Nonlinear maps preserving mixed Lie triple products on factor von Neumann algebras

Abstract

We prove that every bijective map that preserves mixed Lie triple products from a factor von Neumann algebra $\mathcal{M}$ with $dim\mathcal{M}>4$ into another factor von Neumann algebra $\mathcal{N}$ is of the form $A\to ϵ\Psi \left(A\right)$, where $ϵ\in \{1,-1\}$ and $\Psi :\mathcal{M}\to \mathcal{N}$ is a linear $\ast$-isomorphism or a conjugate linear $\ast$-isomorphism. Also, we give the structure of this map when $dim\mathcal{M}=4$.