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

Abstract

In this paper, we review the parametrized strict deformation quantization of C * -bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H 3 -twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrized strict deformation quantization. Rieffel's basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrized strict deformation quantization of Hilbert C * -modules over C * -bundles with fibrewise torus action.2010 Mathematics subject classification: primary 58B34; secondary 81S10, 46L87, 16D90, 53D55.

Moduli spaces of instantons on toric noncommutative manifolds

“…The algebra C ∞ (S 4 ) carries a smooth action of the two-torus T 2 given on generators by 13) where (e 2πit 1 , e 2πit 2 ) ∈ T 2 . This action makes C ∞ (S 4 ) into an algebra in the category V 2 .…”

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

“…As before, we strictly deform quantize with respect to the 2-torus action to get a noncommutative principal torus bundle (NCTP) with total space M 6 θ and constant deformation parameter θ, cf. [20,21]. NCTP bundles occur in the study of T-duality in a background flux [25,26,27] in string theory, and was first described in terms of strict deformation quantization in [20,21].…”

Covariant Fields of C*-Algebras under Rieffel Deformation

“…After the present work was completed, we learnt of the recent articles [9,10] in which a statement comparable with our Theorem 4.3 is included (essentially without proof) with interesting applications. These papers reply on Kasprzak's reformulation [11] of the Rieffel deformation (based on Landstad's characterization of crossed products).…”

“…These papers reply on Kasprzak's reformulation [11] of the Rieffel deformation (based on Landstad's characterization of crossed products). However, in [9,10] there is no comment about how nondegenerecy is proved, and this is crucial in the definition of a C(T )-algebra (cf. Definition 3.1).…”

“…Our detailed proof based on the more traditional approach is intended to support future work on spectral theory of pseudodifferential operators (see also [1,14]), as well as applications to quantum mechanics. The lower-semi-continuity part (contained in Section 4), often described as difficult, is not treated in [9,10] and seems not to have correspondence in the literature.…”