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

TL;DR

This paper investigates restriction theorems for the $p$-analog of the Fourier-Stieltjes algebra, establishing conditions under which restriction maps are surjective isometries or have dense ranges.

Contribution

It proves that restriction maps are surjective isometries for open subgroups and characterizes their density in certain compact or SIN subgroups.

Findings

Restriction map is a surjective isometry for open subgroups.

Range of restriction map is dense in $B_p(H)$ for specific compact subgroups.

Provides new insights into the structure of $p$-analog Fourier-Stieltjes algebras.

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.