Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

TL;DR

This paper characterizes Lie triple isomorphisms on generalized matrix algebras by analyzing trace functions and their properties, providing conditions under which these isomorphisms are almost standard and applying results to various algebra classes.

Contribution

It introduces new conditions for trace functions on generalized matrix algebras and characterizes Lie triple isomorphisms as almost standard, extending previous understanding.

Findings

Characterized the form of trace functions satisfying specific commutator conditions.

Established sufficient conditions for Lie triple isomorphisms to be almost standard.

Applied results to full matrix algebras, triangular algebras, and unital algebras with idempotents.

Abstract

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$ and $\mathcal{Z(G)}$ be the center of $\mathcal{G}$. Suppose that ${\mathfrak q}\colon \mathcal{G}\times \mathcal{G}\longrightarrow \mathcal{G}$ is an $\mathcal{R}$-bilinear mapping and ${\mathfrak T}_{\mathfrak q}\colon \mathcal{G}\longrightarrow \mathcal{G}$ is the trace of $\mathfrak{q}$. We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition $[{\mathfrak T}_{\mathfrak q}(G), G]\in \mathcal{Z(G)}$ for all $G\in \mathcal{G}$. The question of when ${\mathfrak T}_{\mathfrak q}$ has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of $\mathcal{G}$ to be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article.