Total extension groups for unital Kirchberg algebras

TL;DR

This paper introduces a hierarchy and a new invariant called the total extension group for unital Kirchberg algebras with finitely generated K-groups, enabling complete classification of their automorphism groups and related C*-algebras.

Contribution

It defines the total extension group as a new invariant that, along with the hierarchy, fully classifies unital Kirchberg algebras with finitely generated K-groups.

Findings

Total extension group provides a complete invariant for classification.

Hierarchy based on homotopy groups of automorphism groups.

Applicable to classify Cuntz--Krieger algebras.

Abstract

We introduce a hierarchy for unital Kirchberg algebras with finitely generated K-groups by which the first and second homotopy groups of the automorphism groups serve as a complete invariant of classification. We also introduce an invariant called the total extension group which is the direct sum of the strong and weak extension groups. In the case of unital Kirchberg algebras with finitely generated K-groups, the total extension group gives a complete invariant and provides a useful tool to classify the Cuntz--Krieger algebras.