Quasidiagonal Extensions of the Reduced Group C*-algebras of Certain Discrete Groups

  • Alexander Kaplan
  • Steen Pedersen
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)


Let G be countable group containing a free subgroup F of finite index. We show that the reduced group C *-algebra C red * (G) has a quasidiagonal extension. Our proof is based on a result of Haagerup and Thorbjørnsen [HT]_asserting the existence of such an extension of C red * (F) when F is a free group of rank greater than one. A consequence of our result is that if G is a free product of finitely many (non-trivial) cyclic groups and Γ≠Z 2*ℤ2 Z2, then Ext(C red *(Γ)) is not a group.


Quasi diagonal extension reduced group C*-algebra free product of cyclic groups 

Mathematics Subject Classification (2000)

Primary 46L05 Secondary 46L45 46L35 


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  1. [Ar]
    W. Arveson, Notes on extensions of C *-algebras, Duke Math. J. 44 (1977), 329–355.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [Bl]
    B. Blackadar, Operator Algebras: Theory of C *-algebras and von Neumann Algebras, Encyclopaedia of Mathematical Sciences Volume 122, Operator Algebras and Non-Commutative Geometry III, Springer, 2006.Google Scholar
  3. [BDF]
    L.G. Brown, R.G. Douglas and P.A. Fillmore, Extensions of C *-algebra and Khomology, Annals Math. 105 (1977), 265–324.CrossRefMathSciNetGoogle Scholar
  4. [Br]
    N. Brown, On quasidiagonal C *-algebras, Advanced Studies in Pure Mathematics Volume 38, Operator algebras and applications, pp. 19–64, Math. Soc. Japan, Tokyo 2004.Google Scholar
  5. [Ch]
    M.D. Choi, A simple C *-algebra generated by two finite-order unitaries, Canad. J. Math. 31 (1979), 867–880.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [EH]
    E.G. Effros and U. Haagerup, Lifting problems and local reflexivity of C *-algebras, Duke Math. J. 52 (1985), 103–128.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [HT]
    U. Haagerup and S. Thorbjørnsen, A new application of random matrices: Ext(C red *(F 2)) is not a group, Annals Math. 162 (2005), 711–775.zbMATHCrossRefGoogle Scholar
  8. [Ki]
    E. Kirchberg, On subalgebras of the CAR-algebra, J. Funct. Anal. 129 (1995), 35–63.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [Pe]
    G.K. Pedersen, C *-algebras and their automorphism groups, Academic Press, 1979.Google Scholar
  10. [Rie]
    M.A. Rieffel, Induced representations of C *-algebras, Adv. in Math. 13 (1974), 176–257.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Ro]
    J. Rosenberg, Appendix to “Strongly quasidiagonal C *-algebras” by D. Hadwin, Journ. Operator Theory 18 (1987), 3–18.zbMATHGoogle Scholar
  12. [Vo]
    D. Voiculescu, Around quasidiagonal operators, Integral Equat. Operator Theory 17 (1993), 137–149.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Wa1]
    S. Wassermann, C *-algebras associated with groups with Kazhdan’s property T, Annals. Math. 134 (1991), 423–431.CrossRefMathSciNetGoogle Scholar
  14. [Wa2]
    S. Wassermann, A separable quasidiagonal C *-algebra with a nonquasidiagonal quotient by the compact operators, Math. Proc. Cambridge Philos. Soc. 110 (1991), 143–145.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [Wat]
    Y. Watatani, Index for C *-subalgebras, Memoirs Amer. Math. Soc. 424, 1990.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Alexander Kaplan
    • 1
  • Steen Pedersen
    • 1
  1. 1.Department of MathematicsWright State UniversityDaytonUSA

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