The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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REAL RANK OF $C^*$-ALGEBRAS OF TYPE I

  • Sudo, Takahiro (Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus)
  • Received : 2010.06.15
  • Accepted : 2010.11.25
  • Published : 2010.11.30
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Abstract

We estimate the real rank of a composition series of closed ideals of a $C^*$-algebra such that its subquotients have continuous trace, which is equivalent to that the $C^*$-algebra is of type I.

Keywords

References

  1. E. J. Beggs & D. E. Evans: The real rank of algebras of matrix valued functions. Internat. J. Math. 2 (1991), 131-138. https://doi.org/10.1142/S0129167X91000089
  2. L. G. Brown and G. K. Pedersen: $C^*$-algebras of real rank zero. J. Funct. Anal. 99 (1991), 131-149. https://doi.org/10.1016/0022-1236(91)90056-B
  3. J. Dixmier: $C^*$-algebras. North-Holland, 1977.
  4. N. Elhage Hassan: Rang reel de certaines extensions. Proc. Amer. Math. Soc. 123 (1995), 3067-3073.
  5. V. Nistor: Stable range for tensor products of extensions of $\kappa$ by C(X). J. Operator Theory 16 (1986), 387-396.
  6. V. Nistor: Stable rank for a certain class of type I $C^*$-algebras. J. Operator Theory 17 (1987), 365-373.
  7. G. K. Pedersen: $C^*$-Algebras and their Automorphism Groups, Academic Press, 1979.
  8. I. Raeburn & D.P. Williams: Morita Equivalence and Continuous-Trace $C^*$-algebras. SURV 60, AMS (1998).
  9. M.A. Rieffel: Dimension arid stable rank in the K-theory of $C^*$-algebras. Proc. London Math. Soc. 46 (1983), 301-333. https://doi.org/10.1112/plms/s3-46.2.301
  10. M.A. Rieffel: The homotopy groups of the unitary groups of non-commutative tori. J. Operator Theory 17 (1987), 237-254.
  11. A.J-L. Sheu: A cancellation theorem for projective modules over the group $C^*$-algebras of certain nilpotent Lie groups. Canad. J. Math. 39 (1987), 365-427. https://doi.org/10.4153/CJM-1987-018-7
  12. T. Sudo: Stable rank of $C^*$-algebras of type I. Linear Alg. Appl. 383 (2004), 65-76. https://doi.org/10.1016/j.laa.2003.10.010
  13. T. Sudo: The real rank of CCR $C^*$-algebras. Kyungpook Math. J. 48 (2008), 223-232. https://doi.org/10.5666/KMJ.2008.48.2.223
  14. T. Sudo: Real and stable ranks for semigroup crossed products of Toeplitz algebras. to appear.