Lie triple maps on generalized matrix algebras

TL;DR

This paper introduces and characterizes Lie triple centralizers on generalized matrix algebras, providing conditions for their properness and applications to generalized Lie triple derivations.

Contribution

It defines Lie triple centralizers on generalized matrix algebras and characterizes their general form and properness conditions, extending the understanding of these mappings.

Findings

Characterized the form of Lie triple centralizers on generalized matrix algebras.

Established necessary and sufficient conditions for these centralizers to be proper.

Applied results to characterize generalized Lie triple derivations.

Abstract

In this article, we introduce the notion of Lie triple centralizer as follows. Let $\mathcal{A}$ be an algebra, and $\phi:\mathcal{A}\to\mathcal{A}$ be a linear mapping. we say $\phi$ is a Lie triple centralizer whenever $\phi([[a,b],c])=[[\phi(a),b],c]$ for all $a,b,c\in\mathcal{A}$. Then we characterize the general form of Lie triple centralizers on generalized matrix algebra $\mathcal{U}$ and under some mild conditions on $\mathcal{U}$, we present the necessary and sufficient conditions for Lie triple centralizers to be proper. As an application of our results, we characterize generalized Lie triple derivations on generalized matrix algebras.