Unitary equivalence of automorphisms of separable C*-algebras

TL;DR

This paper demonstrates that automorphisms of certain separable C*-algebras are either very simple or highly complex in classification, with no intermediate level, based on their Borel complexity.

Contribution

It establishes a dichotomy in the Borel complexity of automorphism unitary equivalence for separable C*-algebras without continuous trace, showing they are either smooth or not classifiable by countable structures.

Findings

Automorphisms of C*-algebras without continuous trace are not classifiable by countable structures.

The relation of unitary equivalence of automorphisms is either smooth or non-classifiable.

A dichotomy in classification complexity for automorphisms of separable unital C*-algebras.

Abstract

We prove that the automorphisms of any separable C*-algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence. This implies a dichotomy for the Borel complexity of the relation of unitary equivalence of automorphisms of a separable unital C*-algebra: Such relation is either smooth or not even classifiable by countable structures.