On the Axiomatizability of C*-Algebras as Operator Systems

Abstract

We show that the class of unital 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 C * -algebra. Moreover, we prove that the aforementioned class is ∀∃∀-axiomatizable but not ∀∃-axiomatizable nor ∃∀-axiomatizable.