Nonlinear maps on von Neumann algebras preserving the star order

Star order is defined on a C*-algebra in the following way: a?b if a*a=a*b and aa*=ba*. Let ? be a von Neumann algebra without Type I-2 direct summand. Let ?(n) be the set of all normal elements of ?. Suppose that phi: ?(n)?(n) is a continuous bijection that preserves the star order on ?(n) in both directions. Further, let there is a function f: CC and an invertible central element phi(lambda 1) =f(lambda)C for all lambda epsilon C. We show that phi(1)=f(lambda)c for all C. We show that there is a unique Jordan *-isomorphism psi.A -> A such that phi(x) = psi(f(x))c, x is an element of A(u). Ramifications of this result as well as optimality of the assumptions are discussed.