THE STRUCTURE OF MODULE LIE DERIVATIONS ON TRIANGULAR BANACH ALGEBRAS

‎In this paper‎, ‎we introduce the concept of module Lie derivation on Banach algebras and study module Lie derivations on unital triangular Banach algebras $ \mathcal{T}=\Mat{A}{M}{B}$ to its dual‎. ‎Indeed‎, ‎we prove that every module (linear) Lie derivation $ \delta‎: ‎\mathcal{T} \to \mathcal{T}^{\ast}$ can be decomposed as $ \delta = d‎ + ‎\tau $‎, ‎where $ d‎: ‎\mathcal{T} \to \mathcal{T}^{\ast} $ is a module (linear) derivation and $ \tau‎: ‎\mathcal{T} \to Z_{\mathcal{T}}(\mathcal{T}^{\ast}) $ is a module (linear) map vanishing at commutators if and only if this happens for ‎the ‎corner algebras $A$ and $B$‎.