Chinese Annals of Mathematics, Series B

, Volume 40, Issue 1, pp 97–110 | Cite as

Some Properties of Tracially Quasidiagonal Extensions

  • Yile Zhao
  • Xiaochun FangEmail author
  • Xiaoming Xu


Suppose that 0 → IAA/I → 0 is a tracially quasidiagonal extension of C* -algebras. In this paper, the authors give two descriptions of the K0, K1 index maps which are induced by the above extension and show that for any ϵ > 0, any τ in the tracial state space of A/I and any projection \(\bar p \in A/I\) (any unitary \(\bar u \in A/I\), there exists a projection pA (a unitary uA) such that \(|\tau (\bar p) = \tau (\pi (p))| < \in (|\tau (\bar u) = \tau (\pi (u))| < \in )\).


Tracially topological rank Quasidiagonal extension Tracially quasidiagonal extension 

2000 MR Subject Classification

46L05 46L35 


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  1. [1]
    Brown, L. G., Douglas, R. G. and Fillmore, P. A., Unitary equivalence modulo the compact operators and extensions of C*–algebras, Lecture Notes in Math., 345, 1973, 58–128.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Busby, R. C., Double centralizers and extensions of C*–algebras, Trans. Amer. Soc., 132, 1968, 79–99.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Fang, X. and Zhao, Y., Approximately isometric lifting in quasidiagonal extensions, Science in China, 52(3), 2009, 457–467.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Fang, X. and Zhao, Y., The extensions of C*–algebras with tracial topological rank no more than one, Illinois J. Math., 53(2), 2009, 441–462.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Fang, X. and Zhao, Y., The tracial topological rank of certain C*–algebras, Science in China, 54, 2011, 2295–2307.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Hu, S., Lin, H. and Xue, Y., The tracial topological rank of extensions of C*–algebras, Math. Scand., 94, 2004, 125–147.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Lin, H., The tracial topological rank of C*–algebras, Proc. London Math. Soc., 83, 2001, 199–234.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Lin, H., An Introduction to the Classification of Amenable C*–Algebras, World Scientific, Singapore, 2001.CrossRefzbMATHGoogle Scholar
  9. [9]
    Lin, H., Tracially quasidiagonal extensions, Canad. Math. Bull., 46, 2003, 388–399.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Lin, H., Classification of simple C*–algebras with tracial topological rank zero, Duke Math. J., 125, 2004, 91–119.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Lin, H., Simple nuclear C*–algebras of tracial topological rank one, J. Funct. Anal., 251(2), 2007, 601–679.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Lin, H. and Osaka, H., Tracially quasidiagonal extensions and topological stable rank, Illinois J. Math., 47, 2003, 921–937.MathSciNetzbMATHGoogle Scholar
  13. [13]
    Zhao, Y. and Fang, X., The tracial topological rank of extensions of C*–algebras, Complex Analysis and Operator Theory, 251, 2016, 1181–1201.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHangzhou Normal UniversityHangzhouChina
  2. 2.Department of MathematicsTongji UniversityShanghaiChina
  3. 3.School of ScienceShanghai Institute of TechnologyShanghaiChina

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