Advertisement

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)

Abstract

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.

Keywords

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

Mathematics Subject Classification (2000)

Primary 46L05 Secondary 46L45 46L35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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

Personalised recommendations