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 T=bmatrixA & M\\ &Bbmatrix to its dual. Indeed, we prove that every module (linear) Lie derivation δ: T → T^ can be decomposed as δ = d + τ, where d: T → T^ is a module (linear) derivation and τ: T → ZT(T^) is a module (linear) map vanishing at commutators if and only if this happens for the corner algebras A and B.