Maps preserving strong skew lie product on factor von neumann algebras

Let A be a factor von Neumann algebra and Φ be a nonlinear surjective map from A onto itself. We prove that, if Φ satisfies that Φ (A)Φ (B) − Φ (B)Φ (A)* = AB − BA* for all A, B ∈ A, then there exist a linear bijective map Ψ: A → A satisfying Ψ(A)Ψ(B) − Ψ(B)Ψ(A)* = AB − BA* for A, B ∈ A and a real functional h on A with h(0) = 0 such that Φ (A) = Ψ(A) + h(A)I for every A ∈ A. In particular, if A is a type I factor, then, Φ (A) = cA + h(A)I for every A ∈ A, where c = ±1.