Fuzzy sphere: Star product induced from generalized squeezed states

A family of states built from the uncertainty principle on the fuzzy sphere has been shown to reproduce the stereographic projection in the large $j$ limit. These generalized squeezed states are used to construct an associative star product which involves a finite number of derivatives on its primary functional space. It is written in terms of a variable on the complex plane. We show that it actually coincides with the one found by Gross and Presnajder in the simplest cases, endowing the later with a supplementary physical interpretation. We also show how the spherical harmonics emerge in this setting.