Centralizers of Lie Structure of Triangular Algebras

Abstract

Let ${\mathcal {T}} = Tri ({\mathcal {A}},{\mathcal {M}},{\mathcal {B}} )$ be a triangular algebra where ${\mathcal {A}}$ is a unital algebra, ${\mathcal {B}}$ is an algebra which is not necessarily unital, and ${\mathcal {M}}$ is a faithful (${\mathcal {A}}$, ${\mathcal {B}}$)-bimodule which is unital as a left ${\mathcal {A}}$-module. In this paper, under some mild conditions on ${\mathcal {T}}$, we show that if $\phi : {\mathcal {T}} \rightarrow {\mathcal {T}}$ is a linear map satisfying

$$\begin{aligned} A,B \in {\mathcal {T}}, ~~ AB= P \Longrightarrow \phi ( [A,B])=[A,\phi (B) ]=[\phi (A) , B], \end{aligned}$$

where P is the standard idempotent of ${\mathcal {T}}$, then $\phi = \psi +\gamma$ where $\psi :{\mathcal {T}} \rightarrow {\mathcal {T}}$ is a centralizer and $\gamma :{\mathcal {T}}\rightarrow Z( {\mathcal {T}})$ is a linear map vanishing at commutators [A, B] with $AB=P$ whrere $Z( {\mathcal {T}})$ is the center of ${\mathcal {T}}$. Applying our result, we characterize linear maps on ${\mathcal {T}}$ that behave like generalized Lie 2-derivations at idempotent products as an application of above result. Our results are applied to upper triangular matrix algebras and nest algebras.