Lie $n$-centralizers of von Neumann algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914139033567232 |
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| author | Ashraf, Mohammad Ansari, Mohammad Afajal Akhter, Md Shamim Wei, Feng |
| author_facet | Ashraf, Mohammad Ansari, Mohammad Afajal Akhter, Md Shamim Wei, Feng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_03523 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lie $n$-centralizers of von Neumann algebras Ashraf, Mohammad Ansari, Mohammad Afajal Akhter, Md Shamim Wei, Feng Operator Algebras Rings and Algebras 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. |
| title | Lie $n$-centralizers of von Neumann algebras |
| topic | Operator Algebras Rings and Algebras |
| url | https://arxiv.org/abs/2511.03523 |