Approximate Amenability of Fréchet Algebras.

The notion of approximate amenability was introduced by Ghahra-mani and Loy, in the hope that it would yield Banach algebras with-out bounded approximate identity which nonetheless had a form of amenability. So far, however, all known approximately amenable Ba-nach algebras have bounded approximate identities. 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 ap-proximate 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