# Weak separation properties for closed subgroups of locally compact groups

## TL;DR

This paper investigates various separation properties of closed subgroups within locally compact groups, linking them to approximation and amenability conditions, and introduces techniques to analyze these properties through projections and convolution methods.

## Contribution

It characterizes the $H$-separation property using bounded approximate indicators and relates it to amenability and weak amenability, providing new insights into subgroup approximation properties.

## Key findings

- The $H$-separation property is characterized by bounded approximate indicators.
- A discretized analogue of the $H$-separation property is established.
- Conditions involving weak amenability and projections imply the approximability of characteristic functions.

## Abstract

Three separation properties for a closed subgroup $H$ of a locally compact group $G$ are studied: (1) the existence of a bounded approximate indicator for $H$, (2) the existence of a completely bounded invariant projection of $VN\left(G\right)$ onto $VN_{H}\left(G\right)$, and (3) the approximability of the characteristic function $\chi_{H}$ by functions in $M_{cb}A\left(G\right)$ with respect to the weak$^{*}$ topology of $M_{cb}A\left(G_{d}\right)$. We show that the $H$-separation property of Kaniuth and Lau is characterized by the existence of certain bounded approximate indicators for $H$ and that a discretized analogue of the $H$-separation property is equivalent to (3). Moreover, we give a related characterization of amenability of $H$ in terms of any group $G$ containing $H$ as a closed subgroup. The weak amenability of $G$ or that $G_{d}$ satisfies the approximation property, in combination with the existence of a natural projection (in the sense of Lau and \"Ulger), are shown to suffice to conclude (3). Several consequences of (2) involving the cb-multiplier completion of $A\left(G\right)$ are given. Finally, a convolution technique for averaging over the closed subgroup $H$ is developed and used to weaken a condition for the existence of a bounded approximate indicator for $H$.