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

Abstract

We show that the class of unital ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-algebra. Moreover, we prove that the aforementioned class is $\forall \exists \forall$-axiomatizable but not $\forall \exists$-axiomatizable nor $\exists \forall$-axiomatizable.

7 pages

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