# Weak containment by restrictions of induced representations

## TL;DR

This paper investigates the properties of QSIN groups, showing how restrictions of induced representations relate to weak containment, and explores implications for the structure of their C*-algebras.

## Contribution

It establishes a key weak containment result for representations of QSIN groups and analyzes the consequences for their C*-algebra properties.

## Key findings

- If G is a QSIN group with a copy of F_2, then C*(G) is not locally reflexive.
- C*_r(G) does not have the local lifting property.
- Provides tools for studying Fourier and Fourier-Stieltjes spaces.

## 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 $B_\pi$.