Non-global Nonlinear Lie Triple Derivable Maps on Finite von Neumann Algebras

Abstract

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 $$ABC=0$$ .