On (non)commutative products of functions on the sphere

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.