Noncommutative principal torus bundles via parametrised strict deformation quantization

Abstract

Abstract. In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative definition of noncommutative non-principal torus bundles.

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

“…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).…”

“…If A(X ) is a H 3 -twisted NCP T -bundle over X , then we call A ∞ (X ) a fibrewise smooth H 3 -twisted noncommutative principal T -bundle (or fibrewise smooth H 3 -twisted NCP T -bundle) over X . In this paper, we are able to give a complete classification of fibrewise smooth H 3 -twisted NCP T -bundles over X via a parametrized version of Rieffel's theory of strict deformation quantization as derived in [9].…”

“…Parametrized strict deformation quantization of C * -bundles with fibrewise action of T In a nutshell, parametrized strict deformation quantization is a functorial extension of Rieffel's strict deformation quantization from algebras A to C(X )-algebras A(X ), and in particular to C * -bundles over X . Unlike Rieffel's deformation theory [21], the version in [9] starts with multipliers via the Landstad-Kasprzak approach. Here we review the theory for C * -bundles over X .…”

“…Parametrized strict deformation quantization of C * -bundles was introduced by the authors in an earlier paper [9], as a generalization of Rieffel's strict deformation quantization of C * -algebras [21,22]. The particular version of Rieffel's theory that was generalized in [9] was due to Kasprzak [13] and Landstad [14,15].…”

“…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].…”