Principal non-commutative torus bundles

Abstract

Abstract. In this paper we study continuous bundles of C*-algebras which are noncommutative 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 T n -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 T n -bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T -duals.

PARAMETRIZED STRICT DEFORMATION QUANTIZATION OF C*-BUNDLES AND HILBERT C*-MODULES

“…That is, the noncommutative torus bundles (NCT bundles) of this section strictly include the noncommutative principal torus bundles (NCPT bundles) of [6,9].…”

“…A C * -bundle A(X ) over X in the sense of [6] is exactly a C 0 (X )-algebra in the sense of Kasparov [12]. That is, A(X ) is a C * -algebra together with a nondegenerate * -homomorphism [3] Parametrized strict deformation quantization of C * -algebras 27 called the structure map, where Z M(A) denotes the centre of the multiplier algebra M(A) of A.…”

“…For example, we generalize the main application of our results in [9] (as well as those in [6]). More precisely, suppose that A(X ) is a C * -bundle over a locally compact space X with a fibrewise action of a torus T , and that A(X ) T ∼ = CT(X, H 3 ), where CT(X, H 3 ) is a continuous trace algebra with spectrum X and Dixmier-Douady class H 3 ∈ H 3 (X ; Z).…”

“…The particular version of Rieffel's theory that was generalized in [9] was due to Kasprzak [13] and Landstad [14,15]. The results in [9] were used to classify noncommutative principal torus bundles as defined by Echterhoff et al [6], as parametrized strict deformation quantizations of principal torus bundles. These arise as special cases of the continuous fields of noncommutative tori that appear as T -duals to space-times with background H-flux in [16,17].…”

Fibrations with noncommutative fibers

“…We shall recall the basic constructions and properties of C 0 (X)-algebras in the preliminary section below. We refer to [8] for further notations concerning C 0 (X)-algebras.The main problem when studying bundles from the topological point of view is to provide good topological invariants which help to understand the local and global structure of the bundles. A good example is given by the class of separable continuous-trace C*-algebras, which are, up to Morita equivalence, just the section algebras of locally trivial bundles over X with fibres the compact This work was partially supported by the Deutsche Forschungsgemeinschaft (SFB 478).…”