The Corona Factorization property, Stability, and the Cuntz semigroup of a C*-algebra

TL;DR

This paper explores how the Corona Factorization Property of t-unital C*-algebras is fully characterized by their Cuntz semigroup, linking algebraic properties to semigroup comparability conditions and stability criteria.

Contribution

It establishes that the Corona Factorization Property is captured by the Cuntz semigroup and introduces a new intrinsic stability criterion called property (S).

Findings

Corona Factorization Property is characterized by the Cuntz semigroup.

Unital C*-algebras with finite decomposition rank have the Corona Factorization Property.

Property (S) characterizes stability under certain Cuntz semigroup conditions.

Abstract

The Corona Factorization Property, originally invented to study extensions of C*-algebras, conveys essential information about the intrinsic structure of the C*-algebras. We show that the Corona Factorization Property of a \sigma-unital C*-algebra is completely captured by its Cuntz semigroup (of equivalence classes of positive elements in the stabilization of A). The corresponding condition in the Cuntz semigroup is a very weak comparability property termed the Corona Factorization Property for semigroups. Using this result one can for example show that all unital C*-algebras with finite decomposition rank have the Corona Factorization Property. Applying similar techniques we study the related question of when C*-algebras are stable. We give an intrinsic characterization, that we term property (S), of C*-algebras that have no non-zero unital quotients and no non-zero bounded 2-quasitraces. We then show that property (S) is equivalent to stability provided that the Cuntz semigroup of the C*-algebras satisfies another (also very weak) comparability property, that we call the \omega-comparison property.