Lie $n$-centralizers of von Neumann algebras

Abstract

Let $\U$ be a von Neumann algebra with a projection $P\in \U$. For any $A_1,A_2,\ldots,A_n\in\U,$ define $p_1(A_1)=A_1,$ $p_n(A_1,A_2,\dots ,A_n)=[p_{n-1}(A_1,A_2,\dots ,A_{n-1}),A_n]$ for all integers $n\ge 2,$ where $[A,B]=AB-BA$ $(A,B\in\U)$ denotes the usual Lie product. Assume that $ϕ:\U\to\U$ is an additive mapping satisfying \[ϕ(p_n(A_1, A_2, \ldots, A_n)) = p_n(ϕ(A_1), A_2, \ldots, A_n) = p_n(A_1, ϕ(A_2), \ldots, A_n) \] for all $A_1, A_2, \ldots, A_n \in \U$ with $A_1{A}_2=P$ In this article, it is shown that the map $\varphi$ is of the form $\varphi (A)=WA+\xi (A)$ for all $A\in \U$, where $W\in \mathrm{Z}(\U)$, and $ξ:\U \to \Z(\U)$ ($\Z(\U)$ is the center of $\U$) is an additive map such that $\xi (p_n(A_1,A_2,\dots ,A_n))=0$ for any $A_1, A_2, \ldots, A_n \in \U$ with $A_1{A}_2=P$. As an application, we characterize generalized Lie $n$-derivations on arbitrary von Neumann algebras.