Module Lie Derivation‎ of Triangular Banach Algebra to its Dual

Abstract

‎In this paper‎, ‎we introduce the concept of module Lie derivation on triangular Banach algebras‎ ‎$ \mathcal{T}=[\begin{smallmatrix}‎ ‎ A & M \\‎ ‎ & B‎ ‎ \end{smallmatrix}] $ to its Dual‎. ‎ We examine the relationship between module Lie derivations‎ ‎ $ L_{ A}‎ : ‎\mathcal{A} \to \mathcal{A}^{\ast} $ and $ L_{ B}‎ : ‎\mathcal{B} \to \mathcal{B}^{\ast} $ with module Lie derivation‎ ‎ $ L‎ : ‎\mathcal{T} \to \mathcal{T}^{\ast}.$‎