The unital Ext-groups and classification of $C^\ast$-algebras

TL;DR

This paper develops a $K$-theoretic classification framework for unital and non-unital extensions of $C^*$-algebras using unital $ ext{Ext}$-groups, providing a complete classification for certain classes of algebras.

Contribution

It introduces the concept of unital $ ext{Ext}$-groups and applies them to classify extensions of $C^*$-algebras, including a full classification of extensions of UCT Kirchberg algebras by stable AF algebras.

Findings

Complete $K$-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.

Identification of invertible elements in semigroups of unital extensions as unital $ ext{Ext}$-groups.

Application of unital $ ext{Ext}$-groups to classify both unital and non-unital extensions.

Abstract

The semigroups of unital extensions of separable $C^\ast$-algebras come in two flavours: a strong and a weak version. By the unital $\mathrm{Ext}$-groups, we mean the groups of invertible elements in these semigroups. We use the unital $\mathrm{Ext}$-groups to obtain $K$-theoretic classification of both unital and non-unital extensions of $C^\ast$-algebras, and in particular we obtain a complete $K$-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.