Model theory of C*-algebras

A number of significant properties of C ∗ C^* -algebras can be expressed in continuous logic, or at least in terms of definable (in a model-theoretic sense) sets. Certain sets, such as the set of projections or the unitary group, are uniformly definable across all C ∗ C^* -algebras. On the other hand, the definability of some other sets, such as the connected component of the identity in the unitary group of a unital C ∗ C^* -algebra, or the set of elements that are Cuntz–Pedersen equivalent to 0 0 , depends on structural properties of the C ∗ C^* -algebra in question. Regularity properties required in the Elliott programme for classification of nuclear C ∗ C^* -algebras imply the definability of some of these sets. In fact any known pair of separable, nuclear, unital and simple C ∗ C^* -algebras with the same Elliott invariant can be distinguished by their first-order theory. Although parts of the Elliott invariant of a classifiable (in the technical C ∗ C^* -algebraic sense) C ∗ C^* -algebra can be reconstructed from its model-theoretic imaginaries, the information provided by the theory is largely complementary to the information provided by the Elliott invariant. We prove that all standard invariants employed to verify non-isomorphism of pairs of C ∗ C^* -algebras indistinguishable by their K-theoretic invariants (the divisibility properties of the Cuntz semigroup, the radius of comparison, and the existence of finite or infinite projections) are invariants of the theory of a C