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

For a locally compact group $G$ and $1<p<\mathrm{\infty },$ 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 $r(u)=u{\mathrm{\mid }}_H$ for any closed subgroup $H$ of $G\mathrm{.}$ In this article, we prove that the restriction map $r$ is a surjective isometry for any open subgroup $H$ of $G\mathrm{.}$ 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.