On the classification of nuclear C*-algebras

TL;DR

This paper advances the classification of nuclear C*-algebras by developing new existence and uniqueness results using KK-theory and quasidiagonality, leading to classification results for certain locally approximated quasidiagonal algebras.

Contribution

It introduces a unified approach to classify nuclear C*-algebras by leveraging KK-theory and local approximation properties, extending previous methods.

Findings

New classification results for quasidiagonal C*-algebras

Reproves classification of purely infinite nuclear C*-algebras

Develops general existence and uniqueness theorems using KK-theory

Abstract

The mid-seventies' works on C*-algebras of Brown-Douglas-Fillmore and Elliott both contained uniqueness and existence results in a now standard sense. These papers served as keystones for two separate theories -- KK-theory and the classification program -- which for many years parted ways with only moderate interaction. But recent years have seen a fruitful interaction which has been one of the main engines behind rapid progress in the classification program. In the present paper we take this interaction even further. We prove general existence and uniqueness results using KK-theory and a concept of quasidiagonality for representations. These results are employed to obtain new classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin. An important novel feature of these classes is that they are defined by a certain local approximation property, rather than by an inductive limit construction. Our existence and uniqueness results are in the spirit of classical Ext-theory. The main complication overcome in the paper is to control the stabilization which is necessary when one works with finite C*-algebras. In the infinite case, where programs of this type have already been successfully carried out, stabilization is unnecessary. Yet, our methods are sufficiently versatile to allow us to reprove, from a handful of basic results, the classification of purely infinite nuclear C*-algebras of Kirchberg and Phillips. Indeed, it is our hope that this can be the starting point of a unified approach to classification of nuclear C*-algebras.