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Projective Modules over Higher-Dimensional Non-Commutative Tori

Published online by Cambridge University Press:  20 November 2018

Marc A. Rieffel
Marc A. Rieffel*
Affiliation:
University of California, Berkeley, California
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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].)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 2 , 01 April 1988 , pp. 257 - 338
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Anderson, J. and Paschke, W., The rotation algebra, preprint.Google Scholar
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. 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. Blackadar, B., Notes on the structure of projections in simple C*-algebras, Semesterbericht Functionalanalysis, Tubingen, Wintersemester (1982/1983).Google Scholar
5. Bourbaki, N., Algèbre multilineaire, elements de mathématique (Hermann, Paris, 1958).Google Scholar
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. 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. Connes, A., C*-algèbres et géométrie différentielle, C. R. Acad. Se. Paris 290 (1980), 599–604.Google Scholar
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. 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. Connes, A., The Chern character in K homology, Publ. Math. I.H.E.S. 62 (1986), 41–93.Google Scholar
12. Connes, A., De Rham homology and non commutative algebra, Publ. Math. I.H.E.S. 62 (1986), 94–144.Google Scholar
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. Connes, A. and Takesaki, M., The flow of weights on factors of type III, Tôhoku Math. J. 29 (1977), 473–575.Google Scholar
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. 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. 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. Green, P., The local structure of twisted covariance algebras, Acta Math. 140 (1978), 191–250.Google Scholar
19. Elliott, G. A., Square-integrable representations and the dual topology, J. Funct. Anal. 35 (1980), 279–294.Google Scholar
20. Greub, W., Halperin, S. and Vanstone, R., Connections, curvature, and cohomology, Vol. II (Academic Press, New York, 1973).Google Scholar
21. Herman, R. H. and Vaserstein, L. N., The stable range of C* -algebras, Invent. Math. 77 (1984), 553–555.Google Scholar
22. Hewitt, E. and Ross, K. A., Abstract harmonic analysis, I (Springer-Verlag, Berlin-Gottingen-Heidelberg, 1963).Google Scholar
23. Husemoller, D., Fibre bundles (Springer-Verlag, New York-Heidelberg-Berlin, 1966).Google Scholar
24. Igusa, J., Theta functions (Springer-Verlag, Berlin-Heidelberg-New York, 1972).Google Scholar
25. Kleppner, A., Multipliers on Abelian groups, Math. Ann. 158 (1965), 11–34.Google Scholar
26. Kumjian, A., On localizations and simple C*-algebras, Pacific J. Math. 112 (1984), 141–192.Google Scholar
27. Lang, S., Algebra (Addison-Wesley, Reading, Mass., 1965).Google Scholar
28. Loomis, L. H., An introduction to abstract harmonic analysis (Van Nostrand, New York, 1953).Google Scholar
29. Mackey, G. W., Unitary representations of group extensions, I, Acta Math. 99 (1958), 265–311.Google Scholar
30. Menai, P. and Moncasi, J., On regular rings with stable range 2, J. Pure Appl. Algebra 24 (1982), 25–40.Google Scholar
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. Ozeki, H., Chern classes of projective modules, Nagoya Math. J. 23 (1963), 121–152.Google Scholar
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. Packer, J. A., C*-algebras generated by projective representations of the discrete Heisenberg group, I, II, preprints.Google Scholar
35. Pedersen, G. K., C*-algebras and their automorphism groups, London Math. Soc. Monographs 14 (Academic Press, London, 1979).Google Scholar
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. 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. 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. Riedel, N., Classification of the C*-algebras associated with minimal rotations, Pacific J. Math. 101 (1982), 153–161.Google Scholar
40. Riedel, N., On the topological stable rank of irrational rotation algebras, J. Operator Theory 13 (1985), 143–150.Google Scholar
41. Rieffel, M. A., On the uniqueness of the Heisenberg commutation relations, Duke Math. J. 39 (1972), 745–752.Google Scholar
42. Rieffel, M. A., Induced representations of C*-algebras, Adv. Math. 13 (1974), 176–257.Google Scholar
43. Rieffel, M. A., Commutation theorems and generalized commutation relations, Bull. Soc. Math. France 104 (1976), 205–224.Google Scholar
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. 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. Rieffel, M. A., C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415–429.Google Scholar
47. Rieffel, M. A., Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257(1981), 403–418.Google Scholar
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. 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. 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. 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. 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. 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. Swan, R., Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264–277.Google Scholar
55. Taylor, J. L., Banach algebras and topology, in Algebras in analysis (Academic Press, New York, 1975), 118–186.Google Scholar
56. Valette, A., Minimal projections, integrable representations and property (T), Arch. Math. (Basel) 43 (1984), 397–406.Google Scholar
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. Weil, A., Sur certains groupes d'opérateurs unitaires, Acta Math. Ill (1964), 143–21 1.Google Scholar
59. Weyl, H., Uber die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77 (1916), 313–352.Google Scholar
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