On the axiomatizability of $\mathrm{C}^*$-algebras as operator systems

We show that the class of unital $\mathrm{C}^*$-algebras is an elementary class in the language of operator systems. As a result, we have that there is a definable predicate in the language of operator systems that defines the multiplication in any $\mathrm{C}^*$-algebra. Moreover, we prove that the aforementioned class is $\forall\exists\forall$-axiomatizable but not $\forall\exists$-axiomatizable nor $\exists\forall$-axiomatizable.