Lie Triple Derivations on von Neumann Algebras


Lei LIU.Lie Triple Derivations on von Neumann Algebras[J].Chinese Annals of Mathematics B,2018,39(5):817~828
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Lei LIU;


This work was supported by the National Natural Science Foundation of China (No.11401452) and the China Postdoctoral Science Foundation (No.2015M581513).
Abstract: Let $\mathcal{A}$ be a von Neumann algebra with no central abelian projections. It is proved that if an additive map $\delta: \mathcal{A}\rightarrow \mathcal{A}$ satisfies $\delta([[a, b], c])=[[\delta(a), b], c]+[[a, \delta(b)], c]+[[a, b], \delta(c)]$ for any $a, b, c\in \mathcal{A}$ with $ab=0$ (resp. $ab=P$, where $P$ is a fixed nontrivial projection in $\mathcal{A}$), then there exist an additive derivation $d$ from $\mathcal{A}$ into itself and an additive map $f:\mathcal{A}\rightarrow \mathcal{Z}_{\mathcal{A}}$ vanishing at every second commutator $[[a, b], c]$ with $ab=0$ (resp. $ab=P$) such that $\delta(a)=d(a)+f(a)$ for any $a\in \mathcal{A}$.


Derivations, Lie triple derivations, von Neumann algebras


16W25, 47B47
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