Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities

Details Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities

2006-10-17

arxiv.org/abs/math/0610528v1

Mathematics

Functional Analysis

32 pages

Scientific paper

Let A be a locally m-convex Fr\'echet algebra. We give a necessary and sufficient condition for a cyclic Fr\'echet A-module X=A_+/I to be strictly flat, generalizing thereby a criterion of Helemskii and Sheinberg. To this end, we introduce a notion of "locally bounded approximate identity" (a locally b.a.i. for short), and we show that X is strictly flat if and only if the ideal I has a right locally b.a.i. Next we apply this result to amenable algebras and show that a locally m-convex Fr\'echet algebra A is amenable if and only if A is isomorphic to a reduced inverse limit of amenable Banach algebras. We also extend a number of characterizations of amenability obtained by Johnson and by Helemskii and Sheinberg to the setting of locally m-convex Fr\'echet algebras. As a corollary, we show that Connes and Haagerup's theorem on amenable C^*-algebras and Sheinberg's theorem on amenable uniform algebras hold in the Fr\'echet algebra case. We also show that a quasinormable locally m-convex Fr\'echet algebra has a locally b.a.i. if and only if it has a b.a.i. On the other hand, we give an example of a commutative, locally m-convex Fr\'echet-Montel algebra which has a locally b.a.i., but does not have a b.a.i. Some of the results of this paper were announced in ArXiv preprint math.FA/0511132.

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