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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: 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 $A$ to itself must map into the radical. The proof goes via first showing that the separating subspace of a derivation on $A$ must lie in the radical of $A$. In this paper, we exhibit discontinuous derivations on a commutative unital Fréchet algebra $\mathcal{A}$ such that the separating subspace is the whole of $\mathcal{A}$. Thus, the situation on Fréchet algebras is markedly different from that on Banach algebras.


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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
PII: S 0002-9939(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