On the Baum-Connes Assembly Map for Discrete Groups

  • Alain Valette
Part of the Advanced Courses in Mathematics CRM Barcelona book series (ACMBIRK)


In these notes, we study the Baum-Connes analytical assembly maps (or index maps) μ i Γ : RK i Γ (EΓ) → K i (C r * Γ) and \( \bar \mu _i^\Gamma \) ,for a countable group Γ. Here RK i Γ denotes the Γ-equivariant K-homology with Γ-compact supports of the universal space E Γ for proper Γ-actions, while K i (C r * ) (resp. K i (C*Γ) denotes the analytical K-theory of the reduced (resp. full) C*-algebra of Γ. As it is simple and direct, we use the definition of β i Γ suggested by Baum, Connes and Higson in Section 3 of [BCH94]. The Baum-Connes conjecture asserts that, for any group Γ, the map β i Γ is an isomorphism (i= 0, 1). The contents of this paper are as follows:
  1. 1

    We make the necessary changes for constructing \( \bar \mu _i^\Gamma \) , and give a detailed proof that β i Γ and \( \bar \mu _i^\Gamma \) provide K-theory elements of the corresponding C*-algebras.

  2. 2

    We carefully describe the behavior of the left-hand side of the assembly maps under group homomorphisms, and we prove that \( \bar \mu _i^\Gamma \)is natural with respect to arbitrary group homomorphisms. As a consequence, we get a new proof of the fact that, if Γ acts freely on the space X, then the equibvalent K-homology K * Γ (X) is isomorphic to the H-homology K *(Γ\X) of the orbit space.

  3. 3

    To illustrate the non-triviality of the assembly map, we give a direct proof of the Bauam-Connes conjecture for the group ℤ of integers, not appealing to equivariant KK-theory.

  4. 4

    Denote by \( \tilde \kappa _\Gamma :\Gamma \to K_1 \left( {C_r^* \Gamma } \right) \) the homomorphism induced by the canonical inclusion of Γ in the unitary group of C r Γ . We show that there exixts a homomorphism \( \bar \beta _t :\Gamma \to RK_1^\Gamma \left( {\underline E T} \right) \) such that \( \bar \kappa _\Gamma = \mu _i^\Gamma \circ \tilde \beta _t \); this extends a result of Natsume [Nat88] for Γ torsion-free.



Compact Operator Discrete Group Group Homomorphism Chern Character Finite Subgroup 
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  1. [AD87]
    C. Anantharaman-Delaroche. Systèmes dynamiques non commutatifs et Moyennabilité. Math. Ann. 279 (1987), 297–315.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [Arv76]
    W. Arveson. An invitation to C*-algebra. Springer, 1976.Google Scholar
  3. [AS63]
    M.F. Atiyah and I.M. Singer. The index of elliptic operators on compact manifolds. Bull. Amer. Math. Soc. 69 (1963), 422–433.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [BBV99]
    C. Béguin, H. Bettaieb, and A. Valette. K-theory for the C*-algebras of one-relator groups. K-theory 16 (1999), 277–298.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [BC88a]
    P. Baum and A. Connes. Chern character for discrete groups. In A fête of topology (Academic Press, pp. 163–232), 1988.Google Scholar
  6. [BC88b]
    P. Baum and A. Connes. K-theory for discrete groups. In Operator algebras and application (London Math. Soc. lecture notes ser. 135, 1–20), 1988.Google Scholar
  7. [BC00]
    P. Baum and A. Connes. Geometric K-theory for Lie groups and foliations.Enseign. Math. 46 (2000), 3–42. First distributed: 1982.MathSciNetzbMATHGoogle Scholar
  8. [BCdlH94]
    M.E.B. Bekka, M. Cowling, and P. de la Harpe. Some groups whose reduced C*-algebra is simple. Publ. Math. I.H.E.S., 80 (1994), 117–134.Google Scholar
  9. [BCH94]
    P. Baum, A. Connes, and N. Higson. Classifying spaces for proper actions and K-theory of group C*-algebras. In C*-algebras 1943–1993: a fifty year celebration (Contemporary Mathematics 167, 241–291), 1994.MathSciNetGoogle Scholar
  10. [BJ83]
    S. Baaj and P. Julg. Théorie bivariante de Kasparov et opérateurs non bornésdans les C*-modules hilbertiens. C.R.Acad.Sci.Paris 296 (1983), 875–878.MathSciNetzbMATHGoogle Scholar
  11. [Bla86]
    B. Blackadar. K-theory for operator algebras. MSRI publications 5, Springer-Verlag, 1986.zbMATHCrossRefGoogle Scholar
  12. [BV96]
    H. Bettaieb and A. Valette. Sur le groupe K1 des C*-algebres réduites de groupes discrets. C.R. Acad. Sci. Paris, 322 (1996), 925–928.MathSciNetzbMATHGoogle Scholar
  13. [Con94]
    A. Connes. Noncommutative geometry. Academic Press, 1994.Google Scholar
  14. [CS84]
    A. Connes and G. Skandalis. The longitudinal index theorem for foliations.Pub. Res. Inst. Math. Sci. Kyoto Univ. 20 (1984), 1139–1183.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Cun83]
    J. Cuntz. K-theoretic amenability for discrete groups. J. für reine angew.Math. 344 (1983), 180–195.MathSciNetzbMATHGoogle Scholar
  16. [Dix77]
    J. Dixmier. C*-algebras. North Holland, 1977.Google Scholar
  17. [DL98]
    J.F. Davis and W. Lück. Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory. K-theory 15 (1998), 201–252.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [EN87]
    G. Elliott and T. Natsume. A Bott periodicity map for crossed products of C*-algebras by discrete groups. K-theory 1 (1987), 423–435.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [FRR95]
    S.C. Ferry, A. Ranicki, and J. Rosenberg. A history and survey of the Novikov conjecture. In Novikov conjectures, index theorems and rigidity (London Math. Soc. lecture notes ser. 226, pp. 7–66), 1995.Google Scholar
  20. [Gre77]
    P. Green. C*-algebras of transformation groups with smooth orbit space. Pac.J. of Math. 72 (1977), 71–97.zbMATHCrossRefGoogle Scholar
  21. [Gro93]
    M. Gromov. Asymptotic invariants of infinite groups. In Geometric group theory (G.A. Niblo and M.A. Roller, eds.), London Math. Soc. lect. notes 182, Cambridge Univ. Press, 1993.Google Scholar
  22. [HLS02]
    N. Higson, V. Lafforgue, and G. Skandalis. Counterexamples to the Baum-Connes conjecture. Geom. funct. anal. 12 (2002), 330–354.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [HP]
    I. Hambleton and E.K. Pedersen. Identifying assembly maps in K- and L-theory. Preprint, june 2001.Google Scholar
  24. [HR00a]
    N. Higson and J. Roe. Amenable actions and the Novikov conjecture. J. Heine Angew. Math. 519,143–153 519 (2000), 143–153.MathSciNetzbMATHGoogle Scholar
  25. [HR00b]
    N. Higson and J. Roe. Analytic K-homology. Oxford mathematical monographs, 2000.Google Scholar
  26. [JT91]
    K. Knudsen Jensen and K. Thomsen. Elements of KK-theory. Birkhäuser,1991.zbMATHCrossRefGoogle Scholar
  27. [Jul98]
    P. Julg. Travaux de Higson et Kasparov sur la conjecture de Baum-Connes.In Séminaire Bourbaki, Exposé 841, 1998.Google Scholar
  28. [Kas75]
    G.G. Kasparov. Topological invariants of elliptic operators.i.K-homology.Math. USSR Izvestija 9 (1975), 751–792.MathSciNetCrossRefGoogle Scholar
  29. [Kas81]
    G.G. Kasparov. The operator K-functor and extensions of C*-algebras. Math.of the USSR-Izvestija 16 (1981), 513–572.zbMATHCrossRefGoogle Scholar
  30. [Kas83]
    G.G. Kasparov. The index of invariant elliptic operators, K-theory, and Lie group representations. Dokl.Akad. Nauk. USSR 268 (1983), 533–537.MathSciNetGoogle Scholar
  31. [Kas88]
    G.G. Kasparov. Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91 (1988), 147–201.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [Kas95]
    G.G. Kasparov. K-theory, group C*-algebras, and higher signatures (Con-spectus, first distributed 1981). In Novikov conjectures,index theorems and rigidity (London Math. Soc. lecture notes ser. 226, 101–146), 1995.Google Scholar
  33. [Kuc]
    D. Kucerovsky. Making Kasparov products unbounded. Preprint, 1997.Google Scholar
  34. [Kuc94]
    D. Kucerovsky. Kasparov products in KK-theory, and unbounded operators with applications to index theory. PhD thesis, Magdalen College, Oxford, 1994.Google Scholar
  35. [Kuc97]
    D. Kucerovsky. The KK-product of unbounded modules. K-theory 11 (1997),17–34.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [Luc]
    W. Lück. On the functoriality of the source of the Baum-Connes map. Un-published, December 1996.Google Scholar
  37. [Lö2]
    W. Lück. The relation between the Baum-Connes conjecture and the trace conjecture. Inventiones. Math. 149 (2002), 123–152.zbMATHCrossRefGoogle Scholar
  38. [Laf98]
    V. Lafforgue. Une démonstration de la conjecture de Baum-Connes pourles groupes réductifs sur un corps p-adiques et pour certains groupes discrets possédant la propriété (t). C.R. Acad. Sci. Paris 327 (1998), 439–444.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [Lan95]
    E.C. Lance. Hilbert C*-modules, a toolkit for operator algebraists. London Math. Soc. lecture notes ser. 210, 1995.CrossRefGoogle Scholar
  40. [Mat]
    M. Matthey. Low dimensional homology, the Baum-Connes assembly map,delocalization and the Chern character. Preprint, Muenster 2001.Google Scholar
  41. [Mat00]
    M. Matthey. K-theories,C*-algebras and assembly maps. PhD thesis, Université de Neuchâtel, 2000.Google Scholar
  42. [Mis]
    G. Mislin. Equivariant K-homology of the classifying space for proper actions.Proceedings of the Euro summer school on proper group actions, Barcelona, september 2001.Google Scholar
  43. [Nat88]
    T. Natsume. The Baum-Connes conjecture, the commutator theorem, and Rieffel projections. C.R. Math. Rep. Acad. Sci. Canada X (1988), 13–18.MathSciNetGoogle Scholar
  44. [Pal61]
    R. Palais. On the existence of slices for actions of non-compact Lie groups.Ann. of Math. 73 (1961), 295–323.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [Pas85]
    D.S. Passman. The algebraic structure of group rings. Krieger Publishing Company, 1985.Google Scholar
  46. [Ped79]
    G.K. Pedersen. C*-algebras and their automorphism groups. Academic Press,1979.Google Scholar
  47. [Pie00]
    F. Pierrot. K-théorie de C*-algebres pleines de groupes de Lie et formule de multiplicité de Langlands. PhD thesis, Paris VII, 2000.Google Scholar
  48. [Pow75]
    R.T. Powers. A remark on the domain of an unbounded derivation of a C*-algebra. J. Funct. Anal. 18 (1975), 85–95.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [Rie74]
    M.A. Rieffel. Induced representations of C*-algebras. Adv. in Math. 13 (1974),176–257.MathSciNetzbMATHCrossRefGoogle Scholar
  50. [Rie82]
    M.A. Rieffel. Applications of strong Morita equivalence to transformation group C*- algebras. In Operator algebras and applications, Proc. Symp. Pure Math. 38, Part 1, 299–310, 1982.MathSciNetCrossRefGoogle Scholar
  51. [Roe]
    J. Roe. Comparing analytic assembly maps. Preprint, April 2001.Google Scholar
  52. [Roe96]
    J. Roe. Index theory,coarse geometry, and topology of manifolds. CBMS regional conf. ser. in Math. 90, 1996.zbMATHGoogle Scholar
  53. [Ros83]
    J. Rosenberg. C*-algebras, positive scalar curvature, and the Novikov conjecture. Publ. Math. I.H.E.S. 58 (1983), 197–212.Google Scholar
  54. [Ros84]
    J. Rosenberg. Group C*-algebras and topological invariants. In Operator algebras and group representations, vol. II. Pitman, pp. 95–115, 1984.Google Scholar
  55. [Ska99]
    G. Skandalis. Progrès récents sur la conjecture de Baum-Connes. Contribution de Vincent Lafforgue. In Séminaire Bourbaki, Exposé 869, 1999.Google Scholar
  56. [Tay75]
    J.L. Taylor. Banach algebras and topology. In Algebras in analysis, pp. 118–186. Academic Press, 1975.Google Scholar
  57. [Val89]
    A. Valette. The conjecture of idempotents: a survey of the C*-algebraic approach. Bull. Soc. Math. Belg., XLI (1989), 485–521.MathSciNetGoogle Scholar
  58. [Val98]
    A. Valette. On Godement’s characterization of amenability. Bull. Austral.Math. Soc. 57 (1998), 153–158.MathSciNetzbMATHCrossRefGoogle Scholar
  59. [Val02]
    A. Valette. Introduction to the Baum-Connes conjecture. ETHZ lectures in math., Birkhauser, 2002.zbMATHCrossRefGoogle Scholar

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  • Alain Valette

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