Centralizing traces and Lie triple isomorphisms on triangular algebras

Let (Formula presented.) be a triangular 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 that (Formula presented.) is a 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 standard form will be addressed. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism on (Formula presented.) to be almost standard. As applications, we characterize Lie triple isomorphisms of upper triangular matrix algebras and nest algebras. Some further research topics related to current work are proposed at the end of this article.