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Ideal amenability of module extensions of Banach algebras. (English) Zbl 1164.46020
Summary: Let $$\mathcal A$$ be a Banach algebra. $$\mathcal A$$ is called ideally amenable if for every closed ideal $$I$$ of $$\mathcal A$$, the first cohomology group of $$\mathcal A$$ with coefficients in $$I^*$$ is zero, i.e., $$H^1({\mathcal A}, I^*)=\{0\}$$. Some examples show that ideal amenability is different from weak amenability and amenability. Also for $$n\in \mathbb {N}$$, $$\mathcal A$$ is called $$n$$-ideally amenable if for every closed ideal $$I$$ of $$\mathcal A$$, $$H^1({\mathcal A},I^{(n)})=\{0\}$$. In this paper, we find necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.

##### MSC:
 46H20 Structure, classification of topological algebras
##### Keywords:
ideally amenable; Banach algebra; derivation
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