Topological amenability and Köthe co-echelon algebras

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.

The Embedding Theorem of an L0-Prebarreled Module into Its Random Biconjugate Space

We first prove Mazur’s lemma in a random locally convex module endowed with the locally L0-convex topology. Then, we establish the embedding theorem of an L0-prebarreled random locally convex module, which says that if (S,P) is an L0-prebarreled random locally convex module such that S has the countable concatenation property, then the canonical embedding mapping J of S onto J(S)⊂(Ss⁎)s⁎ is an L0-linear homeomorphism, where (Ss⁎)s⁎ is the strong random biconjugate space of S under the locally L0-convex topology.