Derivations into n-th duals of ideals of Banach algebras

TL;DR

This paper introduces new notions of amenability for Banach algebras based on their n-th duals of ideals, exploring relationships between these properties and existing concepts.

Contribution

It defines n-I-weak amenability and n-ideally amenability for Banach algebras and investigates their interrelations and distinctions.

Findings

Established relationships between n-I-weak and m-J-weak amenability.

Characterized conditions under which Banach algebras are n-ideally amenable.

Provided insights into the structure of Banach algebras via dual space properties.

Abstract

We introduce two notions of amenability for a Banach algebra $\cal A$. Let $n\in \Bbb N$ and let $I$ be a closed two-sided ideal in $\cal A$, $\cal A$ is $n-I-$weakly amenable if the first cohomology group of $\cal A$ with coefficients in the n-th dual space $I^{(n)}$ is zero; i.e., $H^1({\cal A},I^{(n)})=\{0\}$. Further, $\cal A$ is n-ideally amenable if $\cal A$ is $n-I-$weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We find some relationships of $n-I-$ weak amenability and $m-J-$ weak amenability for some different m and n or for different closed ideals $I$ and $J$ of $\cal A$.