Nonlinear maps preserving mixed bi-skew Jordan N-product on factor von Neumann algebras

Let [Formula: see text] and [Formula: see text] be two factor von Neumann algebras such that [Formula: see text] contains a nontrivial projection. For any [Formula: see text], we denote [Formula: see text] (bi-skew Jordan product) and [Formula: see text] (skew Lie product). In this paper, we consider a bijective map [Formula: see text] satisfying [Formula: see text] for all [Formula: see text], [Formula: see text] and we prove that [Formula: see text], where [Formula: see text], is a linear ∗-isomorphism or a conjugate linear ∗-isomorphism.