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.