The regular representation of Neretin groups is factorial

TL;DR

This paper proves that the left regular representation of Neretin groups is factorial, introducing a new criterion for factoriality in totally disconnected groups and analyzing the type of the associated factors.

Contribution

It provides the first example of a non-discrete simple group with a factorial regular representation and establishes a new criterion for factoriality in totally disconnected groups.

Findings

Left regular representation of Neretin groups is factorial

Introduces a new criterion for factoriality in totally disconnected groups

Determines the type of the factor L(G) for groups satisfying the criterion

Abstract

We show that the left regular representation of Neretin groups is factorial, providing the first example of a non-discrete simple group with this property. This is based on a new criterion of factoriality for totally disconnected groups. For groups G satisfying the criterion, we determine the type of the factor L(G) and derive factoriality results for crossed products associated to G-actions on von Neumann algebras.