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.

Keywords:

triangular Banach algebra , module Lie derivation , standard Lie derivation