Characterizations of Lie centralizers of generalized matrix algebras

AbstractLet 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)=ZA+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.KEYWORDS: Centralizergeneralized matrix algebraLie centralizer2020 MATHEMATICS SUBJECT CLASSIFICATION: Primary: 16W25Secondary: 47B47 Additional informationFundingThis work is supported by the National Natural Science Foundation of China (No. 12071134) and Natural Science Foundation of Shaanxi Province (No. 2021JM-119).