Centralizers of Lie Structure of Triangular Algebras

Let T = Tri(A, M, B) be a triangular algebra where A is a unital algebra, B is an algebra which is not necessarily unital, and M is a faithful (A, B)-bimodule which is unital as a left A-module. In this paper, under some mild conditions on T, we show that if phi : T -> T is a linear map satisfying A, B is an element of T, AB = P double right arrow phi([A, B]) = [A, phi(B)] = [phi(A), B], where P is the standard idempotent of T, then phi = psi + y where psi : T -> T is a centralizer and gamma : T -> Z (T ) is a linear map vanishing at commutators [A, B] with AB = P whrere Z(T) is the center of T. Applying our result, we characterize linear maps on T that behave like generalized Lie 2-derivations at idempotent products as an application of above result. Our results are applied to upper triangular matrix algebras and nest algebras.