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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-2012-05538-3
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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  • 1. Pere Ara, Francesc Perera, and Andrew S. Toms, K-theory for operator algebras. Classification of C$ ^*$-algebras, Commun. Contemp. Math., To appear.
  • 2. Bruce Blackadar, Matricial and ultramatricial topology, Operator algebras, mathematical physics, and low-dimensional topology (Istanbul, 1991), Res. Notes Math., vol. 5, A K Peters, Wellesley, MA, 1993, pp. 11-38. MR 1259056 (95i:46102)
  • 3. Bruce Blackadar and David Handelman, Dimension functions and traces on C$ ^*$-algebras, J. Funct. Anal. 45 (1982), no. 3, 297-340. MR 650185 (83g:46050)
  • 4. Bruce Blackadar and Mikael Rørdam, Extending states on preordered semigroups and the existence of quasitraces on C$ ^*$-algebras, J. Algebra 152 (1992), no. 1, 240-247. MR 1190414 (93k:46049)
  • 5. Kristofer T. Coward, George A. Elliott, and Cristian Ivanescu, The Cuntz semigroup as an invariant for C$ ^*$-algebras, J. Reine Angew. Math. 623 (2008), 161-193. MR 2458043
  • 6. Maurice J. Dupré, Classifying Hilbert bundles. II, J. Functional Analysis 22 (1976), no. 3, 295-322. MR 0415346 (54:3435)
  • 7. George A. Elliott, Leonel Robert, and Luis Santiago, The cone of lower semicontinuous traces on a C$ ^*$-algebra, Amer. J. Math., To appear.
  • 8. Julien Giol and David Kerr, Subshifts and perforation, J. Reine Angew. Math. 639 (2010), 107-119. MR 2608192
  • 9. I. Hirshberg, M. Rørdam, and W. Winter, $ \mathcal {C}_0(X)$-algebras, stability and strongly self-absorbing $ C^*$-algebras, Math. Ann. 339 (2007), no. 3, 695-732. MR 2336064 (2008j:46040)
  • 10. D. Kucerovsky and P. W. Ng, $ S$-regularity and the corona factorization property, Math. Scand. 99 (2006), no. 2, 204-216. MR 2289022 (2009g:46103)
  • 11. -, A simple C$ ^*$-algebra with perforation and the corona factorization property, J. Operator Theory 61 (2009), no. 2, 227-238. MR 2501002 (2010c:46127)
  • 12. Ping Wong Ng and Wilhelm Winter, A note on subhomogeneous C$ ^*$-algebras, C. R. Math. Acad. Sci. Soc. R. Can. 28 (2006), no. 3, 91-96. MR 2310490 (2008a:46058)
  • 13. Eduard Ortega, Francesc Perera, and Mikael Rørdam, The Corona Factorization Property, stability, and the Cuntz semigroup of a C$ ^*$-algebra, arXiv preprint. http://arxiv.org/abs/0903.2917, March 2009.
  • 14. N. Christopher Phillips, Recursive subhomogeneous algebras, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4595-4623 (electronic). MR 2320643 (2009a:46133)
  • 15. Mikael Rørdam, On the structure of simple C$ ^*$-algebras tensored with a UHF-algebra. II, J. Funct. Anal. 107 (1992), no. 2, 255-269. MR 1172023 (93f:46094)
  • 16. -, Stability of C$ ^*$-algebras is not a stable property, Doc. Math. 2 (1997), 375-386 (electronic). MR 1490456 (98i:46060)
  • 17. -, The stable and the real rank of $ Z$-absorbing C$ ^*$-algebras, Internat. J. Math. 15 (2004), no. 10, 1065-1084. MR 2106263 (2005k:46164)
  • 18. Andrew S. Toms, Flat dimension growth for C$ ^*$-algebras, J. Funct. Anal. 238 (2006), no. 2, 678-708. MR 2253738 (2007j:46098)
  • 19. -, Stability in the Cuntz semigroup of a commutative C$ ^*$-algebra, Proc. Lond. Math. Soc. (3) 96 (2008), no. 1, 1-25. MR 2392313
  • 20. -, Comparison theory and smooth minimal C$ ^*$-dynamics, Comm. Math. Phys. 289 (2009), no. 2, 401-433. MR 2506758
  • 21. Jacob v. B. Hjelmborg and Mikael Rørdam, On stability of C$ ^*$-algebras, J. Funct. Anal. 155 (1998), no. 1, 153-170. MR 1623142 (99g:46079)
  • 22. Wilhelm Winter, Nuclear dimension and $ \mathcal {Z}$-stability of perfect C$ ^*$-algebras, arXiv preprint. http://www.arxiv.org/abs/1006.2731, June 2010.

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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.

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