Nonlinear maps preserving the first mixed Jordan triple product on ∗-algebras

Let [Formula: see text] and [Formula: see text] represent two unital ∗-algebras such that [Formula: see text] possesses a nontrivial projection. In this paper, we study, under certain mild assumptions, 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. Furthermore, this result is investigated on factor von Neumann algebras.