  In their 1976 paper, Nagel and Rudin characterize the closed unitarily and
M\"obius invariant spaces of continuous and $L^p$-functions on a sphere, for
$1\leq p<\infty$. In this paper we provide an analogous characterization for
the weak*-closed unitarily and M\"obius invariant spaces of
$L^\infty$-functions on a sphere. We also investigate the weak*-closed
unitarily and M\"obius invariant algebras of $L^\infty$-functions on a sphere.
