A dichotomy for the Mackey Borel structure

TL;DR

This paper establishes a dichotomy in the classification of pure states of separable C*-algebras, showing they are either smoothly classifiable or as complex as a certain non-classifiable equivalence relation, with implications for longstanding problems.

Contribution

It proves a dichotomy for the Mackey Borel structure, demonstrating the complexity of classifying pure states of separable C*-algebras and connecting to prior independent results.

Findings

Pure states are either smoothly classifiable or as complex as a non-classifiable relation.

The classification complexity is equivalent to a known non-classifiable equivalence relation.

Provides insights related to a 1967 problem of Dixmier.

Abstract

We prove that the equivalence of pure states of a separable C*-algebra is either smooth or it continuously reduces $[0,1]^{\bbN}/\ell_2$ 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.