Nonlinear Maps Preserving the Mixed Product [A ○ B, C]* on Von Neumann Algebras

Let A and B be two von Neumann algebras. For A, B is an element of A, define by [A, B](*) = AB - BA* and A circle B = AB + BA* the new products of A and B. Suppose that a bijective map Phi : A -> B satisfies Phi([A circle B, C](*)) = [Phi(A) circle Phi(B), Phi(C)](*) for all A, B, C is an element of A. In this paper, it is proved that if A and B be two von Neumann algebras with no central abelian projections, then the map Phi(I)Phi is a sum of a linear *-isomorphism and a conjugate linear *-isomorphism, where Phi(I) is a self-adjoint central element in B with Phi(I)(2) = I. If A and B are two factor von Neumann algebras, then Phi is a linear *-isomorphism, or a conjugate linear *-isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.