Characterization of (generalized) Lie derivations on $${\mathcal{J}}$$ -subspace lattice algebras by local action

Abstract

Let \({\mathcal{L}}\) be a \({\mathcal{J}}\)-subspace lattice on a Banach space X over the real or complex field \({\mathbb{F}}\) with dim X ≥ 2 and Alg\({\mathcal{L}}\) be the associated \({\mathcal{J}}\)-subspace lattice algebra. For any scalar \({\xi \in \mathbb{F}}\), there is a characterization of any linear map L : Alg \({\mathcal{L} \rightarrow {\rm Alg} {\mathcal{L}}}\) satisfying \({L([A,B]_\xi) = [L(A),B]_\xi + [A,L(B)]_\xi}\) for any \({A, B \in{\rm Alg} {\mathcal{L}}}\) with AB = 0 (rep. \({[A,B]_ \xi = AB - \xi BA = 0}\)) given. Based on these results, a complete characterization of (generalized) ξ-Lie derivations for all possible ξ on Alg\({\mathcal{L}}\) is obtained.