Nonlinear maps preserving Jordan *-products

Let η be a non-zero scalar. In this paper, we investigate a bijective map ϕ between two von Neumann algebras, one of which has no central abelian projections, satisfying ϕ ( A B + η B A ∗ ) = ϕ ( A ) ϕ ( B ) + η ϕ ( B ) ϕ ( A ) ∗ for all A , B in the domain. It is showed that ϕ is a linear *-isomorphism if η is not real and ϕ is a sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η is real.