Non-commutative differential geometry
Publications Mathématiques de l'IHÉS, Tome 62 (1985), pp. 41-144 
@article{PMIHES_1985__62__41_0,
     author = {Connes, Alain},
     title = {Non-commutative differential geometry},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {41--144},
     year = {1985},
     publisher = {Institut des Hautes Etudes Scientifiques},
     volume = {62},
     mrnumber = {823176},
     zbl = {0592.46056},
     language = {en},
     url = {https://www.numdam.org/item/PMIHES_1985__62__41_0/}
}
TY  - JOUR
AU  - Connes, Alain
TI  - Non-commutative differential geometry
JO  - Publications Mathématiques de l'IHÉS
PY  - 1985
SP  - 41
EP  - 144
VL  - 62
PB  - Institut des Hautes Etudes Scientifiques
UR  - https://www.numdam.org/item/PMIHES_1985__62__41_0/
LA  - en
ID  - PMIHES_1985__62__41_0
ER  - 
%0 Journal Article
%A Connes, Alain
%T Non-commutative differential geometry
%J Publications Mathématiques de l'IHÉS
%D 1985
%P 41-144
%V 62
%I Institut des Hautes Etudes Scientifiques
%U https://www.numdam.org/item/PMIHES_1985__62__41_0/
%G en
%F PMIHES_1985__62__41_0
Connes, Alain. Non-commutative differential geometry. Publications Mathématiques de l'IHÉS, Tome 62 (1985), pp. 41-144. https://www.numdam.org/item/PMIHES_1985__62__41_0/

[I] W. Arveson, The harmonic analysis of automorphism groups, Operator algebras and applications, Proc. Symposia Pure Math. 38 (I982), part I, I99-269. | Zbl | MR

[2] M. F. Atiyah, Transversally elliptic operators and compact groups, Lecture Notes in Math. 401, Berlin-New York, Springer (I974). | Zbl | MR

[3] M. F. Atiyah, Global theory of elliptic operators, Proc. Internat. Conf. on functional analysis and related topics, Tokyo, Univ. of Tokyo Press (I970), 2I-29. | Zbl

[4] M. F. Atiyah, K-theory, Benjamin (I967). | Zbl

[5] M. F. Atiyah and I. Singer, The index of elliptic operators IV, Ann. of Math. 93 (I97I), II9-I38. | Zbl

[6] S. Baaj et P. Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les C* modules Hilbertiens, C. r. Acad. Sci. Paris, Série I, 296 (I983), 875-878. | Zbl | MR

[7] P. Baum and A. Connes, Geometric K-theory for Lie groups and Foliations, Preprint I.H.E.S., I982. | Zbl

[8] P. Baum and A. Connes, Leafwise homotopy equivalence and rational Pontrjagin classes, Preprint I.H.E.S., I983. | Zbl

[9] P. Baum and R. Douglas, K-homology and index theory, Operator algebras and applications, Proc. Symposia Pure Math. 38 (I982), part I, II7-I73. | Zbl | MR

[I0] O. Bratteli, Inductive limits of finite dimensional C*-algebras, Trans. Am. Math. Soc. 171 (I972), I95-234. | Zbl | MR

[II] L. G. Brown, R. Douglas and P. A. Fillmore, Extensions of C*-algebras and K-homology, Ann. of Math. (2) 105 (I977), 265-324. | Zbl | MR

[I2] R. Carey and J. D. Pincus, Almost commuting algebras, K-theory and operator algebras, Lecture Notes in Math. 575, Berlin-New York, Springer (I977). | Zbl | MR

[I3] H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press (I956). | Zbl

[I4] A. Connes, The von Neumann algebra of a foliation, Lecture Notes in Physics 80 (I978), I45-I5I, Berlin-New York, Springer. | Zbl | MR

[I5] A. Connes, Sur la théorie non commutative de l'intégration, Algèbres d'opérateurs, Lecture Notes in Math. 725, Berlin-New York, Springer (I979). | Zbl | MR

[I6] A. Connes, A Survey of foliations and operator algebras, Operator algebras and applications, Proc. Symposia Pure Math. 38 (I982), Part I, 52I-628. | Zbl | MR

[I7] A. Connes, Classification des facteurs, Operator algebras and applications, Proc. Symposia Pure Math. 38 (I982), Part II, 43-I09. | Zbl | MR

[I8] A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Publ. R.I.M.S., Kyoto, 20 (I984), II39-II83. | Zbl | MR

[I9] A. Connes, C* algèbres et géométrie différentielle, C.r. Acad. Sci. Paris, Série I, 290 (I980), 599-604. | Zbl | MR

[20] A Connes, Cyclic cohomology and the transverse fundamental class of a foliation, Preprint I.H.E.S. M/84/7 (I984). | Zbl

[2I] A. Connes, Spectral sequence and homology of currents for operator algebras. Math. Forschungsinstitut Oberwolfach Tagungsbericht 42/8I, Funktionalanalysis und C*-Algebren, 27-9/3-I0-I98I.

[22] J. Cuntz and W. Krieger, A class of C*-algebras and topological Markov chains, Invent. Math. 56 (I980), 25I-268. | Zbl | MR | EuDML

[23] J. Cuntz, K-theoretic amenability for discrete groups, J. Reine Angew. Math. 344 (I983), I80-I95. | Zbl | MR | EuDML

[24] R. Douglas, C*-algebra extensions and K-homology, Annals of Math. Studies 95, Princeton University Press, I980. | Zbl | MR

[25] R. Douglas and D. Voiculescu, On the smoothness of sphere extensions, J. Operator Theory 6 (I) (I98I), I03. | Zbl | MR

[26] E. G. Effros, D. E Handelman and C. L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (I980), 385-407. | Zbl | MR

[27] G. Elliott, On the classification of inductive limits of sequences of semi-simple finite dimensional algebras, J. Alg. 38 (I976), 29-44. | Zbl | MR

[28] E. Getzler, Pseudodifferential operators on supermanifolds and the Atiyah Singer index theorem, Commun. Math. Physics 92 (I983), I63-I78. | Zbl | MR

[29] A. Grothendieck, Produits tensoriels topologiques, Memoirs Am. Math. Soc. 16 (I955). | Zbl

[30] J. Helton and R. Howe, Integral operators, commutators, traces, index and homology, Proc. of Conf. on operator theory, Lecture Notes in Math. 345, Berlin-New York, Springer (I973). | Zbl | MR

[3I] J. Helton and R. Howe, Traces of commutators of integral operators, Acta Math. 135 (I975), 27I-305. | Zbl | MR

[32] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I.H.E.S. 49 (I979). | Numdam | Zbl | EuDML

[33] G. Hochschild, B. Kostant and A. Rosenberg, Differential forms on regular affine algebras, Trans. Am. Math. Soc. 102 (I962), 383-408. | Zbl | MR

[34] L. Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (I979), 359-443. | Zbl

[35] B. Johnson, Cohomology in Banach algebras, Memoirs Am. Math. Soc. 127 (I972). | Zbl | MR

[36] B. Johnson, Introduction to cohomology in Banach algebras, in Algebras in Analysis, Ed. Williamson, New York, Academic Press (I975), 84-99. | Zbl | MR

[37] P. Julg and A. Valette, K-moyennabilité pour les groupes opérant sur les arbres, C. r. Acad. Sci. Paris, Série I, 296 (I983), 977-980. | Zbl | MR

[38] D. S. Kahn, J. Kaminker and C. Schochet, Generalized homology theories on compact metric spaces, Michigan Math. J. 24 (I977), 203-224. | Zbl | MR

[39] M. Karoubi, Connexions, courbures et classes caractéristiques en K-théorie algébrique, Canadian Math. Soc. Proc., Vol. 2, part I (I982), I9-27. | Zbl | MR

[40] M. Karoubi, K-theory. An introduction, Grundlehren der Math., Bd. 226 (I978), Springer Verlag. | Zbl | MR

[4I] M. Karoubi et O. Villamayor, K-théorie algébrique et K-théorie topologique I., Math. Scand. 28 (I97I), 265-307. | Zbl | MR | EuDML

[42] G. Kasparov, K-functor and extensions of C*-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (I980), 57I-636. | Zbl | MR

[43] G. Kasparov, K-theory, group C*-algebras and higher signatures, Conspectus, Chernogolovka (I983).

[44] G. Kasparov, Lorentz groups : K-theory of unitary representations and crossed products, preprint, Chernogolovka, I983. | Zbl

[45] B. Kostant, Graded manifolds, graded Lie theory and prequantization, Lecture Notes in Math. 570, Berlin-New York, Springer (I975). | Zbl | MR

[46] J. L. Loday and D. Quillen, Cyclic homology and the Lie algebra of matrices, C. r. Acad. Sci. Paris, Série I, 296 (I983), 295-297. | Zbl | MR

[47] S. Mac Lane, Homology, Berlin-New York, Springer (I975). | Zbl

[48] J. Milnor, Introduction to algebraic K-theory, Annals of Math. Studies, 72, Princeton Univ. Press. | Zbl | MR

[49] J. Milnor and D. Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton Univ. Press. | Zbl

[50] A. S. Miščenko, Infinite dimensional representations of discrete groups and higher signatures, Math. USSR Izv. 8 (I974), 85-II2. | Zbl

[5I] G. Pedersen, C*-algebras and their automorphism groups, New York, Academic Press (I979). | Zbl

[52] M. Penington, K-theory and C*-algebras of Lie groups and Foliations, D. Phil. thesis, Oxford, Michaelmas, Term., I983.

[53] M. Penington and R. Plymen, The Dirac operator and the principal series for complex semi-simple Lie groups, J. Funct. Analysis 53 (I983), 269-286. | Zbl | MR

[54] M. Pimsner and D. Voiculescu, Exact sequences for K-groups and Ext groups of certain cross-product C*-algebras, J. of operator theory 4 (I980), 93-II8. | Zbl | MR

[55] M. Pimsner and D. Voiculescu, Imbedding the irrational rotation C* algebra into an AF algebra, J. of operator theory 4 (I980), 20I-2II. | Zbl | MR

[56] M. Pimsner and D. Voiculescu, K groups of reduced crossed products by free groups, J. operator theory 8 (I) (I982), I3I-I56. | Zbl | MR

[57] M. Reed and B. Simon, Fourier Analysis, Self adjointness, New York, Academic Press (I975).

[58] M. Rieffel, C*-algebras associated with irrational rotations, Pac. J. of Math. 95 (2) (I98I), 4I5-429. | Zbl

[59] J. Rosenberg, C*-algebras, positive scalar curvature and the Novikov conjecture, Publ. Math. I.H.E.S. 58 (I984), 409-424. | Numdam | Zbl | EuDML

[60] W. Rudin, Real and complex analysis, New York, McGraw Hill (I966). | Zbl

[6I] I. Segal, Quantized differential forms, Topology 7 (I968), I47-I72. | Zbl | MR

[62] I. Segal, Quantization of the de Rham complex, Global Analysis, Proc. Symp. Pure Math. 16 (I970), 205-2I0. | Zbl | MR

[63] B. Simon, Trace ideals and their applications, London Math. Soc. Lecture Notes 35, Cambridge Univ. Press (I979). | Zbl | MR

[64] I. M. Singer, Some remarks on operator theory and index theory, Lecture Notes in Math. 575 (I977), I28-I38, New York, Springer. | Zbl | MR

[65] J. L. Taylor, Topological invariants of the maximal ideal space of a Banach algebra, Advances in Math. 19 (I976), I49-206. | Zbl

[66] A. M. Torpe, K-theory for the leaf space of foliations by Reeb components, J. Funct. Analysis 61 (I985), I5-7I. | Zbl | MR

[67] B. L. Tsigan, Homology of matrix Lie algebras over rings and Hochschild homology, Uspekhi Math. Nauk. 38 (I983), 2I7-2I8. | Zbl

[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 (I984), 334-359. | Zbl

[69] A. Wasserman, Une démonstration de la conjecture de Connes-Kasparov, to appear in C. r. Acad. Sci. Paris.

[70] A. Weil, Elliptic functions according to Eisenstein and Kronecker, Erg. der Math., vol. 88, Berlin-New York, Springer (I976). | Zbl | MR

[7I] R. Wood, Banach algebras and Bott periodicity, Topology 4 (I965-I966), 37I-389. | Zbl | MR