Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

The Real Rank of CCR C*-Algebra

  • Sudo, Takahiro (Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus)
  • Received : 2006.04.17
  • Published : 2008.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

We estimate the real rank of CCR C*-algebras under some assumptions. A applications we determine the real rank of the reduced group C*-algebras of non-compac connected, semi-simple and reductive Lie groups and that of the group C*-algebras of connected nilpotent Lie groups.

Keywords

References

  1. C. A. Akemann, G. K. Pedersen and J. Tomiyama, Multipliers of $C^{*}$-algebras, J. Funct. Anal., 13(1973), 277-301. https://doi.org/10.1016/0022-1236(73)90036-0
  2. E. J. Beggs and 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
  3. 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
  4. J. Dixmier, $C^{*}$-algebras, North-Holland, Amsterdam, 1977.
  5. N. Elhage Hassan, Rang reel de certaines extensions, Proc. Amer. Math. Soc., 123(1995), 3067-3073.
  6. J. Hjelmborg and M. Rørdam, On stability of $C^{*}$-algebras, J. Funct. Anal., 155(1998), 153-170. https://doi.org/10.1006/jfan.1997.3221
  7. E. Kaniuth, Group $C^{*}$-algebras of real rank zero or one, Proc. Amer. Math. Soc., 119(1993), 1347-1354.
  8. K. Nagami, Dimension Theory, Academic Press, New York-London, 1970.
  9. M. Nagisa, H. Osaka and N.C. Phillips, Ranks of algebras of continuous $C^{*}$-algebra valued functions, Canad. J. Math., 53(2001), 979-1030. https://doi.org/10.4153/CJM-2001-039-8
  10. V. Nistor, Stable rank for a certain class of type I $C^{*}$-algebras, J. Operator Theory, 17(1987), 365-373.
  11. G. K. Pedersen, $C^{*}$-Algebras and their Automorphism Groups, Academic Press, London-New York-San Francisco, 1979.
  12. M. A. Rieffel, Dimension and 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
  13. T. Sudo, Stable rank of the reduced $C^{*}$-algebras of non-amenable Lie groups of type I, Proc. Amer. Math. Soc., 125(1997), 3647-3654. https://doi.org/10.1090/S0002-9939-97-04034-3
  14. T. Sudo, Dimension theory of group $C^{*}$-algebras of connected Lie groups of type I, J. Math. Soc. Japan, 52(2000), 583-590. https://doi.org/10.2969/jmsj/05230583
  15. T. Sudo, Survey on the rank and structure theory of group $C^{*}$-algebras of Lie groups, Ryukyu Math. J., 13(2000), 79-95.
  16. T. Sudo, Stable rank of $C^{*}$-algebras of type I, Linear Algebra Appl., 383(2004), 65-76. https://doi.org/10.1016/j.laa.2003.10.010
  17. T. Sudo and H. Takai, Stable rank of the $C^{*}$-algebras of nilpotent Lie groups, Internat. J. Math., 6(1995), 439-446. https://doi.org/10.1142/S0129167X95000158
  18. T. Sudo and H. Takai, Stable rank of the $C^{*}$-algebras of solvable Lie groups of type I, J. Operator Theory, 38(1997), 67-86.
  19. N. E.Wegge-Olsen, K-theory and $C^{*}$-algebras, Oxford Univ. Press, Oxford-New York-Tokyo, 1993.