QDQ vs. UCT

TL;DR

This survey discusses recent advances in the classification of nuclear C*-algebras, emphasizing how the Universal Coefficient Theorem impacts quasidiagonality and related conjectures.

Contribution

It highlights the role of the Universal Coefficient Theorem in resolving quasidiagonality questions and their implications for the classification of nuclear C*-algebras.

Findings

Universal Coefficient Theorem ensures quasidiagonality with faithful traces

Positive solution to Rosenberg's conjecture for amenable groups

Advances support the regularity conjecture in C*-algebra classification

Abstract

This is a survey of recent progress in the structure and classification theory of nuclear C*-algebras. In particular, I outline how the Universal Coefficient Theorem ensures a positive answer to the quasidiagonality question in the presence of faithful traces. This has strong consequences for the regularity conjecture and the classification problem for separable, simple, nuclear C*-algebras. Moreover, it entails a positive solution to Rosenberg's conjecture on quasidiagonality of reduced C*-algebras of discrete amenable groups. This note is largely based on a joint paper with Aaron Tikuisis and Stuart White.