Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

Abstract

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$ and $\mathcal{Z}(\mathcal{G})$ be the center of $\mathcal{G}$. Suppose that $\mathfrak{q}:\mathcal{G}\times \mathcal{G}⟶\mathcal{G}$ is an $\mathcal{R}$-bilinear mapping and ${\mathfrak{T}}_{\mathfrak{q}}:\mathcal{G}⟶\mathcal{G}$ is the trace of $\mathfrak{q}$. We describe the form of ${\mathfrak{T}}_{\mathfrak{q}}$ satisfying the condition $[{\mathfrak{T}}_{\mathfrak{q}}(G),G]\in \mathcal{Z}(\mathcal{G})$ 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.

The last one of a series of three on FI theory of generalized matrix algebras. to appear in Linear Multilinear Algebra. arXiv admin note: substantial text overlap with arXiv:1301.2043, arXiv:1111.6316