THE STRUCTURE OF MODULE LIE DERIVATIONS ON TRIANGULAR BANACH ALGEBRAS

In this paper, we introduce the concept of  module Lie  derivations on Banach algebras and study  module Lie  derivations on unital triangular Banach algebras $ \mathcal{T}=\begin{bmatrix}A & M\\ &B\end{bmatrix}$ to its dual. Indeed, we prove that every module (linear) Lie derivation\linebreak $ \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 $B$.