Nonlinear maps preserving the Jordan triple ∗-product on von Neumann algebras

Abstract

This article investigates a bijective map $\Phi$ between two von Neumann algebras, one of which has no central abelian projections, satisfying $\Phi \left(\right[[A,B{]}_{\ast },C{]}_{\ast })=\left[\right[\Phi \left(A\right),\Phi \left(B\right){]}_{\ast },\Phi \left(C\right){]}_{\ast }$ for all $A,B,C$ in the domain, where $[A,B{]}_{\ast }=AB-B{A}^{*}$ is the skew Lie product of $A$ and $B$. We show that the map $\Phi \left(I\right)\Phi$ is a sum of a linear $\ast$-isomorphism and a conjugate linear $\ast$-isomorphism, where $\Phi \left(I\right)$ is a self-adjoint central element in the range with $\Phi (I{)}^2=I$.