Nonlinear maps preserving bi-skew Lie triple product on factor von Neumann algebras

Abstract Let A $\mathcal{A}$ and B $\mathcal{B}$ be two factor von Neumann algebras with dimensions greater than 1. It is proved that if a bijective map Φ : A → B ${\Phi} : \mathcal{A}\to \mathcal{B}$ satisfies Φ ( [ [ A , B ] ⋄ , C ] ⋄ ) = [ [ Φ ( A ) , Φ ( B ) ] ⋄ , Φ ( C ) ] ⋄ ${\Phi}\left({\left[{\left[A,B\right]}_{\diamond },C\right]}_{\diamond }\right)={\left[{\left[{\Phi}\left(A\right),{\Phi}\left(B\right)\right]}_{\diamond },{\Phi}\left(C\right)\right]}_{\diamond }$ for all A , B , C ∈ A $A,B,C\in \mathcal{A}$ , then Φ is a linear ∗-isomorphism, or a conjugate linear ∗-isomorphism, or the negative of a linear ∗-isomorphism, or the negative of a conjugate linear ∗-isomorphism, where [ A , B ] ⋄ = AB * − BA *.