Ideal amenability of module extensions of Banach algebras

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.