The descriptive set theory of C$^*$-algebra invariants

TL;DR

This paper demonstrates the Borel computability of key C*-algebra invariants, enabling classification results and addressing conjectures within the framework of descriptive set theory.

Contribution

It establishes the Borel computability of the Elliott invariant and Cuntz semigroup, advancing the classification theory of C*-algebras.

Findings

AF algebras are classifiable by countable structures

A conjecture by Winter and the second author cannot be disproved using known Borel structures

Key invariants are shown to be Borel computable

Abstract

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.