Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 Articles in Press 2025 06 26 $\phi$-Johnson amenable Banach algebras and Lie derivations 5776 10.22060/ajmc.2025.24052.1354 EN Hoger Ghahramani Department of Mathematics, Faculty of Science, University of Kurdistan, P.O. Box 416, Sanandaj, Kurdistan, Iran Parvin Zamani Department of Mathematics, Faculty of Science, University of Kurdistan, P.O. Box 416, Sanandaj, Kurdistan, Iran Journal Article 2025 04 09 ‎‎Let $\mathfrak{U}$ be a $\phi$-Johnson amenable Banach algebra where $\phi \in\Delta(\mathfrak{U})$ ($\Delta(\mathfrak{U})$ is the character space of $\mathfrak{U}$)‎. ‎Suppose that $X$ is a Banach $\mathfrak{U}$-bimodule such that $a.x=\phi(a)x$ for all $a\in \mathfrak{U}$‎, ‎$x\in X$ or $x.a=\phi(a)x$ for all $a\in \mathfrak{U}$‎, ‎$x\in X$‎. ‎We show that any Lie derivation (not necessarily continuous) $\delta:\mathfrak{U}\rightarrow X$ with the property that $\mathfrak{S}(\delta)\subseteq \mathcal{Z}_{\mathfrak{U}}(X)$ ($\mathfrak{S}(\delta)$ is the separating space of $\delta$) can be decomposed into the sum of a continuous derivation and a center-valued trace‎.