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

Let A and B be two factor von Neumann algebras. For A, B is an element of A. define by [A, B](*) = AB - BA* the new product of A and B. In this paper, we prove that a nonlinear bijective map phi : A -> B satisfies phi([A, B](*)) = [phi(A), phi(B)](*) for all A, B is an element of A if and only phi is a *-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then phi is a unitary isomorphism or a conjugate unitary isomorphism. (C) 2009 Elsevier Inc. All rights reserved.