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

Abstract

Let T be a factor von Neumann algebra acting on complex Hilbert space with dim(T) ? 2. For any T, T1, T2,..., Tn ? T, define q1(T) = T, q2(T1, T2) = T1 ? T2 = T1T+ 2 + T2T+ 1 and qn(T1,..., Tn) = qn?1(T1,..., Tn?1) ? Tn for all integers n ? 2. In this article, we prove that a map ? : T ? T satisfies ?(qn(T1,..., Tn)) = Pni =1 qn(T1,..., Ti?1, ?(Ti), Ti+1,..., Tn) for all T1,..., Tn ? T if and only if ? is an additive *-derivation.