Uniqueness of rotation invariant norms

Abstract

If $N\ge 2$, then there exist finitely many rotations of the sphere $\mathbb{S}^N$ such that the set of the corresponding rotation operators on $L^p(\mathbb{S}^N)$ determines the norm topology for $1 \leq p \leq\infty, p \neq 1$. For $N=1$ the situation is different: the norm topology of $L^2(\mathbb{S}^1)$ cannot be determined by the set of operators corresponding to the rotations by elements of any `thin' set of rotations of $\mathbb{S}^1$.