Nonlinear Maps Preserving the Mixed Type Product (M N ∘ W) on $$*$$-Algebras

Let S and B be two unital $$*$$ -algebras such that S has a nontrivial projection. In the present article, we demonstrate, under certain restrictions that if a bijective map Δ :S→ B satisfies Δ (M N ∘ W) = Δ (M) Δ (N)∘ Δ (W) for all M, N, W ∈ S , then Δ is a $$*$$ -preserving ring isomorphism. As an application, we will describe these mappings on factor von Neumann algebras.