Lie Derivations of Reflexive Algebras

Abstract

A Lie derivation is called standard if it is a sum of a derivation and a linear map with image in the center vanishing on commutators. In this paper we show that Lie derivations of a reflexive algebra AlgL on a Banach space are standard if L is a nest, or has the non-trivial smallest element, or has the non-trivial greatest element. (2000). Primary 47L35; Secondary 17B40, 17B60. Mathematics Subject Classification

Characterizing Lien-derivations for reflexive algebras

“…Lemma 2.1 [4] Let X be a Banach space and let E, F be non-zero subspaces of X and X * , respectively. Let :…”

“…In [3], Qi and Hou proved that the same is true for additive Lie derivations of nest algebras on Banach spaces. Lu and Liu [4] discussed linear Lie derivations of a reflexive algebra AlgL on a Banach space and proved that every linear Lie derivation on AlgL is standard if L has the non-trivial smallest element or the non-trivial greatest element. 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.…”

$$*$$ ∗ -Lie Derivable Mappings on Von Neumann Algebras

“…Recall that an additive mapping δ : [2,4,[6][7][8][10][11][12]. We say that a mapping (without the additivity or linearity assumption) δ : A → A is Recently, many mathematicians devoted themselves to study the characterizations of Lie derivable mappings [1,5,9,14].…”

Local Lie derivations of nest algebras

“…The classical problem, which has been studied for many years, is to find conditions on A under which each Lie derivation is standard or standard-like. This problem has been investigated for general operator algebras [2,[16][17][18][19]21,22,24].…”