“…Then the following question seem natural: What is the structure of local Lie derivations? Chen et al [4] consider the questions on the algebra B(X ) of all bounded linear operators on Banach space X . They showed that each local Lie derivation from B(X ) into itself is a Lie derivation.…”
Section: More Generally a Linear Map δ Of U Is Called A Lie Derivation If δ([A B]) = [δ(A) B] + [A δ(B)]mentioning
Let G be a generalized matrix algebra. We prove that, under certain conditions, every local Lie derivation δ of G can be written in the form δ = d + h, where d is a derivation of G and h is a linear map from G into Z(G) vanishing on each commutator. The result is then applied to some full matrix algebras and triangular algebras.
“…Then the following question seem natural: What is the structure of local Lie derivations? Chen et al [4] consider the questions on the algebra B(X ) of all bounded linear operators on Banach space X . They showed that each local Lie derivation from B(X ) into itself is a Lie derivation.…”
Section: More Generally a Linear Map δ Of U Is Called A Lie Derivation If δ([A B]) = [δ(A) B] + [A δ(B)]mentioning
Let G be a generalized matrix algebra. We prove that, under certain conditions, every local Lie derivation δ of G can be written in the form δ = d + h, where d is a derivation of G and h is a linear map from G into Z(G) vanishing on each commutator. The result is then applied to some full matrix algebras and triangular algebras.
“…A linear map δ of an algebra A is called a local Lie derivation if, for each A ∈ A, there exists a Lie derivation δ A such that δ(A) = δ A (A). In [1], the authors studied local Lie derivations of operator algebras on Banach spaces. In this paper, as the continuity of the previous work, we study the local Lie derivations of nest algebras on Hilbert spaces.…”
Let N be a nest on a Hilbert space H and Alg N be the associated nest algebra. We prove that every local Lie derivation from Alg N into B(H) is a Lie derivation.
“…[13,Theorem 7.5]). Based on these results, many authors have studied local derivations on operator algebras, for example, see in [2,3,4,5,6,7,8,9,10,11,12,15,16,17,18,19,20,22,23].…”
In the present paper we prove that every additive (not necessarily homogenous) local inner derivation on the algebra of matrices over an arbitrary field is an inner derivation, and every local inner derivation on the ring of matrices over a finite ring generated by the identity element or the ring of integers is an inner derivation. We also prove that every additive local inner derivation on the Jordan algebra of symmetric matrices over an arbitrary field is a derivation, and every local inner derivation on the Jordan ring of symmetric matrices over a finite ring generated by the identity element or the ring of integers is a derivation.
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