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A note on purely infinite corona algebras and extensions

Ping Wong Ng, Cangyuan Wang

Abstract

Let $\mathcal{A}$ be a separable nuclear C*-algebra, and $\mathcal{B}$ be a nonunital separable simple $\mathcal{Z}$-stable C*-algebra. Continuing the work from Gabe-Lin-Ng, we classify all essential extensions, with large complement, of the form $$0 \rightarrow \mathcal{B} \rightarrow \mathcal{E} \rightarrow \mathcal{A} \rightarrow 0,$$ for the following cases: i. $\mathcal{C}(\mathcal{B})$ is properly infinite, and the extension is full. ii. $\mathcal{C}(\mathcal{B})$ is purely infinite (though not necessarily simple). We also have some more general results.

A note on purely infinite corona algebras and extensions

Abstract

Let be a separable nuclear C*-algebra, and be a nonunital separable simple -stable C*-algebra. Continuing the work from Gabe-Lin-Ng, we classify all essential extensions, with large complement, of the form for the following cases: i. is properly infinite, and the extension is full. ii. is purely infinite (though not necessarily simple). We also have some more general results.
Paper Structure (4 sections, 14 theorems, 24 equations)

This paper contains 4 sections, 14 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. Classifying extensions: The full case
  4. Classifying extensions: The case of purely infinite corona algebras

Key Result

Theorem 3.6

Let $\mathcal{A}$ be a separable nuclear C*-algebra, and let $\mathcal{B}$ be a nonunital $\sigma$-unital simple C*-algebra for which $\mathcal{C}(\mathcal{B})$ is properly infinite and $K_1$-injective. Then $\mathbf{Ext}_{fpi}(\mathcal{A}, \mathcal{B})$, with the addition operation given in the pre

Theorems & Definitions (31)

  • Definition 3.3
  • Theorem 3.6
  • proof
  • Lemma 3.7
  • proof
  • Proposition 3.8
  • proof
  • Theorem 3.10
  • proof
  • Definition 3.11
  • ...and 21 more