Centralizing Traces and Lie-Type Isomorphisms on Generalized Matrix Algebras: A New Perspective

Let ℛ be a commutative ring, G be a generalized matrix algebra over ℛ with weakly loyal bimodule and G be the center of G. Suppose that q: G× G→ G is an ℛ-bilinear mapping and that T_q: G→ G is a trace of q. The aim of this article is to describe the form of T_q satisfying the centralizing condition [T_q(x) , x] ∈ Z(G) (and commuting condition [T_q(x) , x] = 0) for all x∈ G. More precisely, we will revisit the question of when the centralizing trace (and commuting trace) T_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 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.