A dichotomy for the Mackey Borel structure

We prove that the equivalence of pure states of a separable C*-algebra is either smooth or it continuously reduces $[0,1{]}^{\text{bbN}}\mathrm{/}{\mathrm{\ell }}_2$ and it therefore cannot be classified by countable structures. The latter was independently proved by Kerr–Li–Pichot by using different methods. We also give some remarks on a 1967 problem of Dixmier.