Commuting Traces and Lie Isomorphisms on Generalized Matrix Algebras

TL;DR

This paper characterizes traces satisfying a commuting condition on generalized matrix algebras and uses this to determine when Lie isomorphisms are nearly standard, with applications to matrix and triangular algebras.

Contribution

It provides a characterization of trace functions satisfying a specific commutation condition and establishes conditions under which Lie isomorphisms are almost standard in generalized matrix algebras.

Findings

Characterized traces satisfying the commuting condition.

Established sufficient conditions for Lie isomorphisms to be almost standard.

Applied results to matrix, triangular, and unital algebras.

Abstract

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$, ${\mathfrak q}\colon \mathcal{G}\times\mathcal{G}\longrightarrow \mathcal{G}$ be an $\mathcal{R}$-bilinear mapping and ${\mathfrak T}_{\mathfrak q}\colon:\mathcal{G}\longrightarrow \mathcal{G}$ be a trace of $\mathfrak{q}$. We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition ${\mathfrak T}_{\mathfrak q}(G)G=G{\mathfrak T}_{\mathfrak q}(G)$ 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 isomorphism of $\mathcal{G}$ to be almost standard. As applications we characterize Lie isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some further research topics related to current work are proposed at the end of this article.