Quantifier elimination in C*-algebras

Abstract

The only unital C∗-algebras that admit elimination of quantifiers in continuous logic in the language of unital C∗-algebras are C, C2, C(Cantor space) and M2(C). We also prove that the theory of C∗-algebras does not have model companion and show that the theory of Mn(On+1) is not ∀∃-axiomatizable for any n ≥ 2.