Multiplicative Jordan triple isomorphisms on the self-adjoint elements of von Neumann algebras

In this paper we consider multiplicative Jordan triple isomorphisms between the sets of self-adjoint elements (respectively the sets of positive elements) of von Neumann algebras. These transformations are the bijective maps which satisfy the equalityϕ(ABA)=ϕ(A)ϕ(B)ϕ(A)ϕ(ABA)=ϕ(A)ϕ(B)ϕ(A)on their domains. We show that all those transformations originate from linear ∗-algebra isomorphisms and linear ∗-algebra antiisomorphisms in the case when the underlying von Neumann algebras do not have commutative direct summands. An application of our results concerning non-linear maps which preserve the absolute value of products is also presented.