Geodesic
Ideal amenability of module extensions of Banach algebras
Archivum mathematicum, Tome 43 (2007) no. 3, pp. 177-184
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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.
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.
Classification : 46Hxx
Keywords: ideally amenable; Banach algebra; derivation 
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Gordji, Eshaghi M.; Habibian, F.; Hayati, B. Ideal amenability of module extensions of Banach algebras. Archivum mathematicum, Tome 43 (2007) no. 3, pp. 177-184. http://geodesic.mathdoc.fr/item/ARM_2007_43_3_a3/

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