The Corona Factorization Property, Stability, and the Cuntz Semigroup of a C*-algebra

Abstract

The Corona Factorization Property, originally invented to study exten- sions of C � -algebras, conveys essential information about the intrinsic structure of the C � -algebra. We show that the Corona Factorization Property of a �-unital C � -algebra is completely captured by its Cuntz semigroup (of equivalence classes of positive ele- ments 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 � -algebra satisfies another (also very weak) comparability property, that we call the !-comparison property.