Certain Invariant Spaces of Bounded Measurable Functions on a Sphere

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.