Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classification of Extensions of Classifiable C*-algebras

Soren Eilers, Gunnar Restorff, Efren Ruiz

TL;DR

The paper develops a six-term K-theory invariant for extensions of classifiable C*-algebras with a single nonzero ideal, proving a classification theorem for full essential extensions using $K$-theory with positive cones. It extends Rørdam’s approach to broader classes via fullness and corona factorization, leveraging KK-theory and the six-term exact sequence to decide when two extensions yield isomorphic middle algebras. It applies the framework to Matsumoto algebras and graph algebras, providing stable isomorphism criteria in terms of ordered $K_0$-data and scale, and discusses the role of extended invariants and potential refinements for non-full cases. Overall, the work offers practical invariants for stable isomorphism of non-simple subquotients and connects dynamical systems and graph C*-algebras to K-theoretic classification.

Abstract

We classify extensions of certain classifiable C*-algebras using the six term exact sequence in K-theory together with the positive cone of the K_0-groups of the distinguished ideal and quotient. We then apply our results to a class of C*-algebras arising from substitutional shift spaces.

Classification of Extensions of Classifiable C*-algebras

TL;DR

The paper develops a six-term K-theory invariant for extensions of classifiable C*-algebras with a single nonzero ideal, proving a classification theorem for full essential extensions using -theory with positive cones. It extends Rørdam’s approach to broader classes via fullness and corona factorization, leveraging KK-theory and the six-term exact sequence to decide when two extensions yield isomorphic middle algebras. It applies the framework to Matsumoto algebras and graph algebras, providing stable isomorphism criteria in terms of ordered -data and scale, and discusses the role of extended invariants and potential refinements for non-full cases. Overall, the work offers practical invariants for stable isomorphism of non-simple subquotients and connects dynamical systems and graph C*-algebras to K-theoretic classification.

Abstract

We classify extensions of certain classifiable C*-algebras using the six term exact sequence in K-theory together with the positive cone of the K_0-groups of the distinguished ideal and quotient. We then apply our results to a class of C*-algebras arising from substitutional shift spaces.

Paper Structure

This paper contains 12 sections, 18 theorems, 30 equations.

Table of Contents

  1. Extensions
  2. Notation
  3. Full extensions
  4. Six term exact sequence in $\mathnormal{K}$-theory
  5. Classification results
  6. Applications
  7. Matsumoto algebras
  8. Graph algebras
  9. Concluding discussion
  10. Extended invariants
  11. Improved classification results
  12. Acknowledgement

Key Result

Lemma 1.3

Let $A$ be a separable $C \sp \ast$-algebra. If $p$ is a norm-full projection in $A \otimes \mathbf{M}_{n} \subset A \otimes \mathcal{K}\xspace$, then there exists a $*$-iso-mor-phism $\varphi$ from $A \otimes \mathcal{K}\xspace$ onto $p ( A \otimes \mathcal{K}\xspace ) p \otimes \mathcal{K}\xspace$

Theorems & Definitions (40)

  • Definition 1.2
  • Lemma 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Proposition 1.6
  • proof
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • ...and 30 more