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2009 Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities
A.Yu. Pirkovskii
Homology Homotopy Appl. 11(1): 81-114 (2009).


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.


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A.Yu. Pirkovskii. "Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities." Homology Homotopy Appl. 11 (1) 81 - 114, 2009.


Published: 2009
First available in Project Euclid: 1 September 2009

zbMATH: 1180.46039
MathSciNet: MR2506128

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

Rights: Copyright © 2009 International Press of Boston


Vol.11 • No. 1 • 2009
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