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 of a stable C-algebra is itself stable, giving a characterization in terms of the radius of comparison. We also use the radius of comparison to quantify the least for which a C-algebra without bounded 2-quasitraces or unital quotients has the property that 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

Email:
bruceb@unr.edu

**Leonel Robert**

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

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

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

Email:
wilhelm.winter@nottingham.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-2012-05538-3

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.