Characterizations of Lie centralizers of triangular algebras

Abstract

Let $\mathcal{A}$ be an unital algebra over the complex field $\mathbb{C}$. A linear map ϕ from $\mathcal{A}$ into itself is called a Lie centralizer at a given point $G\in \mathcal{A}$ if $\varphi ([S,T])=[S,\varphi (T)]=[\varphi (S),T]$ for all $S,T\in \mathcal{A}$ with ST = G. The aim of this paper is to give a description of Lie centralizers at an arbitrary but fixed point on triangular algebras. These results are then applied to nest algebras and upper triangular matrix algebras.

Keywords:

• Lie centralizer
• centralizer
• triangular algebra
• nest algebra

2010 Mathematics Subject Classifications:

• 16W25
• 47B47
• 47L35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 12071134] and the Natural Science Foundation of Shaanxi Province [grant number 2021JM-119].