Nonlinear mappings preserving product [formula omitted] on factor von Neumann algebras

Abstract

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 article, it is proved that a not necessarily linear bijective mapping 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.