Linearity of (generalized) \(*\)-Lie derivations and their structures on \(*\)-algebras

Abstract

Let \( {\mathcal {A}} \) be a unital \(*\)-algebra with characteristic not 2 and containing a nontrivial projection. We show that each nonlinear \(*\)-Lie derivation on \({\mathcal {A}}\) is a linear \(*\)-derivation. Moreover, we characterize nonlinear left \(*\)-Lie centralizers and nonlinear generalized \(*\)-Lie derivations. These results are applied to standard operator algebras and von Neumann algebras in complex Hilbert spaces, which generalize some known results.