$p$-Fourier algebras on compact groups

Abstract

Let $G$ be a compact group. For $1\le p\le \infty$ we introduce a class of Banach function algebras ${\mathcal{A}}^p(G)$ on $G$ which are the Fourier algebras in the case $p=1$, and for $p=2$ are certain algebras discovered by Forrest, Samei and Spronk. In the case $p\ne 2$ we find that ${\mathcal{A}}^p(G)\cong {\mathcal{A}}^p(H)$ 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.