The structure of module Lie derivations on triangular Banach algebras

Abstract

‎Let $\mathfrak A$‎, ‎$A$ and $B$ be Banach algebras such that $A$ and $B$ are Banach $\mathfrak A$-bimodule and let $M$ be a Banach $(A,B)$-$\mathfrak{A}$-module‎. ‎Let $\mathcal{T}={\hbox {Tri}}(A,B,M)$ be triangular Banach algebra‎. ‎In this paper‎, ‎we examine the relationship between $\mathfrak{A}$-module Lie derivations on‎ ‎$ A $ and $ B $ and $ \mathfrak{T} $-module Lie derivation on $\mathcal{T}$ where‎ ‎$ \mathfrak{T} = \{[\begin{smallmatrix}‎ ‎\alpha & \\‎ ‎& \alpha‎ ‎\end{smallmatrix}]‎ : ‎\alpha \in \mathfrak{A}‎ \}.$‎