Restriction theorems for the $p$-analog of the Fourier-Stieltjes algebra

Abstract

For a locally compact group $G$ and $1 < p < \infty,$ let $B_p(G)$ denote the $p$-analog of the Fourier-Stieltjes algebra $B(G) \, (\text{or} \, B_2(G))$. Let $r: B_p(G) \to B_p(H)$ be the restriction map given by $r(u) = u|_H$ for any closed subgroup $H$ of $G.$ In this article, we prove that the restriction map $r$ is a surjective isometry for any open subgroup $H$ of $G.$ Further, we show that the range of the map $r$ is dense in $B_p(H)$ when $H$ is either a compact normal subgroup of $G$ or compact subgroup of an [SIN]$_H$-group.