Characterization of nonlinear Lie type derivations on von Neumann algebras by local action

Let R be a unital ring containing a nontrivial idempotent. In this article, under a mild condition on R, we prove that if a map δ:R→R satisfies δ(Pn(A1,A2,A3,…,An))=∑i=1nPn(A1,…,Ai−1,δ(Ai),Ai+1,…,An) for any A1,A2,A3,…,An∈R with A1A2A3=0, then δ(A+B)−δ(A)−δ(B)∈Z(R) for any A,B∈R. In particular, if R is a von Neumann algebra with no central summands of type I1 or a factor, then δ(x)=d(x)+τ(x) for all x∈R, where d:R→R is an additive derivation and τ:R→Z(R) is a map vanishing on each (n−1)-th commutator pn(A1,A2,A3,…,An) with A1A2A3=0.