Approximate weak amenability of certain Banach algebras

TL;DR

This paper investigates approximate weak amenability in Banach algebras, establishing new theoretical links between group algebras and measure algebras, and introduces novel notions of approximate amenability.

Contribution

It introduces new concepts of approximate weak and cyclic amenability and characterizes bounded ω*-approximately weakly [cyclic] amenable ℓ¹-Munn algebras.

Findings

If L¹(G)** is approximately weakly amenable, then M(G) is also approximately weakly amenable.

New notions of approximate weak and cyclic amenability are defined for Banach algebras.

Characterizations of bounded ω*-approximately weakly [cyclic] amenable ℓ¹-Munn algebras are provided.

Abstract

It is shown that for a locally compact group $G$, if $L^{1}(G)^{**}$ is approximately weakly amenable, then $M(G)$ is approximately weakly amenable. Then, new notions of approximate weak amenability and approximate cyclic amenability for Banach algebras are introduced. Bounded $\omega^{*}$-approximately weakly [cyclic] amenable $\ell^{1}$-Munn algebras are characterized.