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

Let T be a factor von Neumann algebra acting on complex Hilbert space with dim(T) ≥ 2. For any T, T₁, T₂, …, T_n ∈ T, define q₁ (T) = T, q₂ (T₁, T₂) = T₁ ⋄ T₂ = T₁T^∗2^+T2T^{∗ andq}1 n(T₁, …, T_n) = q_n−1 (T₁, …, T_n−1) ⋄ T_n for all integers n ≥ 2. In this article, we prove that a map ζ: T → T satisfies ζ(q_n (T₁, …, T_n)) =^∑n i=1 q_n (T₁, …, T_i−1, ζ(T_i), T_i+1, …, T_n) for all T₁, …, T_n ∈ T if and only if ζ is an additive ∗-derivation.