Israel Journal of Mathematics

, Volume 56, Issue 3, pp 267–279 | Cite as

Crossed products of type I af algebras by abelian groups

  • Elliot C. Gootman
  • Aldo J. Lazar
Article
Received:

Abstract

Let (G, A, α) be a separableC*-dynamical system, withG abelian, and let Γ denote the dual group ofG. We characterize the Γ-invariant ideals of the crossed product algebraG×∩A, and use this characterization to prove that if in additionG is compact andA is type I AF, thenG×∩A is AF also. Finally, assumingG is discrete abelian and bothA andG×∩A are type I. we determine necessary and sufficient conditions, in terms ofA and the isotropy subgroups for the action ofG onÂ, forG×∩A to be AF.

Keywords

Irreducible Representation Dual Group Compact Abelian Group Primitive Ideal Cross Product Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Baggett and A. Kleppner,Multiplier representations of abelian groups, J. Funct. Anal.14 (1973), 299–324.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    O. Bratteli,Crossed products of UHF algebras by product type actions, Duke Math. J.46 (1979), 1–23.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    O. Bratteli and G. A. Elliott,Structure spaces of approximately finite-dimensional C*-algebras. II, J. Funct. Anal.30 (1978), 74–82.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    O. Bratteli, G. A. Elliott and R. H. Herman,On the possible temperatures of a dynamical system, Commun. Math. Phys.74 (1980), 281–295.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    L. G. Brown,Extensions of AF algebras: the projection lifting problem, Proc. Symp. Pure Math.38 (1982), Part 1, 175–176.Google Scholar
  6. 6.
    J. Dixmier,C*-algebras, North-Holland Mathematical Library, Volume 15, Amsterdam, 1977.Google Scholar
  7. 7.
    E. G. Effros,On the structure theory of C*-algebras: some old and new problems, Proc. Symp. Pure Math.38 (1982), Part 1, 19–34.MathSciNetGoogle Scholar
  8. 8.
    G. A. Elliott,Automorphisms determined by multipliers on ideals of a C*-algebra, J. Funct. Anal.23 (1976), 1–10.MATHCrossRefGoogle Scholar
  9. 9.
    J. M. G. Fell,Weak containment and induced representations of groups, Can. J. Math.14 (1962), 237–268.MATHMathSciNetGoogle Scholar
  10. 10.
    J. Glimm,Locally compact transformation groups, Trans. Am. Math. Soc.101 (1961), 124–138.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    E. C. Gootman,Abelian group actions on type I C*-algebras, Lecture Notes in Mathematics No. 1132, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1985, pp. 152–169.Google Scholar
  12. 12.
    P. Green,The local structure of twisted covariance algebras, Acta Math.140 (1978), 191–250.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    P. Green,The structure of imprimitivity algebras, J. Funct. Anal.36 (1980), 88–104.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    R. Y. Lee,On the C*-algebras of operator fields, Indiana Univ. Math. J.25 (1976), 303–314.CrossRefMathSciNetGoogle Scholar
  15. 15.
    D. Olesen,A classification of ideals in crossed products, Math. Scand.45 (1979), 157–167.MATHMathSciNetGoogle Scholar
  16. 16.
    D. Olesen and G. K. Pedersen,Applications of the Connes spectrum to C*-dynamical systems, J. Funct. Anal.30 (1978), 179–197.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    G. K. Pedersen,C*-algebras and their Automorphism Groups, London Math. Soc. Monographs No. 14, Academic Press, London, 1979.MATHGoogle Scholar
  18. 18.
    N. C. Phillips,K-Theoretic Freeness of Actions of Finite Groups on C*-algebras, Lecture Notes in Math., Springer-Verlag, New York, to appear.Google Scholar
  19. 19.
    J. Phillips and I. Raeburn,Crossed products by locally unitary automorphism groups and principal bundles, J. Operator Theory11 (1984), 215–241.MATHMathSciNetGoogle Scholar
  20. 20.
    J. Rosenberg and C. Schochet,The Künneth theorem and the universal coefficient theorem for equivariant K-theory and KK-theory, MSRI preprint 09111-85.Google Scholar
  21. 21.
    D. P. Williams,The topology on the primitive ideal space of transformation group C*-algebras and C. C. R. transformation group C*-algebras, Trans. Am. Math. Soc.266 (1981), 335–359.MATHCrossRefGoogle Scholar

Copyright information

© Hebrew University 1986

Authors and Affiliations

  • Elliot C. Gootman
    • 1
  • Aldo J. Lazar
    • 1
  1. 1.School of Mathematical SciencesTel Aviv UniversityRamat AvivIsrael

Personalised recommendations