Nonlinear Maps Preserving the Jordan Triple 1-\(*\)-Product on Von Neumann Algebras

In this paper, we investigate a bijective map \(\Phi \) between two von Neumann algebras, one of which has no central abelian projections, satisfying \(\Phi (A\bullet B\bullet C)=\Phi (A)\bullet \Phi (B)\bullet \Phi (C)\) for all A, B, C in the domain, where \(A\bullet B=AB+BA^{*}\) is the Jordan 1-\(*\)-product of A and B. It is showed that the map \(\Phi (I)\Phi \) is a sum of a linear \(*\)-isomorphism and a conjugate linear \(*\)-isomorphism, where \(\Phi (I)\) is a self-adjoint central element in the range with \(\Phi (I)^{2}=I\).