Deformation Quantization via Fell Bundles

TL;DR

This paper introduces a unified method for deforming C*-algebras using Fell bundles, encompassing various known non-commutative geometries and establishing a framework for strict deformation quantization in specific cases.

Contribution

It presents a general deformation approach for C*-algebras via Fell bundles, unifying several examples of non-commutative spaces and providing conditions for strict deformation quantization.

Findings

Unified perspective on deformation of C*-algebras using Fell bundles

Examples include non-commutative spheres, lens spaces, and quantum Heisenberg manifolds

Establishment of strict deformation quantization under periodic R^{2d} actions

Abstract

A method for deforming C*-algebras is introduced, which applies to C*-algebras that can be described as the cross-sectional C*-algebra of a Fell bundle. Several well known examples of non-commutative algebras, usually obtained by deforming commutative ones by various methods, are shown to fit our unified perspective of deformation via Fell bundles. Examples are the non-commutative spheres of Matsumoto, the non-commutative lens spaces of Matsumoto and Tomiyama, and the quantum Heisenberg manifolds of Rieffel. In a special case, in which the deformation arises as a result of an action of R^{2d}, assumed to be periodic in the first d variables, we show that we get a strict deformation quantization.