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

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