Omitting types and AF algebras

The model theory of metric structures ([?]) was successfully applied to analyze ultrapowers of C*-algebras in [?] and [?]. Since important classes of separable C*-algebras, such as UHF, AF, or nuclear algebras, are not elementary (i.e., not characterized by their theory—see [?, §6.1]), for a moment it seemed that model theoretic methods do not apply to these classes of C*-algebras. We prove results suggesting that this is not the case. Many of the prominent problems in the modern theory of C*-algebras are concerned with the extent of the class of nuclear C*-algebras. We have the bootstrap class problem (see [?, IV3.1.16]), the question of whether all nuclear C*-algebras satisfy the Universal Coefficient Theorem, UCT, (see [?, §2.4]), and the Toms—Winter conjecture (to the effect that the three regularity properties of nuclear C*-algebras discussed in [?] are equivalent; see [?]). If one could characterize classes of algebras in question—such as nuclear algebras, algebras with finite nuclear dimension, or algebras with finite decomposition rank—as algebras that omit certain sets of types (see §??) then one might use the omitting types theorem ([?, §12]) to construct such algebras, modulo resolving a number of nontrivial technical obstacles. This paper is the first, albeit modest, step in this project. Recall that a unital C*-algebra is UHF (Uniformly HyperFinite) if it is a tensor product of full matrix algebras, Mn(C). Non-unital UHF algebras are direct limits of full matrix algebras, and in the separable, unital case the two definitions are equivalent. A C*-algebra is AF (Approximately Finite) if it is a direct limit of finite-dimensional C*-algebras. These three classes of C*-algebras were the first to be classified by work of Glimm, Dixmier and Elliott (building on Bratteli’s results), respectively. Elliott’s classification of separable AF algebras by the ordered K0 group was a prototype for Elliott’s program for classification of nuclear, simple, separable, unital C*-algebras by their K-theoretic invariants (see [?] or [?]). Types are defined in §??.