  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 $SL(2,\mathbb R)$.
