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Proceedings of the London Mathematical Society Advance Access originally published online on November 25, 2008
Proceedings of the London Mathematical Society 2009 99(1):1-31; doi:10.1112/plms/pdn050
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© 2008 London Mathematical Society

Principal non-commutative torus bundles

Siegfried Echterhoff

Westfälische Wilhelms-Universität Münster
Mathematisches Institut
Einsteinstr. 62
D-48149 Münster
Germany

Ryszard Nest

SNF Center in Non-Commutative
Geometry
University of Copenhagen
Universitetsparken 5
DK-2100 KBH. Ø
Denmark
rnest@math.ku.dk

Herve Oyono-Oyono

Université Blaise Pascal de
Clermont-Ferrand
Laboratoire de Mathématiques
Plateau des Cézeaux
63177 Aubière Cedex
France
oyono@math.cnrs.fr

Received 4 July 2007.

In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally RKK-trivial. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to Tn-equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal Tn-bundles with H-flux, as studied by Mathai and Rosenberg which possess ‘classical’ T-duals.

2000 Mathematics Subject Classification 19K35, 46L55, 46L80, 46L85 (primary), 14DXX, 46L25, 58B34, 81R60, 81T30 (secondary).

This work was partially supported by the Deutsche Forschungsgemeinschaft (SFB 478)


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