Nonlinear maps preserving Jordan *-products

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