An algebraic approach to the radius of comparison

Bruce Blackadar, Leonel Robert, Aaron P. Tikuisis, Andrew S. Toms, Wilhelm Winter

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18 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)3657-3674
Number of pages18
JournalTransactions of the American Mathematical Society
Volume364
Issue number7
Early online date21 Feb 2012
DOIs
Publication statusPublished - Jul 2012

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ASJC Scopus subject areas

  • Analysis

Cite this

Blackadar, B., Robert, L., Tikuisis, A. P., Toms, A. S., & Winter, W. (2012). An algebraic approach to the radius of comparison. Transactions of the American Mathematical Society, 364(7), 3657-3674. https://doi.org/10.1090/S0002-9947-2012-05538-3