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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.

46H20 Structure, classification of topological algebras
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