Strictly flat cyclic Fr\'echet modules and approximate identities

Abstract

Let A be a locally m-convex Fr\'echet algebra. We give a necessary and sufficient condition for a cyclic Fr\'echet 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.i.), and we show that X is strictly flat if and only if the ideal I has a right locally bounded a.i. An example is given of a commutative locally m-convex Fr\'echet algebra that has a locally bounded a.i., but does not have a bounded a.i. On the other hand, we show that a quasinormable locally m-convex Fr\'echet algebra has a locally bounded a.i. if and only if it has a bounded a.i. Some applications to amenable Fr\'echet algebras are also given.