Characterizations of Lie n-derivations of unital algebras with nontrivial idempotents

Abstract

Let A be a unital algebra with a nontrivial idempotent e, and f = 1 − e. Suppose that A satisfies that exe · eAf = {0} = f Ae · exe implies exe = 0 and eAf · f xf = {0} = f xf · f Ae implies f xf = 0 for each x in A. We obtain the (necessary and) sufficient conditions for a Lie n-derivation ϕ on A to be of the form ϕ = d + δ + γ, where d is a derivation on A, δ is a singular Jordan derivation on A and γ is a linear mapping from A into the centre Z(A) vanishing on all (n − 1)−th commutators of A. In particular, we also discuss the (necessary and) sufficient conditions for a Lie n-derivation ϕ on A to be standard, i.e., ϕ = d + γ.

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

“…It was Krylov who initiated the study of linear mappings on generalized matrix algebras from the classifying point of view [34]. Since then many articles are devoted to this topic, and a number of interesting results are obtained (see [1], [3], [6], [7], [12], [19], [23], [24], [40], [41], [42], [60], [61], [62]). Nevertheless, it leaves so much to be desired.…”

“…In an analogous way, taking a 1 = 0 in (3. 19) and using (3.26), we obtain g 24 (a 2 , a 4 )a 4 = a 2 k 44 (a 4 , a 4 ) − f 44 (a 4 , a 4 )a 2 , (3.32)…”

Local Lie derivations on certain operator algebras

“…A Lie n-derivation ϕ on A is called to be standard if ϕ = d + τ , where d is a derivation on A and τ is a linear map from A into its center Z(A) vanishing on all (n−1)-th commutators of A. We use some known results about the standard forms of Lie n-derivations on unital algebras with a nontrivial idempotent and von Neumann algebras (see [2,8,12,13,24,30,33]). In particular, when a Lie n-derivation is standard on nest algebras, triangular algebras, and generalized matrix algebras have been studied in [1,7,28,34,36,37].…”