The isomorphism relation for separable C*-algebras

TL;DR

This paper demonstrates that the isomorphism relation for separable C*-algebras and related structures can be classified using Borel reducibility to orbit equivalence relations of Polish group actions, advancing the understanding of their complexity.

Contribution

It establishes the Borel reducibility of the isomorphism and isometry relations for certain operator structures to orbit equivalence, linking operator algebra classification to descriptive set theory.

Findings

Isomorphism relation for separable C*-algebras is Borel reducible to a Polish group orbit.

Complete and n-isometry relations for operator spaces and systems are similarly reducible.

Provides a framework connecting operator algebra classification with Borel complexity theory.

Abstract

We prove that the isomorphism relation for separable C$^*$-algebras, and also the relations of complete and $n$-isometry for operator spaces and systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a standard Borel space.