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

Abstract

We provide a new proof of the Kirchberg--Phillips theorem by adapting the framework laid out by Carrión--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).

47 pages