Nonlinear mappings preserving Jordan multiple ∗– product on factor von Neumann algebras

Let and be two factor von Neumann algebras. For define by the Jordan multiple product in any fixed operation order. In this article, it is proved that a not necessarily linear bijective mapping satisfies if and only if is a ring isomorphism. In particular, if the von Neumann algebras and are type I factors, then is a unitary isomorphism or a conjugate unitary isomorphism.