Nonlinear Generalized Lie n-Derivations on von Neumann Algebras

Abstract

Let ${\mathcal {M}}$ be a von Neumann algebra without central summands of type $I_1$. Assume that $G:{{\mathcal {M}}}\rightarrow {{\mathcal {M}}}$ is a nonlinear map. It is shown that G is a generalized Lie n-derivation ($n\ge 2$) if and only if $G(A)=\varphi (A)+\tau (A)$ holds for all $A\in {{\mathcal {M}}}$, where $\varphi :{\mathcal M}\rightarrow {{\mathcal {M}}}$ is an additive generalized derivation and $\tau :{{\mathcal {M}}}\rightarrow {{\mathcal {Z}}}({{\mathcal {M}}})$ is a central-valued map annihilating all $(n-1)$th commutators. This generalizes some related known results.