A unified approach for classifying simple nuclear $C^\ast$-algebras

TL;DR

This paper offers a new, unified proof of the Kirchberg--Phillips theorem for classifying simple nuclear $C^*$-algebras, avoiding reliance on Kirchberg's Geneva Theorems and extending the framework to both stably finite and purely infinite cases.

Contribution

It provides a novel proof approach that unifies the classification of simple nuclear $C^*$-algebras, deriving key theorems as corollaries without using Kirchberg's Geneva Theorems.

Findings

Unified proof for classifying simple nuclear $C^*$-algebras

Derives Kirchberg's Geneva Theorems as corollaries

Applicable to both stably finite and purely infinite cases

Abstract

We provide a new proof of the Kirchberg--Phillips theorem by adapting the framework laid out by Carri\'on--Gabe--Schafhauser--Tikuisis--White for classifying separable simple unital nuclear stably finite $\mathcal Z$-stable $C^\ast$-algebras satisfying the UCT. Not only does this give a unified approach to classifying stably finite and purely infinite $C^\ast$-algebras, in contrast to the other proofs of the Kirchberg--Phillips theorem, our proof does not rely on Kirchberg's Geneva Theorems, but instead implies them as corollaries (for nuclear $C^\ast$-algebras).