Bulletin of the
Korean Mathematical Society BKMS

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