2005/11/05 by A. Yu. Pirkovskii, Pirkovskii, A. Yu.
Mathematics · #16D40 #18G50 #46H25 #46M10 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46M18 #Secondary 46A45 #math.FA #msc:16D40 #msc:18G50 #msc:46A45 #msc:46H25 #msc:46M10 #msc:46M18
paper · pdf · doi:10.48550/arxiv.math/0511132
8 pages; version 2: a mistake in Theorem 3 is corrected
arxiv created 2006/05/05 · arxiv updated 2009/12/01
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.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 Fréchet 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 Fréchet algebra has a locally bounded a.i. if and only if it has a bounded a.i. Some applications to amenable Fréchet algebras are also given.