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The connected stable rank
of the purely infinite simple $C^*$-algebras


Author: Yifeng Xue
Journal: Proc. Amer. Math. Soc. 127 (1999), 3671-3676
MSC (1991): Primary 46L05
DOI: https://doi.org/10.1090/S0002-9939-99-05397-6
Published electronically: July 12, 1999
MathSciNet review: 1670439
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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $\mathcal A$ is a unital purely infinite simple $C^*$-algebra. If the class [1] of the unit 1 in $K_0(\mathcal A)$ has torsion, then $\operatorname{csr}(\mathcal A)=\infty$; if [1] is torsion-free in $K_0(\mathcal A)$, then $\operatorname{csr}(\mathcal A)=2$. If $\mathcal A$ is a non-unital purely infinite simple $C^*$-algebra, then $\operatorname{csr}(\mathcal A)=2$.


References [Enhancements On Off] (What's this?)

  • [Bk] B. Blackadar, K-Theory for Operator Algebras, MSRI Publications No. 5, Springer-Verlag/New York/Berlin/Heidelberg/London/Paris/Tokyo, 1986. MR 88g:46082
  • [BP] L. G. Brown and G. K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), 131-149. MR 92m:46086
  • [Cu] J. Cuntz, K-Theory for Certain $C^*$-Algebras, Ann. of Math. 113 (1981), 181-197. MR 84c:46058
  • [Lin] H. Lin, Approximation by normal elements with finite spectra in $C^*$-algebras of real rank zero, Pacific J. Math. 173 (1995), 443-489. MR 98h:46059
  • [LZ] H. Lin and S. Zhang, On infinite simple $C^*$-algebras, J. Funct. Anal. 100 (1991), 221-231. MR 92m:46088
  • [Ni] V. Nistor, Stable range for tension products of extensions of $\mathcal K$ by $C(X)$, J. Operator Theory 16 (1986), 387-396. MR 88b:46085
  • [Rf] M. A. Rieffel, Dimensional and stable rank in the $K$-theory of $C^*$-Algebras, Proc. London Math. Soc. 46, no. (3) (1983), 301-333. MR 84g:46085
  • [Sr] H. Schröder, On the homotopy type of the regular group of a $W^*$-algebra, Math. Ann. 167 (1992), 171-277.
  • [Zh] S. Zhang, $C^*$-algebras with real rank zero and their multiplier algebras, I, Pacific J. Math. 155 (1992), 169-197.

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Additional Information

Yifeng Xue
Affiliation: Institute of Fundamental Education, East China University of Science and Technology, Shanghai 200237, People’s Republic of China
Address at time of publication: Department of Mathematics, East China University of Science and Technology, Shanghai 200237, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-99-05397-6
Keywords: Purely infinite simple $C^*$-algebras, connected stable rank, $K$-group of the $C^*$-algebras.
Received by editor(s): August 1, 1997
Received by editor(s) in revised form: February 24, 1998
Published electronically: July 12, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society

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