An algebraic approach to the radius of comparison
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- by Bruce Blackadar, Leonel Robert, Aaron P. Tikuisis, Andrew S. Toms and Wilhelm Winter PDF
- Trans. Amer. Math. Soc. 364 (2012), 3657-3674 Request permission
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 $\mathrm {M}_n(D)$ is stable.References
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Additional Information
- Bruce Blackadar
- Affiliation: Department of Mathematics and Statistics, University of Nevada, Ansari Business Building, 601 – Mail Stop 084, Reno, Nevada 89557-0084
- Email: bruceb@unr.edu
- Leonel Robert
- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada L4J 3A4
- MR Author ID: 716339
- Email: leonel.robert@gmail.com
- Aaron P. Tikuisis
- Affiliation: Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
- MR Author ID: 924851
- Email: aptikuis@math.toronto.edu
- Andrew S. Toms
- Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067
- Email: atoms@purdue.edu
- Wilhelm Winter
- Affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
- MR Author ID: 671014
- Email: wilhelm.winter@nottingham.ac.uk
- Received by editor(s): August 20, 2010
- Published electronically: February 21, 2012
- Additional Notes: The second author was supported by an NSERC CGS-D scholarship
The third author was supported by NSF grant DMS-0969246
The fourth author was supported by EPSRC First Grant EP/G014019/1 - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 3657-3674
- MSC (2010): Primary 46L80
- DOI: https://doi.org/10.1090/S0002-9947-2012-05538-3
- MathSciNet review: 2901228