Geodesic
Projective Modules over Higher-Dimensional Non-Commutative Tori
Canadian journal of mathematics, Tome 40 (1988) no. 2, pp. 257-338

Voir la notice de l'article provenant de la source Cambridge University Press

The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus Tn with its algebra, C(Tn), of continuous complex-valued functions under pointwise multiplication. But C(Tn) is the universal C*-algebra generated by n commuting unitary operators. By definition, [15, 16, 50], a non-commutative n-torus is the universal C*-algebra generated by n unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [8, 10], these algebras have a natural differentiable structure, defined by a natural ergodic action of Tn as a group of automorphisms. The non-commutative tori behave in inany ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [37] that the K-groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are KK-equivalent to ordinary tori by Corollary 7.5 of [52].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [17].) 
Rieffel, Marc A. Projective Modules over Higher-Dimensional Non-Commutative Tori. Canadian journal of mathematics, Tome 40 (1988) no. 2, pp. 257-338. doi: 10.4153/CJM-1988-012-9
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[1] 1. Anderson, J. and Paschke, W., The rotation algebra, preprint. Google Scholar

[2] 2. Barnes, B. A., The role of minimal idempotents in the representation theory of locally compact groups, Proc. Edinburgh Math. Soc. 23 (1980), 229–238. Google Scholar

[3] 3. Bellissard, J., Lima, R. and Testard, D., Almost periodic Schrodinger operators, Mathematics + Physics, Lectures on Recent Results 1, (World Scientific, Singapore/Philadelphia, 1985), 1–64. Google Scholar

[4] 4. Blackadar, B., Notes on the structure of projections in simple C*-algebras, Semesterbericht Functionalanalysis, Tubingen, Wintersemester (1982/1983). Google Scholar

[5] 5. Bourbaki, N., Algèbre multilineaire, elements de mathématique (Hermann, Paris, 1958). Google Scholar

[6] 6. Bratteli, O., Elliott, G. A. and Jorgensen, P. E. T., Decomposition of unbounded derivations into invariant and approximately inner parts, J. reine ang. Math. 346 (1984), 166–193. Google Scholar

[7] 7. Bruhat, F., Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes p-adiques, Bull. Soc. Math. France 89 (1961), 43–75. Google Scholar

[8] 8. Connes, A., C*-algèbres et géométrie différentielle, C. R. Acad. Se. Paris 290 (1980), 599–604. Google Scholar

[9] 9. Connes, A., An analogue of the Thorn isomorphism for crossed products of a C*-algebra by an action of R, Adv. Math. 39 (1981), 31–55. Google Scholar

[10] 10. Connes, A., A survey of foliations and operator algebras, in Operator algebras and applications, Proc. Symp. Pure Math. 38 (Amer. Math. Soc, Providence, 1982), 521–628. Google Scholar

[11] 11. Connes, A., The Chern character in K homology, Publ. Math. I.H.E.S. 62 (1986), 41–93. Google Scholar

[12] 12. Connes, A., De Rham homology and non commutative algebra, Publ. Math. I.H.E.S. 62 (1986), 94–144. Google Scholar

[13] 13. Connes, A. and Rieffel, M. A., Yang-Mills for non-commutative two-tori, Proceedings of Conf. on Operator Algebras and Mathematical Physics, University of Iowa (1985), Contemporary Math. 62 (1987), 237–266. Google Scholar

[14] 14. Connes, A. and Takesaki, M., The flow of weights on factors of type III, Tôhoku Math. J. 29 (1977), 473–575. Google Scholar

[15] 15. Cuntz, J., Elliott, G. A., Goodman, F. M. and Jorgensen, P. E. T., On the classification of noncommutative tori, II, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), 189–194. Google Scholar

[16] 16. Disney, S., Elliott, G. A., Kumjian, A. and Raeburn, I., On the classification of noncommutative tori, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), 137–141. Google Scholar

[17] 17. Elliott, G. A., On the K-theory of the C*-algebra generated by a projective representation of a torsion-free discrete abelian group, in Operator algebras and group representations 1 (Pitman, London, 1984), 157–184. Google Scholar

[18] 18. Green, P., The local structure of twisted covariance algebras, Acta Math. 140 (1978), 191–250. Google Scholar

[19] 19. Elliott, G. A., Square-integrable representations and the dual topology, J. Funct. Anal. 35 (1980), 279–294. Google Scholar

[20] 20. Greub, W., Halperin, S. and Vanstone, R., Connections, curvature, and cohomology, Vol. II (Academic Press, New York, 1973). Google Scholar

[21] 21. Herman, R. H. and Vaserstein, L. N., The stable range of C* -algebras, Invent. Math. 77 (1984), 553–555. Google Scholar

[22] 22. Hewitt, E. and Ross, K. A., Abstract harmonic analysis, I (Springer-Verlag, Berlin-Gottingen-Heidelberg, 1963). Google Scholar

[23] 23. Husemoller, D., Fibre bundles (Springer-Verlag, New York-Heidelberg-Berlin, 1966). Google Scholar

[24] 24. Igusa, J., Theta functions (Springer-Verlag, Berlin-Heidelberg-New York, 1972). Google Scholar

[25] 25. Kleppner, A., Multipliers on Abelian groups, Math. Ann. 158 (1965), 11–34. Google Scholar

[26] 26. Kumjian, A., On localizations and simple C*-algebras, Pacific J. Math. 112 (1984), 141–192. Google Scholar

[27] 27. Lang, S., Algebra (Addison-Wesley, Reading, Mass., 1965). Google Scholar

[28] 28. Loomis, L. H., An introduction to abstract harmonic analysis (Van Nostrand, New York, 1953). Google Scholar

[29] 29. Mackey, G. W., Unitary representations of group extensions, I, Acta Math. 99 (1958), 265–311. Google Scholar

[30] 30. Menai, P. and Moncasi, J., On regular rings with stable range 2, J. Pure Appl. Algebra 24 (1982), 25–40. Google Scholar

[31] 31. Olesen, D., Pedersen, G. K. and Takesaki, M., Ergodic actions of compact Abelian groups, J. Operator Theory 3 (1980), 237–269. Google Scholar

[32] 32. Ozeki, H., Chern classes of projective modules, Nagoya Math. J. 23 (1963), 121–152. Google Scholar

[33] 33. Packer, J. A., K-theoretic invariants for C*-algebras associated to transformations and induced flows, J. Funct. Anal. 67 (1986), 25–59. Google Scholar

[34] 34. Packer, J. A., C*-algebras generated by projective representations of the discrete Heisenberg group, I, II, preprints. Google Scholar

[35] 35. Pedersen, G. K., C*-algebras and their automorphism groups, London Math. Soc. Monographs 14 (Academic Press, London, 1979). Google Scholar

[36] 36. Pimsner, M. V., Range of traces on K0 of reduced crossed products by free groups, Proc. Conf. on Operator Algebras, Connections with Topology and Ergodic Theory, Lecture Notes Math. 1132 (Springer-Verlag, Berlin-Heidelberg, 1985), 374–408. Google Scholar

[37] 37. Pimsner, M. V. and Voiculescu, D., Exact sequences for K-groups and Ext-groups of certain crossed product C* -algebras, J. Operator Theory 4 (1980), 93–118. Google Scholar

[38] 38. Poguntke, D., Simple quotients of group C*-algebras for two step nilpotent groups and connected Lie groups, Ann. Sci. Ec. Norm. Sup. 16 (1983), 151–172. Google Scholar

[39] 39. Riedel, N., Classification of the C*-algebras associated with minimal rotations, Pacific J. Math. 101 (1982), 153–161. Google Scholar

[40] 40. Riedel, N., On the topological stable rank of irrational rotation algebras, J. Operator Theory 13 (1985), 143–150. Google Scholar

[41] 41. Rieffel, M. A., On the uniqueness of the Heisenberg commutation relations, Duke Math. J. 39 (1972), 745–752. Google Scholar

[42] 42. Rieffel, M. A., Induced representations of C*-algebras, Adv. Math. 13 (1974), 176–257. Google Scholar

[43] 43. Rieffel, M. A., Commutation theorems and generalized commutation relations, Bull. Soc. Math. France 104 (1976), 205–224. Google Scholar

[44] 44. Rieffel, M. A., Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner, Studies in Analysis, Adv. Math. Suppl. Series 4 (1979), 43–82. Google Scholar

[45] 45. Rieffel, M. A., Morita equivalence for operator algebras, in Operator algebras and applications, Proc. Symp. Pure Math. 38 (Amer. Math. Soc, Providence, 1982), 285–298. Google Scholar

[46] 46. Rieffel, M. A., C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415–429. Google Scholar

[47] 47. Rieffel, M. A., Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257(1981), 403–418. Google Scholar

[48] 48. Rieffel, M. A., Dimension and stable rank in the K-theory of C*-algebras, Proc. London Math. Soc. 46(1983), 301–333. Google Scholar

[49] 49. Rieffel, M. A., The cancellation theorem for projective modules over irrational rotation C*-algebras, Proc. London Math. Soc. 47 (1983), 285–302. Google Scholar

[50] 50. Rieffel, M. A., “Vector bundles” over higher dimensional “non-commutative tori”, Proc. Conf. on Operator Algebras, Connections with Topology and Ergodic Theory, Lecture Notes Math. 1132 (Springer-Verlag, Berlin-Heidelberg, 1985), 456–467. Google Scholar

[51] 51. Rieffel, M. A., K-theory of crossed products of C*-algebras by discrete groups, Proc. 1984 Conf. on Group Actions on Rings, Contemporary Math. 43 (1985), 253–265. Google Scholar

[52] 52. Rosenberg, J. and Schochet, C., The Kunneth and the universal coefficient theorem for Kasparov's generalized K-functor, Duke Math. J. 55 (1987), 431–474. Google Scholar

[53] 53. Sheu, A. J.-L., The cancellation property for modules over the group C*-algebras of certain nilpotent Lie groups, doctoral dissertation, University of California, Berkeley (1985). Google Scholar

[54] 54. Swan, R., Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264–277. Google Scholar

[55] 55. Taylor, J. L., Banach algebras and topology, in Algebras in analysis (Academic Press, New York, 1975), 118–186. Google Scholar

[56] 56. Valette, A., Minimal projections, integrable representations and property (T), Arch. Math. (Basel) 43 (1984), 397–406. Google Scholar

[57] 57. Warfield, R. B., Cancellation of modules and groups and stable range of endomorphism rings, Pacific J. Math. 91 (1980), 457–485. Google Scholar

[58] 58. Weil, A., Sur certains groupes d'opérateurs unitaires, Acta Math. Ill (1964), 143–21 1. Google Scholar

[59] 59. Weyl, H., Uber die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77 (1916), 313–352. Google Scholar

[60] 60. Zeller-Meier, G., Produits croisés d'une C*-algèbre par un groupe d'automorphismes, J. Math, pures appl. 47 (1968), 101–239. Google Scholar

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