The corona factorization property, stability, and the cuntz semigroup of a C*-algebra

The Corona Factorization Property, originally invented to study extensions of C*-algebras, conveys essential information about the intrinsic structure of the C*-algebra. We show that the Corona Factorization Property of a σ-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 a 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 nonzero unital quotients and no nonzero bounded 2-quasitraces. We then show that property (S) is equivalent to stability provided that the Cuntz semigroup of the C*-algebra has another (also very weak) comparability property, that we call the ω-comparison property. © 2011 The Author(s). Published by Oxford University Press. All rights reserved.