In this paper we characterize spaces of continuous and L p -functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work generalizes Nagel and Rudin's 1976 results concerning unitarily and Möbius invariant spaces of continuous and measurable functions defined on the unit sphere in C n .

We explore how the higher order Hochschild cohomology controls a deformation theory when the simplicial set models the 3-sphere. Besides generalizing to the d-sphere for any d ≥ 1 , we also investigate a deformation theory corresponding to the tertiary Hochschild cohomology, which naturally reduces to those studied for the secondary and usual Hochschild cohomologies under certain conditions.

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In their 1976 paper, Nagel and Rudin characterize the closed unitarily and Möbius invariant spaces of continuous and L p -functions on a sphere, for 1 ≤ p < ∞. In this paper we provide an analogous characterization for the weak*-closed unitarily and Möbius invariant spaces of L ∞ -functions on a sphere. We also investigate the weak*-closed unitarily and Möbius invariant algebras of L ∞ -functions on a sphere.