Lie triple derivations on nest algebras

Abstract

Let δ be a Lie triple derivation from a nest algebra A into an A-bimodule M. We show that if M is a weak* closed operator algebra containing A then there are an element S ∈ M and a linear functional f on A such that δ(A) = SA − AS + f (A)I for all A ∈ A, and if M is the ideal of all compact operators then there is a compact operator K such that δ(A) = KA − AK for all A ∈ A. As applications, Lie derivations and Jordan derivations on nest algebras are characterized.

Characterizations of Lie Higher and Lie Triple Derivations on Triangular Algebras

“…Lie triple derivations have received a fair amount of attention in recent years, see for example [7,12,15] and references therein. A central problem is whether a Lie triple derivation can be decomposed into a sum of a derivation and a linear map…”

“…Results related to Lie triple derivations on prime algebras were considered in [3]. Lu [7] considered similar questions on nest algebras. In Section 4, we characterize Lie triple derivations on triangular algebras and give necessary and sufficient conditions such that every Lie triple derivation is proper.…”

Characterizing Lien-derivations for reflexive algebras

“…For Lie triple derivations, Miers [5] showed that every linear Lie triple derivation on a von Neumann algebra without central summands of type I 1 is of the form τ + h, where τ is a linear derivation and h is a Recently, Wang and Lu [6] described the structure of linear Lie triple derivations on J -subspace lattice algebras. For other results, see [7][8][9][10] and the references therein.…”

A note on f-derivations of triangular algebras

“…Similarly, every Jordan triple derivation is an 3 ]. The Lie triple derivations on triangular algebras were studied in [1,7,8,13,16]. These articles are the main motivation for the study of f -derivations on triangular algebras.…”