Characterizations of Lie centralizers of generalized matrix algebras

Abstract

Let G be a generalized matrix algebra. A linear map ϕ : G → G is said to be a left (right) Lie centralizer at E ∈ G if ϕ ( [ S , T ] ) = [ ϕ ( S ) , T ] ( ϕ ( [ S , T ] ) = [ S , ϕ ( T ) ] ) holds for all S , T ∈ G with ST = E. ϕ is of a standard form if ϕ ( A ) = Z A + γ ( A ) for all A ∈ G , where Z is in the center of G and γ 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 ϕ . It is shown that, under some suitable assumptions on G , ϕ has a standard form.