Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

Let (Formula presented.) be a generalized matrix algebra over a commutative ring (Formula presented.) and (Formula presented.) be the centre of (Formula presented.). Suppose that (Formula presented.) is an (Formula presented.) -bilinear mapping and (Formula presented.) is the trace of (Formula presented.). We describe the form of (Formula presented.) satisfying the condition (Formula presented.) for all (Formula presented.). The question of when (Formula presented.) has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of (Formula presented.) 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.