Weak containment by restrictions of induced representations

Abstract

A QSIN group is a locally compact group $G$ whose group algebra $L^1(G)$ admits a quasi-central bounded approximate identity. Examples of QSIN groups include every amenable group and every discrete group. It is shown that if $G$ is a QSIN group, $H$ is a closed subgroup of $G$, and $\pi$ is a unitary representation of $H$, then $\pi$ is weakly contained in $(\mathrm{Ind}_H^G\pi)|_H$. This provides a powerful tool in studying the C*-algebras of QSIN groups. In particular, it is shown that if $G$ is a QSIN group which contains a copy of $\mathbb F_2$ as a closed subgroup, then $C^*(G)$ is not locally reflexive and $C^*_r(G)$ does not admit the local lifting property. Further applications are drawn to the "(weak) extendability" of Fourier spaces $A_\pi$ and Fourier-Stieltjes spaces $B_\pi$.