Compact metrizable structures and classification problems

TL;DR

This paper develops a framework for compact metric structures and applies it to analyze classification problems in analysis, including C*-algebras and Choquet simplices, providing new proofs and insights.

Contribution

It introduces a new framework for compact metric structures and applies it to key classification problems, offering simplified proofs and broader understanding.

Findings

Established the completeness of the isomorphism relation for certain C*-algebras

Unified approach to classification problems in analysis

Provided simplified proof of Sabok's result on C*-algebras

Abstract

We introduce and study the framework of compact metric structures and their associated notions of isomorphisms such as homeomorphic and bi-Lipschitz isomorphism. This is subsequently applied to model various classification problems in analysis such as isomorphism of C*-algebras and affine homeomorphism of Choquet simplices, where among other things we provide a simple proof of the completeness of the isomorphism relation of separable, simple, nuclear C*-algebras recently established by M. Sabok.