  The only C*-algebras that admit elimination of quantifiers in continuous
logic are $\mathbb{C}, \mathbb{C}^2$, $C($Cantor space$)$ and
$M_2(\mathbb{C})$. We also prove that the theory of C*-algebras does not have
model companion and show that the theory of $M_n(\mathcal {O_{n+1}})$ is not
$\forall\exists$-axiomatizable for any $n\geq 2$.
