A note on purely infinite corona algebras and extensions

TL;DR

This paper classifies essential extensions of certain nuclear C*-algebras with purely infinite corona algebras, extending previous work and providing new classification results for these algebraic structures.

Contribution

It extends the classification of essential extensions to cases where the corona algebra is purely infinite, broadening the understanding of these algebraic extensions.

Findings

Classified all essential extensions with large complement for purely infinite corona algebras.

Extended previous classification results to non-simple purely infinite corona algebras.

Provided general results applicable to a wider class of C*-algebra extensions.

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.