Maps preserving product XY-YX∗ on factor von Neumann algebras

Let A and B be two factor von Neumann algebras. For A, B∈A, define by [A,B]∗=AB-BA∗ the new product of A and B. In this paper, we prove that a nonlinear bijective map Φ:A→B satisfies Φ([A,B]∗)=[Φ(A),Φ(B)]∗ for all A,B∈A if and only if Φ is a *-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism.