$$*$$-Lie–Jordan-Type Maps on $$C^*$$-Algebras

Abstract

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