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), 36573674
MSC (2010):
Primary 46L80
Published electronically:
February 21, 2012
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Abstract: The radius of comparison is an invariant for unital Calgebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general Calgebras, and give an algebraic (as opposed to functionaltheoretic) 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 Calgebras 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 Calgebra 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 Calgebra without bounded 2quasitraces or unital quotients has the property that is stable.
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 1.
 Pere Ara, Francesc Perera, and Andrew S. Toms, Ktheory for operator algebras. Classification of Calgebras, Commun. Contemp. Math., To appear.
 2.
 Bruce Blackadar, Matricial and ultramatricial topology, Operator algebras, mathematical physics, and lowdimensional topology (Istanbul, 1991), Res. Notes Math., vol. 5, A K Peters, Wellesley, MA, 1993, pp. 1138. MR 1259056 (95i:46102)
 3.
 Bruce Blackadar and David Handelman, Dimension functions and traces on Calgebras, J. Funct. Anal. 45 (1982), no. 3, 297340. MR 650185 (83g:46050)
 4.
 Bruce Blackadar and Mikael Rørdam, Extending states on preordered semigroups and the existence of quasitraces on Calgebras, J. Algebra 152 (1992), no. 1, 240247. MR 1190414 (93k:46049)
 5.
 Kristofer T. Coward, George A. Elliott, and Cristian Ivanescu, The Cuntz semigroup as an invariant for Calgebras, J. Reine Angew. Math. 623 (2008), 161193. MR 2458043
 6.
 Maurice J. Dupré, Classifying Hilbert bundles. II, J. Functional Analysis 22 (1976), no. 3, 295322. MR 0415346 (54:3435)
 7.
 George A. Elliott, Leonel Robert, and Luis Santiago, The cone of lower semicontinuous traces on a Calgebra, Amer. J. Math., To appear.
 8.
 Julien Giol and David Kerr, Subshifts and perforation, J. Reine Angew. Math. 639 (2010), 107119. MR 2608192
 9.
 I. Hirshberg, M. Rørdam, and W. Winter, algebras, stability and strongly selfabsorbing algebras, Math. Ann. 339 (2007), no. 3, 695732. MR 2336064 (2008j:46040)
 10.
 D. Kucerovsky and P. W. Ng, regularity and the corona factorization property, Math. Scand. 99 (2006), no. 2, 204216. MR 2289022 (2009g:46103)
 11.
 , A simple Calgebra with perforation and the corona factorization property, J. Operator Theory 61 (2009), no. 2, 227238. MR 2501002 (2010c:46127)
 12.
 Ping Wong Ng and Wilhelm Winter, A note on subhomogeneous Calgebras, C. R. Math. Acad. Sci. Soc. R. Can. 28 (2006), no. 3, 9196. MR 2310490 (2008a:46058)
 13.
 Eduard Ortega, Francesc Perera, and Mikael Rørdam, The Corona Factorization Property, stability, and the Cuntz semigroup of a Calgebra, 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, 45954623 (electronic). MR 2320643 (2009a:46133)
 15.
 Mikael Rørdam, On the structure of simple Calgebras tensored with a UHFalgebra. II, J. Funct. Anal. 107 (1992), no. 2, 255269. MR 1172023 (93f:46094)
 16.
 , Stability of Calgebras is not a stable property, Doc. Math. 2 (1997), 375386 (electronic). MR 1490456 (98i:46060)
 17.
 , The stable and the real rank of absorbing Calgebras, Internat. J. Math. 15 (2004), no. 10, 10651084. MR 2106263 (2005k:46164)
 18.
 Andrew S. Toms, Flat dimension growth for Calgebras, J. Funct. Anal. 238 (2006), no. 2, 678708. MR 2253738 (2007j:46098)
 19.
 , Stability in the Cuntz semigroup of a commutative Calgebra, Proc. Lond. Math. Soc. (3) 96 (2008), no. 1, 125. MR 2392313
 20.
 , Comparison theory and smooth minimal Cdynamics, Comm. Math. Phys. 289 (2009), no. 2, 401433. MR 2506758
 21.
 Jacob v. B. Hjelmborg and Mikael Rørdam, On stability of Calgebras, J. Funct. Anal. 155 (1998), no. 1, 153170. MR 1623142 (99g:46079)
 22.
 Wilhelm Winter, Nuclear dimension and stability of perfect Calgebras, 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 895570084
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 479072067
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:
http://dx.doi.org/10.1090/S000299472012055383
PII:
S 00029947(2012)055383
Received by editor(s):
August 20, 2010
Published electronically:
February 21, 2012
Additional Notes:
The second author was supported by an NSERC CGSD scholarship
The third author was supported by NSF grant DMS0969246
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.
