## Homology, Homotopy and Applications

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 1 September 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1251832561

**Mathematical Reviews number (MathSciNet)**

MR2506128

**Zentralblatt MATH identifier**

1180.46039

**Subjects**

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]

**Keywords**

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

#### Citation

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