Lie $n$-centralizers of von Neumann algebras
Mohammad Ashraf, Mohammad Afajal Ansari, Md Shamim Akhter, Feng Wei
TL;DR
The paper addresses the problem of characterizing Lie $n$-centralizers on von Neumann algebras. It employs a block decomposition via central projections and a core-free projection to derive that any additive Lie $n$-centralizer has the form $\phi(A)=W A+\xi(A)$ with $W\in Z(\mathcal{U})$ and $\xi:\mathcal{U}\to Z(\mathcal{U})$ such that $\xi(p_n(A_1,\dots,A_n))=0$ when $A_1A_2=P$, providing a complete structure theorem. This leads to a corollary for arbitrary von Neumann algebras and an application: generalized Lie $n$-derivations decompose as $G_{\mathcal{L}}(A)=W A+\delta(A)+\chi(A)$ with $\delta$ a derivation and $\chi$ central-valued. The results extend known factor-algebra cases (no central summands of type $I_1$) to broader von Neumann algebras, and enable further characterizations of related mappings.
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 $p_1(A_1)=A_1,$ $p_n (A_1,A_2,\ldots,A_n)=[p_{n-1} (A_1,A_2,\ldots,A_{n-1}),A_n]$ for all integers $n\geq 2,$ 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 $A_1A_2=P$ In this article, it is shown that the map $φ$ is of the form $φ(A)=WA+ξ(A)$ 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 $ξ(p_n(A_1, A_2, \ldots, A_n) )=0$ for any $A_1, A_2, \ldots, A_n \in \U$ with $A_1A_2=P$. As an application, we characterize generalized Lie $n$-derivations on arbitrary von Neumann algebras.