LAUSR

Let $${\mathcal {M}}$$M be a von Neumann algebra without central summands of type $$I_1$$I1. Assume that $$G:{{\mathcal {M}}}\rightarrow {{\mathcal {M}}}$$G:M→M is a nonlinear map. It is shown that G is a generalized Lie n-derivation ($$n\ge 2$$n≥2) if and only if $$G(A)=\varphi (A)+\tau (A)$$G(A)=φ(A)+τ(A) holds for all $$A\in {{\mathcal {M}}}$$A∈M, where $$\varphi :{\mathcal M}\rightarrow {{\mathcal {M}}}$$φ:M→M is an additive generalized derivation and $$\tau :{{\mathcal {M}}}\rightarrow {{\mathcal {Z}}}({{\mathcal {M}}})$$τ:M→Z(M) is a central-valued map annihilating all $$(n-1)$$(n-1)th commutators. This generalizes some related known results.