Forcing axioms and coronas of $C^*$-algebras

TL;DR

This paper demonstrates that under the Proper Forcing Axiom, certain corona algebras exhibit rigidity, confirming a conjecture for a broad class of separable $C^*$-algebras with specific properties.

Contribution

It proves a conjecture regarding the rigidity of corona algebras for separable $C^*$-algebras with the metric approximation property under the Proper Forcing Axiom.

Findings

Rigidity results for large classes of corona algebras

Confirmation of Coskey and Farah's conjecture for specific $C^*$-algebras

Applicability to separable $C^*$-algebras with the metric approximation property

Abstract

We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable $C^*$-algebras with the metric approximation property and an increasing approximate identity of projections.