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

Let I be a factor von Neumann algebra acting on complex Hilbert space with dim(I) >= 2. For any T, T-1, T-2, ..., T-n is an element of I, define (q1)(T) = T, q(2)(T-1, T-2) = T-1 lozenge T-2 = T1T*(2) + T2T*(1) and q(n)(T-1,..., T-n) = q(n-1)(T-1, . . . ,Tn-1) lozenge Tn for all integers n >= 2. In this article, we prove that a map zeta : I -> I satisfies zeta(q(n)(T-1, . . . , T-n)) = E-i=1(n) q(n)(T-1, . . . ,Ti-1, zeta(T-i),Ti+1, . . . , T-n) for all T-1, ..., T-n is an element of I if and only if zeta is an additive *-derivation.