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Non-global Nonlinear Lie Triple Derivable Maps on Finite von Neumann Algebras
Bulletin of the Iranian Mathematical Society ( IF 0.7 ) Pub Date : 2021-01-27 , DOI: 10.1007/s41980-020-00493-4
Xingpeng Zhao , Haixia Hao

Let \({\mathcal {M}}\) be a finite von Neumann algebra with no central summands of type \(I_{1}\). Assume that \(\delta :{\mathcal {M}}\rightarrow {\mathcal {M}}\) is a nonlinear map satisfying \(\delta ([[A,B],C])=[[\delta (A),B],C]+[[A,\delta (B)],C]+[[A,B],\delta (C)]\) for any \(A,B,C\in {\mathcal {M}}\) with \(ABC=0\). Then, we prove that there exists an additive derivation \(d:{\mathcal {M}}\rightarrow {\mathcal {M}}\), such that \(\delta (A)=d(A)+\tau (A)\) for any \(A\in {\mathcal {M}}\), where \(\tau :{\mathcal {M}}\rightarrow {\mathcal {Z}}_{{\mathcal {M}}}\) is a nonlinear map, such that \(\tau ([[A,B],C])=0\) for any \(A,B,C\in {\mathcal {M}}\) with \(ABC=0\).



中文翻译:

有限von Neumann代数上的非全局非线性李三重导数

\({\ mathcal {M}} \)是一个不具有\(I_ {1} \)类型中心求和的有限冯诺依曼代数。假设\(\ delta:{\ mathcal {M}} \ rightarrow {\ mathcal {M}} \)是一个满足\(\ delta([[A,B],C])= [[\ delta (A),B],C] + [[A,\ delta(B)],C] + [[A,B],\ delta(C)] \)对于任何\(A,B,C \ in {\ mathcal {M}} \)\(ABC = 0 \)。然后,我们证明存在一个加法导数\(d:{\ mathcal {M}} \ rightarrow {\ mathcal {M}} \\),使得\(\ delta(A)= d(A)+ \ tau (A)\)对于任何\(A \ in {\ mathcal {M}} \\),其中\(\ tau:{\ mathcal {M}} \ rightarrow {\ mathcal {Z}} _ {{\ mathcal { M}}} \)是一个非线性映射,对于{\ mathcal {M}} \\中的任何\(A,B,C \\)\(ABC是\\ tau([[A,B],C])= 0 \)= 0 \)

更新日期:2021-01-28
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