Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities
| dc.creator | Pirkovskii, A. Yu. | |
| dc.date | 2006-10-17 | |
| dc.date.accessioned | 2026-07-07T07:29:09Z | |
| dc.date.available | 2026-07-07T07:29:09Z | |
| dc.description | 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. Some of the results of this paper were announced in ArXiv preprint math.FA/0511132. | |
| dc.description | 32 pages | |
| dc.identifier | https://arxiv.org/abs/math/0610528 | |
| dc.identifier | http://arxiv.org/abs/math/0610528 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/117997 | |
| dc.subject | Functional Analysis | |
| dc.subject | Primary 46M18, 46M10, 46H25; Secondary 46A45, 16D40, 18G50 | |
| dc.title | Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities | |
| dc.type | text |