Lifting derivations and $n-$weak amenability of the second dual of a Banach algebra

TL;DR

This paper investigates the relationship between the $n$-weak amenability of a Banach algebra's second dual and the algebra itself, providing new results and simplified proofs for specific cases.

Contribution

It establishes that for $n eq 1$, $n$-weak amenability of the second dual implies that of the algebra, and offers a unified proof approach for the case $n=1$.

Findings

For $n eq 1$, $n$-weak amenability of $ extbf{A}^{**}$ implies that of $ extbf{A}$.

For $n=1$, the paper confirms the implication, sharpening previous results.

Provides simplified proofs for known results in the case $n=1$.

Abstract

We show that for $n\geq 2$ the $n-$weak amenability of the second dual $\A^{**}$ of a Banach algebra $\A$ implies that of $\A$. We also provide a positive answer for the case $n=1,$ which sharpens some older results. Our method of proof also provides a unified approach to give short proofs for some known results in the case where $n=1$.