$L^p$-Fourier and Fourier-Stieltjes algebras for locally compact groups

Abstract

Let $G$ be a locally compact group and $1\leq p<\infty$. A continuous unitary representation $\pi\!: G\to B(\mathcal{H})$ of $G$ is an $L^p$-representation if the matrix coefficient functions $s\mapsto \langle \pi(s)x,x\rangle$ lie in $L^p(G)$ for sufficiently many $x\in \mathcal{H}$. Brannan and Ruan defined the $L^p$-Fourier algebra $A_{L^p}(G)$ to be the set of matrix coefficient functions of $L^p$-representations. Similarly, the $L^p$-Fourier-Stieltjes algebra $B_{L^p}(G)$ is defined to be the weak*-closure of $A_{L^p}(G)$ in the Fourier-Stieltjes algebra $B(G)$. These are always ideals in the Fourier-Stieltjes algebra containing the Fourier algebra. In this paper we investigate how these spaces reflect properties of the underlying group and study the structural properties of these algebras. As an application of this theory, we characterize the Fourier-Stieltjes ideals of $SL(2,\mathbb R)$.