The Corona Factorization property, Stability, and the Cuntz semigroup of a C*-algebra
By E. Ortega, F. Perera, M. Rørdam. 2009
66 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9454ba0516841565bb6f91d9bf313d6e31994401
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In Algebra and Number Theory
In Mathematical Physics
- David Kerr
- W. Winter
- Leonel Robert
- M. Rørdam
- Andrew S. Toms
- Yasuhiko Sato
- G'abor Szab'o
- Stuart White
Citations in the corpus
- Nuclear dimension and -stability of pure C ∗ -algebras (2010, background): “1] that decomposition rank at most m implies m-comparison (in the simple and unital case; it was observed in [23] that essentially the same proof holds also in the general situation).”
- Divisibility properties for C*‐algebras (2011, methodology): “i=1 z(n) i = XN i1,i2,...,in=1 zi1,i2,...,in. 19 Recall that a C∗-algebra A has the Corona Factorization Property if and only if all full projections in M(A⊗K) are properly infinite. It was shown in [OPR] that if A is a separable C∗-algebra, then A and all its closed two-sided ideals have the Corona”
- Dimension, comparison, and almost finiteness (2017, background): “ A⊆X. Thus σ([1A]) <σ([1B]) for every state σon S(X,G). Since the action is minimal, Xis covered by finitely many translates of B, so that [1A] ≤m[1B] for some m∈N. It follows by Proposition 2.1 of [35] that there exists an n∈Nfor which (n+ 1)[1A] ≤n[1B]. Almost unperforation then yields [1A] ≤[1B”