Mappings Preserving Sum of Products ab + b ◦ a^∗ on Factor Von Neumann Algebras

Let A and B be two factor von Neumann algebras.In this paper, we proved that a bijective mapping : A B satisfies (ab+ba * ) = (a)(b)+(b)(a) * (where is the special Jordan product on A and B), for all elements a, b A, if and only if is a * -ring isomorphism.In particular, if the von Neumann algebras A and B are type I factors, then is a unitary isomorphism or a conjugate unitary isomorphism.