Nonlinear maps preserving mixed products on von Neumann algebras

Abstract

Let $\mathcal{M}$ be a von Neumann algebra with no central summands of type $\uppercase\expandafter{\romannumeral1}_{1}$ and $\Phi:\mathcal{M}\rightarrow\mathcal{M}$ be a nonlinear bijective map preserving mixed products satisfying that $\Phi([a\bullet b,c])=[\Phi(a)\bullet\Phi(b),\Phi(c)]$ for all $a,b,c\in\mathcal{M}$.  Then there exists $z\in\mathcal{Z}_{\mathcal{M}}$ with $z^{2}=I$ such that $\Phi$ is of the form $\Phi=z\Psi$, where $\Psi:\mathcal{M}\rightarrow\mathcal{M}$ is the sum of a linear $\ast$-isomorphism and a conjugate linear $\ast$-isomorphism.