# Ozawa's class $\mathcal S$ for locally compact groups and unique prime factorization

## TL;DR

This paper characterizes a class of locally compact groups with specific boundary actions, provides new examples, and establishes unique prime factorization results for their von Neumann algebras, also showing invariance under measure equivalence.

## Contribution

It introduces a new characterization of class $$ for locally compact groups, broadens the class with new examples, and proves unique prime factorization for associated von Neumann algebras.

## Key findings

- Characterization of class $$ via amenable actions on boundaries
- New examples of groups in class $$
- Unique prime factorization results for group von Neumann algebras

## Abstract

We study class $\mathcal S$ for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class $\mathcal S$ and prove unique prime factorization results for group von Neumann algebras of products of locally compact groups in this class. We also prove that class $\mathcal S$ is a measure equivalence invariant.