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

Abstract

Let $\mathfrak{T}$ be a factor von Neumann algebra acting on complex Hilbert space  with dim($\mathfrak{T})\geq 2.$ For any $T, T_1, T_2, \dots, T_n \in\mathfrak{T},$ define $q_1(T)=T,$ $q_2(T_1, T_2)=T_1\diamond T_2=T_1T_2^\ast+T_2T_1^\ast$  and $q_n(T_1, \cdots, T_n)=q_{n-1}(T_1, \cdots, T_{n-1})\diamond T_n$ for all  positive integers  $n\geq 2.$  In this article, we prove that a  map $\zeta:\mathfrak{T}\rightarrow\mathfrak{T}$ satisfies $\zeta(q_n(T_1, \cdots, T_n))=\sum_{i=1}^{n} q_n(T_1,\cdots, T_{i-1}, \zeta(T_i), T_{i+1}, \cdots, T_n)$ for all $T_1, \cdots T_n\in\mathfrak{T}$ if and only if $\zeta$ is an additive $\ast$-derivation.