Mappings Preserving Sum of Products a∘b−ba∗ on Factor von Neumann Algebras

Let A and B be two factor von Neumann algebras. In this paper, we proved that a bijective mapping Φ:A→B satisfies Φ(a∘b−ba∗)=Φ(a)∘Φ(b)−Φ(b)Φ(a)∗ (where ∘ is the special Jordan product on A and B, respectively), for all elements 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.