Abstract We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra script upper M $\mathcal {M}$ M is never model complete if its direct integral decomposition contains upper I upper I Subscript 1 $\mathrm {II}_1$ I I 1 factors script upper N $\mathcal {N}$ N such that upper M 2 left parenthesis script upper N right parenthesis $M_2(\mathcal {N})$ M 2 ( N ) embeds into an ultrapower of script upper N $\mathcal {N}$ N . The proof in the case of upper I upper I Subscript 1 $\mathrm {II}_1$ I I 1 factors uses an explicit construction based on random matrices and quantum expanders.

Abstract We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II1factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé 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.