On the stable rank of simple C*-algebras

Villadsen, Jesper

doi:10.1090/S0894-0347-99-00314-8

On the stable rank of simple C*-algebras

Author: Jesper Villadsen
Journal: J. Amer. Math. Soc. 12 (1999), 1091-1102
MSC (1991): Primary 46L05; Secondary 46L80, 46M20
DOI: https://doi.org/10.1090/S0894-0347-99-00314-8
Published electronically: June 15, 1999
MathSciNet review: 1691013
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Abstract: It is shown that there exists a simple, finite C*-algebra whose topological (or Bass) stable rank is any given natural number or $\infty$.

Edwin J. Beggs and David E. Evans, The real rank of algebras of matrix valued functions, Internat. J. Math. 2 (1991), no. 2, 131–138. MR 1094700, DOI https://doi.org/10.1142/S0129167X91000089
Bruce Blackadar, A stable cancellation theorem for simple $C^{\ast } $-algebras, Proc. London Math. Soc. (3) 47 (1983), no. 2, 303–305. Appendix to: “The cancellation theorem for projective modules over irrational rotation $C^{\ast } $-algebras” [Proc. London Math. Soc. (3) 47 (1983), no. 2, 285–302] by M. A. Rieffel. MR 703982, DOI https://doi.org/10.1112/plms/s3-47.2.303
Bruce Blackadar, Comparison theory for simple $C^*$-algebras, Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press, Cambridge, 1988, pp. 21–54. MR 996438
Bruce Blackadar, Marius Dădărlat, and Mikael Rørdam, The real rank of inductive limit $C^*$-algebras, Math. Scand. 69 (1991), no. 2, 211–216 (1992). MR 1156427, DOI https://doi.org/10.7146/math.scand.a-12379
Bruce Blackadar, Alexander Kumjian, and Mikael Rørdam, Approximately central matrix units and the structure of noncommutative tori, $K$-Theory 6 (1992), no. 3, 267–284. MR 1189278, DOI https://doi.org/10.1007/BF00961466
Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI https://doi.org/10.1016/0022-1236%2891%2990056-B
Joachim Cuntz, $K$-theory for certain $C^{\ast } $-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. MR 604046, DOI https://doi.org/10.2307/1971137
Marius Dădărlat, Gabriel Nagy, András Némethi, and Cornel Pasnicu, Reduction of topological stable rank in inductive limits of $C^*$-algebras, Pacific J. Math. 153 (1992), no. 2, 267–276. MR 1151561
A. Dold, Lectures on algebraic topology, Springer-Verlag, New York-Berlin, 1972 (German). Die Grundlehren der mathematischen Wissenschaften, Band 200. MR 0415602
J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233–280. MR 164248, DOI https://doi.org/10.1007/BF02545788
Victor Guillemin and Alan Pollack, Differential topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0348781
Richard H. Herman and Leonid N. Vaserstein, The stable range of $C^{\ast } $-algebras, Invent. Math. 77 (1984), no. 3, 553–555. MR 759256, DOI https://doi.org/10.1007/BF01388839
Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247
John W. Milnor and James D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. MR 0440554
Victor Nistor, Stable rank for a certain class of type ${\rm I}$ $C^\ast $-algebras, J. Operator Theory 17 (1987), no. 2, 365–373. MR 887231
Marc A. Rieffel, Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. MR 693043, DOI https://doi.org/10.1112/plms/s3-46.2.301
Marc A. Rieffel, The cancellation theorem for projective modules over irrational rotation $C^{\ast } $-algebras, Proc. London Math. Soc. (3) 47 (1983), no. 2, 285–302. MR 703981, DOI https://doi.org/10.1112/plms/s3-47.2.285
Marc A. Rieffel, The homotopy groups of the unitary groups of noncommutative tori, J. Operator Theory 17 (1987), no. 2, 237–254. MR 887221
Mikael Rørdam, On the structure of simple $C^*$-algebras tensored with a UHF-algebra, J. Funct. Anal. 100 (1991), no. 1, 1–17. MR 1124289, DOI https://doi.org/10.1016/0022-1236%2891%2990098-P

Villadsen, J., Simple C*-algebras with perforation, J. Funct. Anal., 154, (1998), pp. 110–116.