Quasidiagonality of nuclear C*-algebras

TL;DR

This paper proves that faithful traces on certain nuclear C*-algebras are quasidiagonal, leading to classification results, insights into the Toms-Winter conjecture, and confirming the Rosenberg conjecture for amenable groups.

Contribution

It establishes quasidiagonality of faithful traces on separable, nuclear C*-algebras in the UCT class, connecting classification, the Toms-Winter conjecture, and group C*-algebras.

Findings

Faithful traces on separable, nuclear C*-algebras in the UCT class are quasidiagonal.

Classification of certain nuclear C*-algebras is now complete.

Discrete, amenable groups have quasidiagonal C*-algebras.

Abstract

We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C*-algebras.