Document Type : Original Manuscript
Authors
Department of Mathematics, University of Birjand, P.O. Box 9717434765, Birjand, Iran.
Abstract
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 $B$.
Keywords