Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-29T03:50:47.735Z Has data issue: false hasContentIssue false
  • English
  • Français

The K-theory of the C*-algebra of foliations by slope components

Published online by Cambridge University Press:  28 May 2008

Catherine Oikonomides
Catherine Oikonomides
Affiliation:
Department of Mathematics, Keio University, 3-14-1 Hiyoshi Kohoku-ku, Yokohama, Japan, [email protected].
Get access

Abstract

We compute the K-theory of the C*-algebra for a large class of foliations of the 3-torus, which contains in particular all smooth foliated circle bundles over the 2-torus. This generalizes a well-known result of Torpe. We show that the rank of the K-theory groups reflect part of the geometrical aspect of the foliation. To illustrate these results, we compute some concrete examples, including a case where both K-theory groups have infinite rank.

Keywords

FoliationsC*-algebrasK-theory
Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 2 , April 2009 , pp. 221 - 260
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Camacho, C. and Neto, A. Lins, Geometric Theory of Foliations, Birkhäuser (1985)Google Scholar
2.Candel, A., C*-algebras of proper foliations, Proc. Am. Math. Soc. 124 (1996), 899–905CrossRefGoogle Scholar
3.Connes, A., Sur la théorie non commutative de l'intégration, Lecture Notes in Math. 725 (1979), 19–143Google Scholar
4.Connes, A., A survey of foliations and operator algebras, in Operator algebras and applications, Proc. Symp. in Pure Math. A.M.S. 38 Part I (1982), 521–628Google Scholar
5.Connes, A., Non-commutative differential geometry. Part II: De Rham homology and non commutative algebra, Publ. Math. IHES 62 (1985), 257–360Google Scholar
6.Connes, A., Cyclic cohomology and the transverse fundamental class of a foliation, inGeometric methods in operator algebras, Araki, H. and Effros, G. ed. (1986), 52–144Google Scholar
7.Green, P., C*algebras of transformation groups with smooth orbit space, Pacific J. of Math. 72 (1977), 71–97Google Scholar
8.Hector, G., Groupoïdes, feuilletages et C*-algèbres, inGeometric study of foliations, Tokyo 1993, Mizutani, T. et al. ed. (1994), 3–34Google Scholar
9.Herman, M., Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES 49 (1978), 5–234CrossRefGoogle Scholar
10.Hilsum, M. and Skandalis, G., Stabilité des C*-algèbres de feuilletages, Ann. Inst. Fourier 33 (1983), 201–208Google Scholar
11.Kopell, N., Commuting diffeomorphisms, inGlobal Analysis, Proc. of Symp. in Pure Math. (1970), 165–184CrossRefGoogle Scholar
12.Moussu, R. and Roussarie, R., Relations de conjugaison et de cobordisme entre certains feuilletages, Publ. Math. IHES 43 (1974), 143–168CrossRefGoogle Scholar
13.Oikonomides, C., The Godbillon-Vey cyclic cocycle for PL-foliations, J. of Functional Analysis 234 Issue 1 (2006), 127–151CrossRefGoogle Scholar
14.Pimsner, M. and Voiculescu, D., Exact sequences for K-groups and Ext-groups of certain cross product C*algebras, J. Operator Theory 4 (1980), 93–118Google Scholar
15.Putnam, I., Schmidt, K. and Skau, C., C*-algebras associated with Denjoy homeomorphisms of the circle, J. Operator Theory 16 (1986), 99–126Google Scholar
16.Rieffel, M., C*-algebras associated with irrational rotations, Pacific J. of Math. 93 No. 2 (1981), 415–429Google Scholar
17.Rieffel, M., Applications of strong Morita equivalence to transformation group C*-algebras, Proc. Symp. Pure Math. 38 (1982), Part I, 299–310Google Scholar
18.Torpe, A-M., K-theory for the leaf space of foliations by Reeb components, J. of Functional Analysis 61 (1985), 15–71Google Scholar
19.Tsuboi, T., Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle, J. Math. Soc. Japan 47 No. 1 (1995), 1–30Google Scholar
20.Wegge-Olsen, N.E., K-theory and C*-algebras, a friendly approach, Oxford University press (1993)Google Scholar