Characterizations of Lie centralizers of generalized matrix algebras

Let G be a generalized matrix algebra. A linear map phi:G -> G is said to be a left (right) Lie centralizer at E is an element of G if phi([S,T])=[phi(S),T] (phi([S,T])=[S,phi(T)]) holds for all S,T is an element of G with ST = E. phi is of a standard form if phi(A)=ZA+gamma(A) for all A is an element of G, where Z is in the center of G and gamma is a linear map from G into its center vanishing on each commutator [S,T] whenever ST = E. In this paper, we give a complete characterization of phi. It is shown that, under some suitable assumptions on G,phi has a standard form.