On \(n\)-ideal amenability of certain Banach algebras (Q2907025)

In 2010, \textit{M. Alaghmandan, R. Nasr-Isfahani} and the second author [Bull. Aust. Math. Soc. 82, 274--281 (2010; Zbl 1209.46024)] characterized the \(\phi\)-amenability and \(\phi\)-contractibility of an abstract Segal algebra \(B\) with respect to a Banach algebra \(A\) for every character \(\phi\) on \(A\). They proved that \(A\) is \(\phi\)-amenable if and only if \(B\) is \(\phi_{|B}\)-amenable. They also characterized the character amenability of the Segal algebra \(S^1(G)\) with respect to \(L^1(G)\), where \(G\) is a locally compact group. \textit{M. Eshaghi Gordji} and \textit{T. Yazdanpanah} [Proc. Indian Acad. Sci., Math. Sci. 114, 399--408 (2004; Zbl 1073.46039)] introduced and studied the notion of ideal amenability, \(n\)-ideal amenability and \(n\)-\(I\)-weak amenability of a Banach algebra.NEWLINENEWLINEIn the present paper, the authors prove that if \(A\) and \(B\) are Banach algebras, \(A\) is \(n\)-\(I\)-weakly amenable and \(\tau:A\to B\) and \(\phi:B\to A\) are bounded homomorphisms such that \(\tau\circ\phi=I_B\), and if \(I,J\) are closed two-sided ideals of \(A\) and \(B\), respectively, such that \(\tau(I)\subseteq J\) and \(\phi(J)\subseteq I\), then for each \(n\in \mathbb{N} \), \(B\) is \(n\)-\(J\)-weakly amenable.NEWLINENEWLINEThey also investigate the relation between \(n\)-ideal amenability of \(A\) and approximate \(n\)-ideal amenability of abstract Segal algebras in \(A\).