Strictly flat cyclic Fréchet modules and approximate identities

Details Strictly flat cyclic Fréchet modules and approximate identities Strictly flat cyclic Fréchet modules and approximate identities

2005-11-05

arxiv.org/abs/math/0511132v2

Mathematics

Functional Analysis

8 pages; version 2: a mistake in Theorem 3 is corrected

Scientific paper

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.

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