Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The real rank zero property of crossed product


Author: Xiaochun Fang
Journal: Proc. Amer. Math. Soc. 134 (2006), 3015-3024
MSC (2000): Primary 46L05; Secondary 46L35, 46L40
DOI: https://doi.org/10.1090/S0002-9939-06-08357-2
Published electronically: May 8, 2006
MathSciNet review: 2231627
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A$ be a unital $ C^*$-algebra, and let $ (A, G, \alpha)$ be a $ C^*$-dynamical system with $ G$ abelian and discrete. In this paper, we introduce the continuous affine map $ R$ from the trace state space $ T(A\times_{\alpha}G)$ of the crossed product $ A\times_{\alpha}G$ to the $ \alpha$-invariant trace state space $ T(A)_{\alpha^*}$ of $ A$. If $ A\times_{\alpha}G$ is of real rank zero and $ \hat{G}$ is connected, we have proved that $ R$ is homeomorphic. Conversely, if $ R$ is homeomorphic, we also get some properties and real rank zero characterization of $ A\times_{\alpha}G$. In particular, in that case, $ A\times_{\alpha}G$ is of real rank zero if and only if each unitary element in $ A\times_{\alpha}G$ with the form $ u_{_A}\prod_{i=1}^n x_i^*y_i^*x_iy_i$ can be approximated by the unitary elements in $ A\times_{\alpha}G$ with finite spectrum, where $ u_{_A}\in U_0(A)$, $ x_i,y_i\in C_c(G,A)\cap U_0(A\times_{\alpha}G)$, and if moreover $ A$ is a unital inductive limit of the direct sums of non-elementary simple $ C^*$-algebras of real rank zero, then the $ u_{_A}$ above can be cancelled.


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

  • 1. De la Harpe, P. and Skandalis, G., Dèterminant Associè á une Trace sur une Algebrè de Banach, Ann. Inst. Fourier (Grenoble), 34-1(1984), 241-260. MR 0743629 (87i:46146a)
  • 2. Blackadar, B., K-theory for Operator Algebras, Springer-Verlag, New York, 1986. MR 0859867 (88g:46082)
  • 3. Bratteli, O., Evans, D. E. and Kishimoto, A., The Rokhlin Property for Quasi-Free Automomorphism of the Fermion Algebra, Proc. London Math. Soc., 71(1995), 675-694. MR 1347409 (97g:46083)
  • 4. Brown, P. L. and Pedersen, G. K., $ C^*$-algebras of Real Rank Zero, J. Funct. Anal. 99(1991), 131-149. MR 1120918 (92m:46086)
  • 5. Elliott, G. A., On the Classification of $ C^*$-algebras of Real Rank Zero, J. Reine Angew. Math., 443(1993), 179-219. MR 1241132 (94i:46074)
  • 6. Elliott, G. A., Gong, G. and Li, L., On the Classification of Simple Inductive Limit $ C^*$-algebras, II: The Isomorphism Theorem, preprint.
  • 7. Elliott, G. A. and Fang, X., Simple Inductive Limits of C*-algebras with Building Blocks from Spheres of Odd Dimension, Contemp. Math.(228), 'Operator Algebra and Operator Theory', 1998, 79-86. MR 1667655 (2000k:46077)
  • 8. Fang, X., The Simplicity and Real Rank Zero Property of the Inductive Limit of Continuous Trace C*-algebras, Analysis, 19 (1999), 377-389. MR 1743530 (2001k:46085)
  • 9. Kishimoto, A., The Rokhlin Property for Automorphism of the UHF Algebras, J. Reine Angew. Math., 465(1995), 183-196. MR 1344136 (96k:46114)
  • 10. Lin, H., Almost Multiplicative Morphisms and Some Applications, J. Operator Theory 37(1997), 121-154. MR 1438204 (98b:46091)
  • 11. Lin, H., An Introduction to the Classification of Amenable $ C^*$-algebras, World Scientific, Singapore, 2001. MR 1884366 (2002k:46141)
  • 12. Pedersen, G. K., $ C^*$-algebras and Their Automorphism Groups, Academic Press, London and New York, 1979. MR 0548006 (81e:46037)
  • 13. Rieffel, M. Dimension and Stable Rank in the K-theory of $ C^*$-algebras, Proc. London Math. Soc., 46(1983), 301-333. MR 0693043 (84g:46085)
  • 14. Rördam, M. Classification of Inductive Limits of Cuntz Algebras, J. Reine Angew. Math., 440(1993), 175-200. MR 1225963 (94k:46120)
  • 15. Thomsen, K., Diagonalization in Inductive Limits of Circle Algebras, J. Operator Theory, 27(1992), 325-340. MR 1249649 (95f:46098)
  • 16. Thomsen, K., Trace, Unitary Characters and Crossed Products by Z, Publ. RIMS. Kyoto Univ., 31(1995), 1011-1029. MR 1382564 (97a:46074)
  • 17. Villadsen, J., Simple $ C^*$-algebras with Perforation, J. Funct. Anal. 154(1998), 110-116. MR 1616504 (99j:46069)
  • 18. Zhang, S., Matricial Structure and Homotopy Type of Simple $ C^*$-algebras with Real Rank Zero, J. Operator Theory 26(1991), 283-312. MR 1225518 (94f:46075)
  • 19. Zhang, S., On the Homotopy Type of the Unitary Group and the Grassmann Space of Purely Infinite Simple $ C^*$-algebras, $ K$-Theory 24(2001), 203-225. MR 1876798 (2002m:46088)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L05, 46L35, 46L40

Retrieve articles in all journals with MSC (2000): 46L05, 46L35, 46L40


Additional Information

Xiaochun Fang
Affiliation: Department of Applied Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China
Email: xfang@mail.tongji.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-06-08357-2
Keywords: Real rank zero, crossed product, trace state space
Received by editor(s): January 3, 2005
Received by editor(s) in revised form: May 9, 2005
Published electronically: May 8, 2006
Additional Notes: This article was supported by the National Natural Science Foundation of China (10271090).
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society