Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-06-08T14:40:50.610Z Has data issue: false hasContentIssue false
  • English
  • Français

A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups

Published online by Cambridge University Press:  20 November 2018

Albert Jeu-Liang Sheu
Albert Jeu-Liang Sheu*
Affiliation:
University of Kansas, Lawrence, Kansas
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projective modules over A with direct summation as the binary operation. The semigroup gives an order structure on K0(A) and is usually called the positive cone in K0(A).

Around 1980, the work of Pimsner and Voiculescu [18] and of A. Connes [4] provided effective ways to compute the K-groups of C*-algebras. Then the classification of finitely generated projective modules over certain unital C*-algebras up to stable isomorphism could be done by computing their K0-groups as ordered groups. Later on, inspired by A. Connes's development of non-commutative differential geometry on finitely generated projective modules [2], the deeper question of classifying such modules up to isomorphism and hence the so-called cancellation question were raised (cf. [21] ).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 2 , 01 April 1987 , pp. 365 - 427
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Atiyah, M. F., K-theory (Benjamin, New York, 1964).Google Scholar
2. Connes, A., C*-algèbres et géométrie différentielle, C.R. Acad. Sci. Paris 290 (1980), 599–604.Google Scholar
3. Connes, A., A survey of foliations and operator algebras, Proceedings of Symposia in Pure Mathematics 38 (1982), Part 1, 521–628.CrossRefGoogle Scholar
4. 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
5. Cuntz, J., K-theory for certain C*-algebras, Ann. of Math. 113 (1981), 181–197.Google Scholar
6. Dixmier, J., C*-algebras (North-Holland, Amsterdam-New York-Oxford, 1977).Google Scholar
7. Fell, J. M. G., The structure of operator fields, Acta Math. 106 (1961), 233–280.Google Scholar
8. Green, P., C*-algebras of transformation groups with smooth orbit space, Pacific J. Math. 72 (1977), 71–97.Google Scholar
9. Hu, S. T., Homotopy theory (Academic Press, New York, 1959).Google Scholar
10. Hochschild, G., The structure of Lie groups (Holden-Day, San Francisco-London-Amsterdam, 1965).Google Scholar
11. Husemoller, D., Fibre bundles (McGraw-Hill, New York, 1966).CrossRefGoogle Scholar
12. Karoubi, M., K-theory (Springer-Verlag, Berlin-Heidelberg-New York, 1978).CrossRefGoogle Scholar
13. Kasparov, G. G., Operator K-functor and extensions of C*-algebras, Izv. Akad. Nauk, SSSR. Ser. Mat. 44 (1980), 571–636.Google Scholar
14. Kirillov, , Unitary representations of nilpotent Lie groups, Russian Math. Surveys 17 (1962), 53–104.Google Scholar
15. Lee, R. Y., Full algebras of operator fields trivial except at one point, Indiana U. Math. J. 26 (1977), 351–372.Google Scholar
16. Mackey, G. W., The theory of unitary group representations (The University of Chicago Press, Chicago-London, 1976).Google Scholar
17. Pedersen, G., C*-algebras and their automorphism groups (Academic Press, London-New York-San Francisco, 1979).Google Scholar
18. Pimsner, M. 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
19. Pukanszky, L., Leçons sur les représentations des groupes (Dunod, Paris, 1966).Google Scholar
20. Rieffel, M. A., Dimension and stable rank in the K-theory of C*-algebras, Proc. London Math. Soc. 46 (1983), 301–333.Google Scholar
21. Rieffel, M. A., The cancellation theorem for projective modules over irrational rotation C*-algebras, Proc. London Math. Soc. 47 (1983), 285–302.Google Scholar
22. Rosenberg, J., The role of K-theory in non-commutative algebraic topology, Amer. Math. Soc. Contemporary Math. 70 (1982), 155–182.Google Scholar
23. Rosenberg, J., Homological invariants of extensions of C*-algebras, Proceedings of Symposia in Pure Mathematics 38 (1982), Part 1, 35–75.Google Scholar
24. Rosenberg, J. and Schochet, C., The classification of extensions of C*-algebras, Bull. Amer. Math. Soc. 4 (1981), 105–110.Google Scholar
25. Rosenberg, J. and Schochet, C., The Künneth and the universal coefficient theorem for Kasparov's generalized K-functor, preprint.Google Scholar
26. Sheu, A. J. L., The cancellation property for modules over the group C*-algebras of certain nilpotent Lie groups, thesis, University of California at Berkeley (1985).Google Scholar
27. Sheu, A. J. L., Classification of projective modules over the unitized group C*-algebras of certain solvable Lie groups, preprint.Google Scholar
28. Steenrod, N., The topology of fiber bundles (Princeton University Press, Princeton, New Jersey, 1951).CrossRefGoogle Scholar
29. Swan, R. W., Vector bundles and projective modules, Trans. Amer. Math. Soc. 705 (1962), 264–277.Google Scholar
30. Taylor, J., Banach algebras and topology, in Algebras in analysis (Academic Press, New York, 1975).Google Scholar
31. Voiculescu, D., Remarks on the singular extension in the C*-algebra of the Heisenberg group, J. Operator Theory 5 (1981), 147–170.Google Scholar
32. Kervaire, M. A., Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960), 161–169.Google Scholar