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 $\boldsymbol R^{2d}$, assumed to be periodic in the first $d$ variables, we show that we get a strict deformation quantization.