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

Let $A$ be a locally $m$-convex Fréchet algebra. We give a necessary and sufficient condition for a cyclic Fréchet $A$-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 Fréchet 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 Fréchet 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 Fréchet algebra case. We also show that a quasinormable locally $m$-convex Fréchet 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 Fréchet-Montel algebra which has a locally b.a.i., but does not have a b.a.i.

Keywords

flat Fréchet module, cyclic Fréchet module, amenable Fréchet algebra, locally $m$-convex algebra, approximate identity, approximate diagonal, Köthe space, quasinormable Fréchet space

2010 Mathematics Subject Classification

Primary 46H25, 46M10, 46M18. Secondary 16D40, 18G50, 46A45.

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Published 1 January 2009