Structure of nuclear C*-algebras: From quasidiagonality to classification, and back again

TL;DR

This paper reviews recent advances in the structure and classification of separable, simple, nuclear C*-algebras, emphasizing quasidiagonality, amenability, and the regularity conjecture's role in understanding their properties.

Contribution

It synthesizes recent progress connecting quasidiagonality and amenability to classification, highlighting the significance of the regularity conjecture in C*-algebra theory.

Findings

Quasidiagonality is crucial for classifying nuclear C*-algebras.

The regularity conjecture links internal and external approximation properties.

Recent developments have advanced the understanding of nuclear C*-algebra structure.

Abstract

I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the regularity conjecture and its interplay with internal and external approximation properties.