Nonlinear maps preserving the mixed product

Let A and B be two von Neumann algebras. For A, B ∈ A, define by A • B = A∗B + B∗A and A ◦ B = A∗B − B∗A the new products of A and B. Suppose that a bijective map Φ : A → B satisfies Φ(A • B ◦ C) = Φ(A) • Φ(B) ◦ Φ(C) for all A, B, C ∈ 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 Φ(I)Φ is sum of a ∗-isomorphism and a conjugate linear ∗-isomorphism, where Φ(I) is a self-adjoint central element in B with Φ(I) 2 = I.