TY - JOUR

T1 - An algebraic approach to the radius of comparison

AU - Blackadar, Bruce

AU - Robert, Leonel

AU - Tikuisis, Aaron P.

AU - Toms, Andrew S.

AU - Winter, Wilhelm

PY - 2012/7

Y1 - 2012/7

N2 - The radius of comparison is an invariant for unital C*-algebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general C*-algebras, and give an algebraic (as opposed to functional-theoretic) reformulation. This yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces. We then give several applications of these results. New examples of C*-algebras with finite radius of comparison are given, and the question of when the Cuntz classes of finitely generated Hilbert modules form a hereditary subset of the Cuntz semigroup is addressed. Most interestingly, perhaps, we treat the question of when a full hereditary subalgebra B of a stable C*-algebra A is itself stable, giving a characterization in terms of the radius of comparison. We also use the radius of comparison to quantify the least n for which a C*-algebra D without bounded 2-quasitraces or unital quotients has the property that M(D) is stable.

AB - The radius of comparison is an invariant for unital C*-algebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general C*-algebras, and give an algebraic (as opposed to functional-theoretic) reformulation. This yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces. We then give several applications of these results. New examples of C*-algebras with finite radius of comparison are given, and the question of when the Cuntz classes of finitely generated Hilbert modules form a hereditary subset of the Cuntz semigroup is addressed. Most interestingly, perhaps, we treat the question of when a full hereditary subalgebra B of a stable C*-algebra A is itself stable, giving a characterization in terms of the radius of comparison. We also use the radius of comparison to quantify the least n for which a C*-algebra D without bounded 2-quasitraces or unital quotients has the property that M(D) is stable.

UR - http://www.scopus.com/inward/record.url?scp=84859091041&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2012-05538-3

DO - 10.1090/S0002-9947-2012-05538-3

M3 - Article

AN - SCOPUS:84859091041

SN - 0002-9947

VL - 364

SP - 3657

EP - 3674

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 7

ER -