Principal noncommutative torus bundles

Siegfried Echterhoff; Ryszard Nest; Herve Oyono-Oyono

Principal noncommutative torus bundles

Siegfried Echterhoff, Ryszard Nest, Herve Oyono-Oyono

Department of Mathematical Sciences

Abstract

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 trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to 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 torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T-duals.

Original language	English
Journal	Proceedings of the London Mathematical Society
ISSN	0024-6115
Publication status	Published - 2008

Cite this

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title = "Principal noncommutative torus bundles",

abstract = "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 trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to 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 torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess {"}classical{"} T-duals.",

author = "Siegfried Echterhoff and Ryszard Nest and Herve Oyono-Oyono",

note = "Keywords: math.KT; math.OA; 19K35; 46L55; 46L80; 46L85",

year = "2008",

language = "English",

journal = "Proceedings of the London Mathematical Society",

issn = "0024-6115",

publisher = "Oxford University Press",

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TY - JOUR

T1 - Principal noncommutative torus bundles

AU - Echterhoff, Siegfried

AU - Nest, Ryszard

AU - Oyono-Oyono, Herve

N1 - Keywords: math.KT; math.OA; 19K35; 46L55; 46L80; 46L85

PY - 2008

Y1 - 2008

N2 - 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 trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to 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 torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T-duals.

AB - 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 trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to 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 torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T-duals.

M3 - Journal article

SN - 0024-6115

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

ER -

Principal noncommutative torus bundles

Abstract

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