski]**Oscar Zariski,*Studies in equisingularity. I. Equivalent singularities of plane algebroid curves*, Amer. J. Math.**87**(1965), 507–536. MR**0177985**

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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

DOI:
https://doi.org/10.1090/S0002-9947-03-03251-3

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