The descriptive classification of some classes of C∗-algebras

We introduce here a parametrization of separable C*-algebras by a standard Borel space and study the descriptive complexity of various canonical classes of C*-algebras in this parametrization. This can be viewed as providing an analog of the corresponding classification of classes of von Neumann algebras (acting on a fixed separable Hilbert space) in the Effros Borel space of von Neumann algebras (see for example Nielsen [13]). However, in contrast with the von Neumann case, where most interesting classes (like: factors, type I, II, III, hyperfinite) turn out to be Borel, in the C*-algebra case many important classes turn out to be co-analytic but not Borel. This makes the situation more interesting from the set-theoretic point of view, and leads to further questions, like the construction of canonical co-analytic norms, which we also address here. Finally, we relate the above to work of Sutherland [17] on parametrization of Polish groups and indicate how one might be able to show that various familiar classes of (second countable) locally compact groups are non-Borel as well.