$p$-Fourier algebras on compact groups

TL;DR

This paper introduces and analyzes a family of Banach function algebras called $p$-Fourier algebras on compact groups, exploring their structure, operator space properties, and amenability, with special cases like $SU(2)$ providing new algebraic insights.

Contribution

It defines $p$-Fourier algebras for all $p$, characterizes their isomorphisms, and investigates their operator space structures, amenability, and regularity, extending Fourier analysis on compact groups.

Findings

$A^p(G)$ is isomorphic iff $G$ and $H$ are isomorphic for $p eq 2$.

$A^p(G)$ admits natural operator space structures and weighted versions.

Certain $p$-Beurling-Fourier algebras are shown to be operator algebras under specific conditions.

Abstract

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.