Restriction Theorems for the p-Analog of the Fourier–Stieltjes Algebra

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 \(r: B_{p}(G) \rightarrow B_p(H)\) 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.