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 thata bijective map Φ : A −→ B satisfies Φ(A • B ◦ C) = Φ(A) • Φ(B) ◦ Φ(C) for allA,B,C ∈ A. In this paper, it is proved that if A and B be two von Neumann algebraswith no central abelian projections, then the map Φ(I)Φ is sum of a ∗-isomorphismand a conjugate linear ∗-isomorphism, where Φ(I) is a self-adjoint central element inB with Φ(I)۲ = I.