Local Lie derivations of nest algebras

Abstract

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.

Characterization of 2-Local Derivations and Local Lie Derivations on Some Algebras

“…It is known that every element in M can be written in the form λI+ n i=1 [A i , B i ], where λ ∈ C and A i , B i ∈ M(see, [7]). If the unit I can not be written in the form The following result is proved in [8]. We show it by Theorem 3.1 as an alternative method.…”

“…Also in this paper [9], the authors characterize 2-local Lie derivations on B(X). Later, L. Chen, and F. Lu [8] prove that every local Lie derivation from a nest algebra algN into B(H) is a Lie derivation, where N is a nest on the Hilbert space H. Quite recently, L. Liu [21] characterizes 2-local Lie derivations on a semi-finite factor von Neumann algebra with dimension greater than 4.…”

On local Lie derivations of generalized matrix algebras

“…They showed that each local Lie derivation from B(X ) into itself is a Lie derivation. Later, Chen and Lu [3] proved that each local Lie derivation of nest algebras on Hilbert spaces is a Lie derivation. In [14,15], Liu and Zhang proved that the same is true for local Lie derivations on von Neumann algebras and a certain class of triangular algebras, respectively.…”

Local Lie derivations on certain operator algebras

“…Moreover, it was proved in [7,Corollary 6.4] that every Lie n-derivation on T (N ) is standard. Chen and Lu [11] showed that every local Lie derivation (n = 2) from T (N ) into B(H) is a Lie derivation. In fact, each local derivation on the nest algebra is an inner derivation [14,Theorem 2.6].…”

“…Then the following question seems natural: When is the local Lie n-derivation as Lie n-derivation? If n = 2, it was studied in [11,25,26,27] on the nest algebras, generalized matrix algebras, von Neumann algebras, triangular algebras, respectively. To the best of our knowledge, there is not any article treating the situation of n ≥ 3.…”