Group Bundle Duality, Invariants for Certain C*-algebras, and Twisted Equivariant K-theory

  • Ezio Vasselli
Part of the Trends in Mathematics book series (TM)


A duality is discussed for Lie group bundles vs. certain tensor C*-categories with non-simple identity, in the setting of Nistor-Troitsky gaugeequivariant K-theory. As an application, we study C*-algebra bundles with fibre a fixed-point algebra of the Cuntz algebra: a classification is given, and a cohomological invariant is assigned, representing the obstruction to perform an embedding into a continuous bundle of Cuntz algebras. Finally, we introduce the notion of twisted equivariant K-theory.


Vector Bundle Compact Group Tensor Category Continuous Section Hilbert Module 
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.


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  1. [1]
    M.F. Atiyah, G. Segal: Twisted K-Theory, arXiv:math.KT/0407054 (2004).Google Scholar
  2. [2]
    M.F. Atiyah: K-Theory, Benjamin, New York, 1967.Google Scholar
  3. [3]
    B. Blackadar: K-Theory of Operator Algebras, MSRI Publications, 1995.Google Scholar
  4. [4]
    E. Blanchard: Déformations de C*-algèbres de Hopf, Bull. Soc. math. France 124 (1996), 141–215.zbMATHMathSciNetGoogle Scholar
  5. [5]
    J. Cuntz: Simple C*-algebras Generated by Isometries, Comm. Math. Phys. 57 (1977), 173–185.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    S. Doplicher, J.E. Roberts: Duals of Compact Lie Groups Realized in the Cuntz Algebras and Their Actions on C*-Algebras, J. Funct. Anal. 74 (1987) 96–120.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    S. Doplicher, J.E. Roberts: A New Duality Theory for Compact Groups, Inv. Math. 98 (1989), 157–218.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. Gierz: Bundles of topological vector spaces and their duality, Lecture Notes in Mathematics, 955, Springer-Verlag, 1982.Google Scholar
  9. [9]
    D. Husemoller: Fibre Bundles, Mc Graw-Hill Series in Mathematics, 1966.Google Scholar
  10. [10]
    M. Karoubi: K-Theory, Springer Verlag, Berlin-Heidelberg-New York, 1978.zbMATHGoogle Scholar
  11. [11]
    M. Nilsen: C*-Bundles and C 0(X)-algebras, Indiana Univ. Math. J. 45 (1996), 463–477.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    V. Nistor, E. Troitsky: An index for gauge-invariant operators and the Dixmier-Douady invariant, Trans. AMS. 356 (2004), 185–218zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Pimsner: A Class of C*-algebras Generalizing both Cuntz-Krieger algebras and Cross Product by ℤ, in: Free Probability Theory, D.-V. Voiculescu Ed., AMS, 1993.Google Scholar
  14. [14]
    G. Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151.zbMATHCrossRefGoogle Scholar
  15. [15]
    E. Vasselli: Continuous Fields of C*-algebras Arising from Extensions of Tensor C*-Categories, J. Funct. Anal. 199 (2003), 122–152.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    E. Vasselli: The C*-algebra of a vector bundle and fields of Cuntz algebras, J. Funct. Anal. 222(2) (2005), 491–502.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    E. Vasselli: Bundles of C*-categories and Duality, arXiv math.CT/0510594, (2005); Bundles of C*-categories, J. Funct. Anal. 247 (2007), 351–377.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Ezio Vasselli
    • 1
  1. 1.Dipartimento di MatematicaUniversità La Sapienza di RomaRomaItaly

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