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Approximate amenability of Fréchet algebras

Published online by Cambridge University Press:  01 September 2008

P. LAWSON  and
C. J. READ
P. LAWSON
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT. e-mail: paull@maths.leeds.ac.uk, read@maths.leeds.ac.uk
C. J. READ
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT. e-mail: paull@maths.leeds.ac.uk, read@maths.leeds.ac.uk
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Abstract

The notion of approximate amenability was introduced by Ghahramani and Loy, in the hope that it would yield Banach algebras without bounded approximate identity which nonetheless had a form of amenability. So far, however, all known approximately amenable Banach algebras have bounded approximate identities (b.a.i.). In this paper we define approximate amenability and contractibility of Fréchet algebras, and we prove the analogue of the result for Banach algebras that these properties are equivalent. We give examples of Fréchet algebras which are approximately contractible, but which do not have a bounded approximate identity. For a good many Fréchet algebras without b.a.i., we find either that the algebra is approximately amenable, or it is “obviously” not approximately amenable because it has continuous point derivations. So the situation for Fréchet algebras is quite close to what was hoped for Banach algebras.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 2 , September 2008 , pp. 403 - 418
Copyright
Copyright © Cambridge Philosophical Society 2008

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