Nonlinear maps preserving the mixed type product [M ◇ N,W]∗ on ∗-algebras

Let [Formula: see text] and [Formula: see text] be unital ∗-algebras such that [Formula: see text] has a nontrivial projection. In this paper, we demonstrate under certain restrictions that if a bijective map [Formula: see text] satisfies [Formula: see text] for all [Formula: see text], then [Formula: see text] is a ∗-preserving ring isomorphism. Also, this result is applied to factor von Neumann algebras.