Nonlinear bi-skew Lie-type derivations on factor von Neumann algebras

Let A be a factor von Neumann algebra with dimA≥2. For any X1,X2,⋯,Xn∈A, define p1(X1)=X1, p2(X1,X2)=X1,X2•=X1X2∗−X2X1∗, and pn(X1,X2,⋯,Xn)=pn−1(X1,X2,⋯,Xn−1),Xn• for all integers n≥2. In this article, we prove that a map L:A→A satisfies L(pn(X1,X2,⋯,Xn))=∑i=1npn(X1,X2,⋯,Xi−1,L(Xi),Xi+1,⋯,Xn) for all X1,X2,⋯,Xn∈A if and only if L is an additive *-derivation.