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Ideal amenability of module extensions of Banach algebras

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M. E. Gordji, F. Habibian, B. Hayati
*

** Address.**

M. E. Gordji, Department of Mathematics,
University of Semnan, Semnan, Iran

Department of
Mathematics, Shahid Beheshti University, Tehran, Iran

F. Habibian, Department of Mathematics,
Isfahan University, Isfahan, Iran

B. Hayati, Department of Mathematics, Shahid Beheshti
University, Tehran, Iran

** E-mail:**

maj_ess@yahoo.com

fhabibian@math.ui.ac.ir

bahmanhayati@yahoo.com

**Abstract.**
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.\ $H^1({\cal A}, I^*)=\{0\}$. Some examples show that ideal
amenability is different from weak amenability and amenability. Also
for $n\in \Bbb {N}$, $\cal A$ is called $n$-ideally amenable if for
every closed ideal $I$ of $\cal A$, $H^1({\cal A},I^{(n)})=\{0\}$.
In this paper we find the necessary and sufficient conditions for a
module extension Banach algebra to be 2-ideally amenable.

**AMSclassification. ** Primary 46HXX.

**Keywords. ** Ideally amenable, Banach algebra, derivation.