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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)



An algebraic approach to the radius of comparison

Authors: Bruce Blackadar, Leonel Robert, Aaron P. Tikuisis, Andrew S. Toms and Wilhelm Winter
Journal: Trans. Amer. Math. Soc. 364 (2012), 3657-3674
MSC (2010): Primary 46L80
Published electronically: February 21, 2012
MathSciNet review: 2901228
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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.

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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

Leonel Robert
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada L4J 3A4

Aaron P. Tikuisis
Affiliation: Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4

Andrew S. Toms
Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067

Wilhelm Winter
Affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom

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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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