Derivations with large separating subspace

Author:
C. J. Read

Journal:
Proc. Amer. Math. Soc. **130** (2002), 3671-3677

MSC (2000):
Primary 46H20, 46H05; Secondary 46H40, 13A02, 13A10, 46M05, 13A05

Published electronically:
April 22, 2002

MathSciNet review:
1920047

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Abstract | References | Similar Articles | Additional Information

Abstract: In his famous paper *The image of a derivation is contained in the radical*, Marc Thomas establishes the (commutative) Singer-Wermer conjecture, showing that derivations from a commutative Banach algebra to itself must map into the radical. The proof goes via first showing that the separating subspace of a derivation on must lie in the radical of . In this paper, we exhibit discontinuous derivations on a commutative unital Fréchet algebra such that the separating subspace is the whole of . Thus, the situation on Fréchet algebras is markedly different from that on Banach algebras.

**[D]**H. G. Dales,*Banach Algebras and Automatic Continuity*, London Mathematical Society Monographs No. 24, (2001) CMP**2001:09****[DE]**P. G. Dixon and J. Esterle,*Michael’s problem and the Poincaré-Fatou-Bieberbach phenomenon*, Bull. Amer. Math. Soc. (N.S.)**15**(1986), no. 2, 127–187. MR**854551**, 10.1090/S0273-0979-1986-15463-7**[L]**K. B. Laursen,*Some remarks on automatic continuity*, Spaces of analytic functions (Sem. Functional Anal. and Function Theory, Kristiansand, 1975) Springer, Berlin, 1976, pp. 96–108. Lecture Notes in Math., Vol. 512. MR**0500203****[S]**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****[SW]**I. M. Singer and J. Wermer,*Derivations on commutative normed algebras*, Math. Ann.**129**(1955), 260–264. MR**0070061****[T]**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**[T2]**M. P. Thomas,*Local Power Series Quotients of Commutative Banach and Fréchet Algebras*, preprint, 2001.

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Additional Information

**C. J. Read**

Affiliation:
Faculty of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom

Email:
read@maths.leeds.ac.uk

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06485-7

Keywords:
Derivation,
Fr\'{e}chet algebra,
Banach algebra,
radical,
separating subspace,
automatic continuity

Received by editor(s):
March 13, 2001

Received by editor(s) in revised form:
July 24, 2001

Published electronically:
April 22, 2002

Communicated by:
N. Tomczak-Jaegermann

Article copyright:
© Copyright 2002
American Mathematical Society