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

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.