English

Abstract

Let ℛ be a commutative ring, $$\mathcal{G}$$ be a generalized matrix algebra over ℛ with weakly loyal bimodule and $$\mathcal{G}$$ be the center of $$\mathcal{G}$$ . Suppose that $$\mathfrak{q} : \mathcal{G} \times \mathcal{G} \rightarrow \mathcal{G}$$ is an ℛ-bilinear mapping and that $$\mathfrak{T}_{\mathfrak{q}} : \mathcal{G} \rightarrow \mathcal{G}$$ is a trace of $$\mathfrak{q}$$ . The aim of this article is to describe the form of $$\mathfrak{T}_{\mathfrak{q}}$$ satisfying the centralizing condition $$\left[\mathfrak{T}_{\mathfrak{q}}(x), x\right] \in \mathcal{Z}(\mathcal{G})$$ (and commuting condition $$\left[\mathfrak{T}_{\mathfrak{q}}(x), x\right]=0 )$$ for all $$x \in \mathcal{G}$$ . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) $$\mathfrak{T}_{\mathfrak{q}}$$ has the so-called proper form from a new perspective. Using the aforementioned trace function, we establish sufficient conditions for each Lie-type isomorphism of $$\mathcal{G}$$ to be almost standard. As applications, centralizing (commuting) traces of bilinear mappings and Lie-type isomorphisms on full matrix algebras and those on upper triangular matrix algebras are totally determined.