Star product on $L^2(S^n)$, $n=2,3,5$

Abstract

We consider the bounded linear operators with domain in the Hilbert space $L^2(S^n)$, $n=2,3,5$ and describe its symbolic calculus defined by the Berezin quantization. In particular, we derive an explicit formula for the composition of Berezin's symbols and thus a noncommutative invariant star product, which in turn is invariant under the action of the group $SU(2)$, $SU(2)\times SU(2)$ and $SU(4)$ on ${\mathbb{C}}^2$ , ${\mathbb{C}}^4$ and ${\mathbb{C}}^8$ respectively.