Lie centralizers and commutant preserving maps on generalized matrix algebras

Abstract

Let G be a 2-torsion free unital generalized matrix algebra with center Z(G), and Φ be a linear mapping on G satisfying the condition X,Y∈G,XY=YX=0⇒[Φ(X),Y]=0. This paper is devoted to the study of the structure of Φ under some mild assumptions on G. We provide the necessary and sufficient conditions for Φ to be in the form Φ(X)=λX+μ(X) (∀X∈G), where λ∈Z(G) and μ:G→Z(G) is a linear mapping. Then we apply our results to characterize linear mappings on G that are commutant preservers or double commutant preservers.