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ideally amenable; Banach algebra; derivation
Let $\cal A$ be a Banach algebra. $\cal A$ is called ideally amenable if for every closed ideal $I$ of $\cal A$, the first cohomology group of $\cal A$ with coefficients in $I^*$ is zero, i.e. $H^1({\cal A}, I^*)=\lbrace 0\rbrace $. Some examples show that ideal amenability is different from weak amenability and amenability. Also for $n\in {N}$, $\cal A$ is called $n$-ideally amenable if for every closed ideal $I$ of $\cal A$, $H^1({\cal A},I^{(n)})=\lbrace 0\rbrace $. In this paper we find the necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.
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