p-Fourier algebras on compact groups

Let G be a compact group. For 1 <= p <= infinity we introduce a class of Banach function algebras A(p)(G) on G which axe the Fourier algebras in the case p = 1, and for p = 2 are certain algebras discovered by Forrest, Samei and Spronk. In the case p not equal 2 we find that A(p)(G) congruent to A(p)(H) if and only if G and H are isomorphic compact groups. These algebras admit natural operator space structures, and also weighted versions, which we call p-Beurling-Fourier algebras. We study various amenability and operator amenability properties, Arens regularity and representability as operator algebras. For a connected Lie G and p > 1, our techniques of estimation of when certain p-Beurling-Fourier algebras are operator algebras rely more on the fine structure of G, than in the case p = 1. We also study restrictions to subgroups. In the case that G = SU(2), restrict to a torus and obtain some exotic algebras of Laurent series. We study amenability properties of these new algebras, as well.