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

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