On (non)commutative products of functions on the sphere

TL;DR

This paper explores the properties of a specific noncommutative product of functions on the sphere, revealing its graded commutativity and connection to deformation quantization, with implications for semiclassical analysis.

Contribution

It completes the construction of the skewed product on the sphere and links it to geometric and deformation quantization frameworks.

Findings

The skewed product is Z_2 graded commutative.

It is nontrivial only for functions with correct antipodal parity.

Provides a geometric interpretation of Rieffel's deformation quantization of the sphere.

Abstract

We investigate the commutativity of global products of functions on the two-sphere from the point of view of a construction started in [RT] and named the skewed product. We complete the construction of the skewed product of functions on the sphere and show that it is Z_2 graded commutative and nontrivial only as a product of functions with correct parity under the antipodal mapping. These properties are valid for a more general class of integral products of functions on the sphere, with integral kernel of a special WKB-type that is natural from semiclassical considerations. We argue that our construction provides a simple geometrical explanation for an old theorem by Rieffel [Rf] on equivariant strict deformation quantization of the two-sphere.