Abstract
We introduce a notion of a topologically flat locally convex module, which extends the notion of a flat Banach module and which is well adapted to the nonmetrizable setting (and especially to the setting of DF-modules). Using this notion, we introduce topologically amenable locally convex algebras and we show that a complete barrelled DF-algebra is topologically amenable if and only if it is Johnson amenable, extending thereby Helemskii–Sheinberg’s criterion for Banach algebras. As an application, we completely characterize topologically amenable Köthe co-echelon algebras.
Highlights
The paper is devoted to the study of amenability properties in the framework of DFalgebras
We define topologically amenable algebras in terms of topologically flat modules, and we show that topological amenability for complete barrelled DF-algebras is equivalent to amenability in Johnson’s sense
Helemskii and Sheinberg proved that a Banach algebra A is amenable in Johnson’s sense if and only if the unitization of A is a flat Banach A-bimodule
Summary
The paper is devoted to the study of amenability properties in the framework of DFalgebras. These are algebras with jointly continuous multiplication whose underlying topological vector spaces are DF-spaces. The category of DF-spaces contains spaces of distributions, e.g. tempered distributions or distributions with compact support. Duals of Fréchet spaces belong to this category.
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