∗-Lie–Jordan-Type Maps on C∗-Algebras

Let $\A$ and $\A $ be two $C^*$-algebras with identities $I_{\A}$ and $I_{\A }$, respectively, and $P_1$ and $P_2 = I_{\A} - P_1$ nontrivial projections in $\A$. In this paper we study the characterization of multiplicative $*$-Lie-Jordan-type maps, where the notion of these maps arise here. In particular, if $\mathcal{M}_{\A}$ is a von Neumann algebra relative $C^{*}$-algebra $\A$ without central summands of type $I_1$ then every bijective unital multiplicative $*$-Lie-Jordan-type maps are $*$-ring isomorphisms.