Approximate semi-amenability of Banach algebras

TL;DR

This paper explores the concept of approximate semi-amenability in Banach algebras, introducing a new property that extends approximate amenability and applies to classes like Segal algebras on amenable SIN-groups.

Contribution

It defines and investigates approximate semi-amenability, showing its relation to approximate amenability and its applicability to certain non-amenable Banach algebras.

Findings

Approximate semi-amenability coincides with approximate amenability under certain conditions.

The new notion applies to Segal algebras on amenable SIN-groups.

It broadens the scope of amenability concepts in Banach algebra theory.

Abstract

In recent work of the authors the notion of a derivation being approximately semi-inner arose as a tool for investigating (approximate) amenability questions for Banach algebras. Here we investigate this property in its own right, together with the consequent one of approximately semi-amenability. Under certain hypotheses regarding approximate identities this new notion is the same as approximate amenability, but more generally it covers some important classes of algebras which are not approximately amenable, in particular Segal algebras on amenable SIN-groups.