Non-commutative differential geometry

  • Alain Connes
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Bibliography

  1. [1]
    W. Arveson, The harmonic analysis of automorphism groups, Operator algebras and applications,Proc. Symposia Pure Math. 38 (1982), part I, 199–269.Google Scholar
  2. [2]
    M. F. Atiyah, Transversally elliptic operators and compact groups,Lecture Notes in Math. 401, Berlin-New York, Springer (1974).Google Scholar
  3. [3]
    M. F. Atiyah, Global theory of elliptic operators,Proc. Internat. Conf. on functional analysis and related topics, Tokyo, Univ. of Tokyo Press (1970), 21–29.Google Scholar
  4. [4]
    M. F. Atiyah, K-theory, Benjamin (1967).Google Scholar
  5. [5]
    M. F. Atiyah andI. Singer, The index of elliptic operators IV,Ann. of Math. 93 (1971), 119–138.CrossRefGoogle Scholar
  6. [6]
    S. Baaj etP. Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les C* modules Hilbertiens,C. r. Acad. Sci. Paris, Série I,296 (1983), 875–878.MATHGoogle Scholar
  7. [7]
    P. Baum andA. Connes,Geometric K-theory for Lie groups and Foliations, Preprint I.H.E.S., 1982.Google Scholar
  8. [8]
    P. Baum andA. Connes,Leafwise homotopy equivalence and rational Pontrjagin classes, Preprint I.H.E.S., 1983.Google Scholar
  9. [9]
    P. Baum andR. Douglas, K-homology and index theory, Operator algebras and applications,Proc. Symposia Pure Math. 38 (1982), part I, 117–173.Google Scholar
  10. [10]
    O. Bratteli, Inductive limits of finite dimensional C*-algebras,Trans. Am. Math. Soc. 171 (1972), 195–234.MATHCrossRefGoogle Scholar
  11. [11]
    L. G. Brown, R. Douglas andP. A. Fillmore, Extensions of C*-algebras and K-homology,Ann. of Math. (2)105 (1977), 265–324.CrossRefGoogle Scholar
  12. [12]
    R. Carey andJ. D. Pincus, Almost commuting algebras, K-theory and operator algebras,Lecture Notes in Math. 575, Berlin-New York, Springer (1977).CrossRefGoogle Scholar
  13. [13]
    H. Cartan andS. Eilenberg,Homological algebra, Princeton University Press (1956).Google Scholar
  14. [14]
    A. Connes, The von Neumann algebra of a foliation,Lecture Notes in Physics 80 (1978), 145–151, Berlin-New York, Springer.Google Scholar
  15. [15]
    A. Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs,Lecture Notes in Math. 725, Berlin-New York, Springer (1979).Google Scholar
  16. [16]
    A. Connes, A Survey of foliations and operator algebras, Operator algebras and applications,Proc. Symposia Pure Math. 38 (1982), Part I, 521–628.Google Scholar
  17. [17]
    A. Connes, Classification des facteurs, Operator algebras and applications,Proc. Symposia Pure Math. 38 (1982), Part II, 43–109.Google Scholar
  18. [18]
    A. Connes andG. Skandalis, The longitudinal index theorem for foliations,Publ. R.I.M.S., Kyoto,20 (1984), 1139–1183.MATHGoogle Scholar
  19. [19]
    A. Connes, C* algèbres et géométrie différentielle,C.r. Acad. Sci. Paris, Série I,290 (1980), 599–604.MATHGoogle Scholar
  20. [20]
    A Connes,Cyclic cohomology and the transverse fundamental class of a foliation, Preprint I.H E.S. M/84/7 (1984).Google Scholar
  21. [21]
    A. Connes, Spectral sequence and homology of currents for operator algebras. Math. Forschungsinstitut Oberwolfach Tagungsbericht 42/81,Funktionalanalysis und C*-Algebren, 27-9/3-10-1981.Google Scholar
  22. [22]
    J. Cuntz andW. Krieger, A class of C*-algebras and topological Markov chains,Invent. Math. 56 (1980), 251–268.MATHCrossRefGoogle Scholar
  23. [23]
    J. Cuntz, K-theoretic amenability for discrete groups,J. Reine Angew. Math. 344 (1983), 180–195.MATHGoogle Scholar
  24. [24]
    R. Douglas, C*-algebra extensions and K-homology,Annals of Math. Studies 95, Princeton University Press, 1980.Google Scholar
  25. [25]
    R. Douglas andD. Voiculescu, On the smoothness of sphere extensions,J. Operator Theory 6 (1) (1981), 103.MATHGoogle Scholar
  26. [26]
    E. G. Effros, D. E Handelman andC. L. Shen, Dimension groups and their affine representations,Amer. J. Math. 102 (1980), 385–407.MATHCrossRefGoogle Scholar
  27. [27]
    G. Elliott, On the classification of inductive limits of sequences of semi-simple finite dimensional algebras,J. Alg. 38 (1976), 29–44.MATHCrossRefGoogle Scholar
  28. [28]
    E. Getzler, Pseudodifferential operators on supermanifolds and the Atiyah Singer index theorem,Commun. Math. Physics 92 (1983), 163–178.MATHCrossRefGoogle Scholar
  29. [29]
    A. Grothendieck, Produits tensoriels topologiques,Memoirs Am. Math. Soc. 16 (1955).Google Scholar
  30. [30]
    J. Helton andR. Howe, Integral operators, commutators, traces, index and homology,Proc. of Conf. on operator theory, Lecture Notes in Math. 345, Berlin-New York, Springer (1973).CrossRefGoogle Scholar
  31. [31]
    J. Helton andR. Howe, Traces of commutators of integral operators,Acta Math. 135 (1975), 271–305.MATHCrossRefGoogle Scholar
  32. [32]
    M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,Publ. Math. I.H.E.S. 49 (1979).Google Scholar
  33. [33]
    G. Hochschild, B. Kostant andA. Rosenberg, Differential forms on regular affine algebras,Trans. Am. Math. Soc. 102 (1962), 383–408.MATHCrossRefGoogle Scholar
  34. [34]
    L. Hörmander, The Weyl calculus of pseudodifferential operators,Comm. Pure Appl. Math. 32 (1979), 359–443.MATHCrossRefGoogle Scholar
  35. [35]
    B. Johnson, Cohomology in Banach algebras,Memoirs Am. Math. Soc. 127 (1972).Google Scholar
  36. [36]
    B. Johnson, Introduction to cohomology in Banach algebras, inAlgebras in Analysis, Ed. Williamson, New York, Academic Press (1975), 84–99.Google Scholar
  37. [37]
    P. Julg andA. Valette, K-moyennabilité pour les groupes opérant sur les arbres,C. r. Acad. Sci. Paris, Série I,296 (1983), 977–980.MATHGoogle Scholar
  38. [38]
    D. S. Kahn, J. Kaminker andC. Schochet, Generalized homology theories on compact metric spaces,Michigan Math. J. 24 (1977), 203–224.MATHCrossRefGoogle Scholar
  39. [39]
    M. Karoubi, Connexions, courbures et classes caractéristiques en K-théorie algébrique,Canadian Math. Soc. Proc., Vol. 2, part I (1982), 19–27.Google Scholar
  40. [40]
    M. Karoubi, K-theory. An introduction,Grundlehren der Math., Bd.226 (1978), Springer Verlag.Google Scholar
  41. [41]
    M. Karoubi etO. Villamayor, K-théorie algébrique et K-théorie topologique I.,Math. Scand. 28 (1971), 265–307.MATHGoogle Scholar
  42. [42]
    G. Kasparov, K-functor and extensions of C*-algebras,Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 571–636.MATHGoogle Scholar
  43. [43]
    G. Kasparov, K-theory, group C*-algebras and higher signatures, Conspectus, Chernogolovka (1983).Google Scholar
  44. [44]
    G. Kasparov,Lorentz groups: K-theory of unitary representations and crossed products, preprint, Chernogolovka, 1983.Google Scholar
  45. [45]
    B. Kostant, Graded manifolds, graded Lie theory and prequantization,Lecture Notes in Math. 570, Berlin-New York, Springer (1975).Google Scholar
  46. [46]
    J. L. Loday andD. Quillen, Cyclic homology and the Lie algebra of matrices,C. r. Acad. Sci. Paris, Série I,296 (1983), 295–297.MATHGoogle Scholar
  47. [47]
    S. Mac Lane,Homology, Berlin-New York, Springer (1975).MATHGoogle Scholar
  48. [48]
    J. Milnor, Introduction to algebraic K-theory,Annals of Math. Studies,72, Princeton Univ. Press.Google Scholar
  49. [49]
    J. Milnor andD. Stasheff, Characteristic classes,Annals of Math. Studies 76, Princeton Univ. Press.Google Scholar
  50. [50]
    A. S. Miščenko, Infinite dimensional representations of discrete groups and higher signatures,Math. USSR Izv. 8 (1974), 85–112.CrossRefGoogle Scholar
  51. [51]
    G. Pedersen, C*-algebras and their automorphism groups, New York, Academic Press (1979).Google Scholar
  52. [52]
    M. Penington, K-theory and C*-algebras of Lie groups and Foliations, D. Phil. thesis, Oxford, Michaelmas, Term., 1983.Google Scholar
  53. [53]
    M. Penington andR. Plymen, The Dirac operator and the principal series for complex semi-simple Lie groups,J. Funct. Analysis 53 (1983), 269–286.MATHCrossRefGoogle Scholar
  54. [54]
    M. Pimsner andD. Voiculescu, Exact sequences for K-groups and Ext groups of certain cross-product C*-algebras,J. of operator theory 4 (1980), 93–118.MATHGoogle Scholar
  55. [55]
    M. Pimsner andD. Voiculescu, Imbedding the irrational rotation C* algebra into an AF algebra,J. of operator theory 4 (1980), 201–211.MATHGoogle Scholar
  56. [56]
    M. Pimsner andD. Voiculescu, K groups of reduced crossed products by free groups,J. operator theory 8 (1) (1982), 131–156.MATHGoogle Scholar
  57. [57]
    M. Reed andB. Simon,Fourier Analysis, Self adjointness, New York, Academic Press (1975).MATHGoogle Scholar
  58. [58]
    M. Rieffel, C*-algebras associated with irrational rotations,Pac. J. of Math. 95 (2) (1981), 415–429.Google Scholar
  59. [59]
    J. Rosenberg, C*-algebras, positive scalar curvature and the Novikov conjecture,Publ. Math. I.H.E.S. 58 (1984), 409–424.Google Scholar
  60. [60]
    W. Rudin,Real and complex analysis, New York, McGraw Hill (1966).MATHGoogle Scholar
  61. [61]
    I. Segal, Quantized differential forms,Topology 7 (1968), 147–172.MATHCrossRefGoogle Scholar
  62. [62]
    I. Segal, Quantization of the de Rham complex,Global Analysis, Proc. Symp. Pure Math. 16 (1970), 205–210.Google Scholar
  63. [63]
    B. Simon, Trace ideals and their applications,London Math. Soc. Lecture Notes 35, Cambridge Univ. Press (1979).Google Scholar
  64. [64]
    I. M. Singer, Some remarks on operator theory and index theory,Lecture Notes in Math. 575 (1977), 128–138, New York, Springer.Google Scholar
  65. [65]
    J. L. Taylor, Topological invariants of the maximal ideal space of a Banach algebra,Advances in Math. 19 (1976), 149–206.MATHCrossRefGoogle Scholar
  66. [66]
    A. M. Torpe, K-theory for the leaf space of foliations by Reeb components,J. Funct. Analysis 61 (1985), 15–71.MATHCrossRefGoogle Scholar
  67. [67]
    B. L. Tsigan, Homology of matrix Lie algebras over rings and Hochschild homology,Uspekhi Math. Nauk. 38 (1983), 217–218.Google Scholar
  68. [68]
    A. Valette, K-Theory for the reduced C*-algebra of semisimple Lie groups with real rank one,Quarterly J. of Math., Oxford, Série 2,35 (1984), 334–359.Google Scholar
  69. [69]
    A. Wasserman, Une démonstration de la conjecture de Connes-Kasparov, to appear inC. r. Acad. Sci. Paris.Google Scholar
  70. [70]
    A. Weil, Elliptic functions according to Eisenstein and Kronecker,Erg. der Math. vol. 88, Berlin-New York, Springer (1976).MATHGoogle Scholar
  71. [71]
    R. Wood, Banach algebras and Bott periodicity,Topology 4 (1965–1966), 371–389.CrossRefGoogle Scholar

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© Publications Mathématiques de L’I.É.E.S. 1985

Authors and Affiliations

  • Alain Connes
    • 1
    • 2
  1. 1.Collège de FranceParis Cedex 05
  2. 2.Institut des Hautes Études scientifiquesBures-sur-Yvette

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