Non-global nonlinear skew Lie n -derivations on *-algebras

Let A be a unital prime *-algebra (over the complex field C) with a nontrivial projection. For any S1,S2,…,Sn∈A, define q1(S1)=S1,q2(S1,S2)=[S1,S2]⋄=S1S2−S2S1∗ and qn(S1,S2,…,Sn)=[qn−1(S1,S2,…,Sn−1),Sn]⋄ for all integers n≥3. In this article, it is shown that if a map ξ:A→A (not necessarily linear) satisfies ξ(qn(S1,S2,…,Sn))=∑i=1nqn(S1,…,Si−1,ξ(Si),Si+1,…,Sn) (n≥3) for all S1,S2,…,Sn∈A with S1∗S2∗⋯Sn∗=0, then ξ is an additive *-derivation. Finally, this result is applied to standard operator algebras and factor von Neumann algebras.