$p$-Fourier algebras on compact groups

Let $G$ be a compact group. For $1\leq p\leq\infty$ we introduce a class of Banach function algebras $\mathrm{A}^p(G)$ on $G$ which are the Fourier algebras in the case $p=1$, and for $p=2$ are certain algebras discovered in \cite{forrestss1}. In the case $p\not=2$ we find that $\mathrm{A}^p(G)\cong \mathrm{A}^p(H)$ 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.