The descriptive set theory of C$^*$-algebra invariants
  We establish the Borel computability of various C$^*$-algebra invariants, including the Elliott invariant and the Cuntz semigroup. As applications we deduce that AF algebras are classifiable by countable structures, and that a conjecture of Winter and the second author for nuclear separable simple C*-algebras cannot be disproved by appealing to known standard Borel structures on these algebras. 
Ilijas Farah, Andrew S. Toms and Asger T\"ornquist
math.OA math.LO