Homology, Homotopy and Applications

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

A.Yu. Pirkovskii

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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.

Article information

Homology Homotopy Appl., Volume 11, Number 1 (2009), 81-114.

First available in Project Euclid: 1 September 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46M18: Homological methods (exact sequences, right inverses, lifting, etc.) 46M10: Projective and injective objects [See also 46A22] 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
Secondary: 16D40: Free, projective, and flat modules and ideals [See also 19A13] 18G50: Nonabelian homological algebra 46A45: Sequence spaces (including Köthe sequence spaces) [See also 46B45]

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


Pirkovskii, A.Yu. Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities. Homology Homotopy Appl. 11 (2009), no. 1, 81--114. https://projecteuclid.org/euclid.hha/1251832561

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