Nonlinear Maps Preserving the Mixed Type Product \((M\diamond N \circ W)\) on \(*\)-Algebras

Let \({\mathcal {S}}\) and \({\mathfrak {B}}\) be two unital \(*\)-algebras such that \({\mathcal {S}}\) has a nontrivial projection. In the present article, we demonstrate, under certain restrictions that if a bijective map \(\Delta :{\mathcal {S}}\rightarrow {\mathfrak {B}}\) satisfies \(\Delta (M\diamond N \circ W) = \Delta (M)\diamond \Delta (N)\circ \Delta (W)\) for all \(M, N, W \in {\mathcal {S}}\), then \(\Delta\) is a \(*\)-preserving ring isomorphism. As an application, we will describe these mappings on factor von Neumann algebras.