On the Axiomatizability of C∗-algebras as Operator Systems

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. Recall that a C∗-algebra is a ∗-subalgebra of B(H), the ∗-algebra of bounded operators on a complex Hilbert space, that is closed in the operator norm topology. In this note, we assume that all C∗-algebras are unital, namely that they contain the identity operator. As shown in [5, Proposition 3.3], there is a natural (continuous) first-order language LC∗ in which KC∗ , the class of LC∗-structures that are unital C∗-algebras, is an elementary class, meaning that there is a (universal) LC∗-theory TC∗ for which KC∗ is the class of models of TC∗ ; in symbols, KC∗ = Mod(TC∗). (The authors only treat not necessarily unital C∗-algebras, but one just adds a constant to name the identity with no additional complications.) An operator system is a ∗-closed subspace of B(H) that is closed in the operator norm topology. The appropriate morphisms between operator systems are the unital completely positive linear maps. There is a natural first-order language Los in which the class of operator systems is universally axiomatizable; see [3, Subsection 3.3] and [7, Appendix B]. Since the operator system structure on a C∗-algebra is uniformly quantifier-free definable, we may assume that Los ⊆ LC∗. For a C∗-algebra A, we let A|Los denote the reduct of A to Los, which simply means that we view A merely as an operator system rather than as a C∗-algebra. Set KC∗ |Los := {A|Los : A ∈ KC∗}. In [6], the following question was raised: is KC∗ |Los an elementary class? The main result of this note is to give an affirmative answer to this question. We first need a lemma, which is nearly identical to [2, Theorem 6.1]. Some notation: for a C∗-algebra B and x, y, z, b ∈ B, let φ(x, y, z, b) be the Los-formula