A note on derivations into annihilators of the ideals of Banach algebras

Abstract

In a recent study, Teymouri ‎\textit{et al.}‎ [Teymouri A., Bodaghi A., Bagha D. E., Derivations into annihilators of the ideals of Banach algebras, Demonstr. Math.‎, 52(1) (2019), 949–958] introduced the notions of $\frac{A}{J}$-weak amenability and quotient ideal amenability for a Banach algebra $A$ relative to a closed two-sided ideal $J$. They investigated the connection between the $\frac{A}{J}$-weak amenability of $A$ and the weak amenability of $\frac{A}{J}$. However, their primary theorem relied on a flawed conclusion, and Theorem 2.13 in their work includes an incorrect result. In this paper, we present counterexamples to highlight these issues, then refine and establish their main theorem under less restrictive assumptions. Additionally, we provide alternative results and extend the analysis of these concepts to specific classes of Banach algebras, particularly triangular Banach algebras‎‏.