Nonlinear Lie Triple Centralizers of Unital Algebras with Idempotents

Let A=A11+A12+A21+A22 be a unital algebra with a nontrivial idempotent e1 that satisfies that A12 is a faithful (A11,A22)-bimodule, A12A21=A21A12=0, Z(A)e1=Z(A11) and Z(A)e2=Z(A22). Moreover, let L be a nonlinear Lie triple centralizer on A. Then, there exist a center element λ∈Z(A) and a map ξ from A to Z(A) sending each second commutator to zero such that L(a)=λa+ξ(a) for all a∈A. As an application, we characterize nonlinear Lie triple centralizers on triangular algebras.