Topological amenability and K\"othe co-echelon algebras

TL;DR

This paper introduces a new concept of topologically flat modules for locally convex algebras, extends amenability criteria to DF-algebras, and characterizes topologically amenable K"othe co-echelon algebras.

Contribution

It develops a novel notion of topologically flat modules, extends amenability criteria to a broader class of algebras, and provides a complete characterization of topologically amenable K"othe co-echelon algebras.

Findings

A new notion of topologically flat modules is introduced.

Complete characterization of topologically amenable K"othe co-echelon algebras.

Equivalence of topological amenability and Johnson amenability for DF-algebras.

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). By 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 K\"othe co-echelon algebras.