Deformation Quantization via Fell Bundles

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.