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 $forallexists$-axiomatizable for any $ngeq 2$.
We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II$_{1}$ factor as Fraisse limits of suitable classes of structures. Moreover by means of Fraisse theory we provide new examples of AF algebras with strong homogeneity properties.
As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.
We prove that the isomorphism relation for separable C$^*$-algebras, and also the relations of complete and $n$-isometry for operator spaces and systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a standard Borel space.
