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

TY - JOUR

T1 - The Corona Factorization property, Stability, and the Cuntz semigroup of a C*-algebra

AU - Ortega Esparza, Edvardo

AU - Perera, Francesc

AU - Rørdam, Mikael

PY - 2012/1/1

Y1 - 2012/1/1

N2 - 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.

AB - 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.

M3 - Journal article

SN - 1073-7928

VL - 2012

SP - 34

EP - 66

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 1

ER -