Lie triple derivations on factor von Neumann algebras
Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 581-591
Published online March 31, 2015
Lei Liu
Xidian University
Abstract : Let $\mathcal{A}$ be a factor von Neumann algebra with dimension greater than 1. We prove that if a linear 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 of $\mathcal{A}$), then there exist an operator $T\in \mathcal{A}$ and a linear map $f:\mathcal{A}\rightarrow \mathbb{C}I$ vanishing at every second commutator $[[a, b], c]$ with $ab=0$ (resp. $ab=P$) such that $\delta(a)=aT-Ta+f(a)$ for any $a\in \mathcal{A}$.
Keywords : Lie derivations, Lie triple derivations, factor von Neumann algebras
MSC numbers : Primary 16W25; Secondary 47B47
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