On local Lie derivations of generalized matrix algebras

Abstract

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.

Ternary derivations of nest algebras

“…A linear map δ on an algebra A is called a local (generalized) derivation if for any a ∈ A there is a (generalized) derivation δ a : A → A (depending on a) such that δ(a) = δ a (a). There have been many papers in the literature investigating when local (generalized) derivations are (generalized) derivations, see [5,8,19,20,25,28,29,36] and the references therein. In the following corollary, we characterize local generalized derivations on nest algebras.…”

Local Lie derivations on certain operator algebras

“…In what follows, we denote by J (A) the subalgebra of A generated by all idempotents in A. Wang and Wang [34] described the standard form of Lie n-derivations for a certain class of generalized matrix algebras. Liu [25] proved that each local Lie derivation (n = 2) is a Lie derivation on generalized matrix algebras under some conditions. As a consequence of Theorem 2.3, we have the following result: Let M s (A), s ≥ 2, be a matrix algebra, where A is a unital algebra, and {e ij | i, j = 1, 2, • • • , n} be the system of matrix unit of M s (A).…”

“…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.…”