Derivations into n-th duals of ideals of Banach algebras

Abstract

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

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