A generalization of the weak amenability of some Banach algebra

TL;DR

This paper explores conditions under which the weak amenability of higher duals of a Banach algebra implies the weak amenability of the algebra itself, extending known results to more general dual spaces.

Contribution

It introduces new conditions that ensure weak amenability of a Banach algebra based on the weak amenability of its higher duals, generalizing previous results.

Findings

Weak amenability of $A^{**}$ implies that of $A$ under new conditions.

Generalization to $(n+2)$-th duals and $T-S$-weak amenability.

Establishes a link between higher duals and original algebra's properties.

Abstract

Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We show that by some new conditions, $A$ is weakly amenable whenever $A^{**}$ is weakly amenable. We will study this problem under generalization, that is, if $(n+2)-th$ dual of $A$, $A^{(n+2)}$, is $T-S-$weakly amenable, then $A^{(n)}$ is $T-S-$weakly amenable where $T$ and $S$ are continuous linear mappings from $A^{(n)}$ into $A^{(n)}$.