On ergodicity of linear actions on $\mathbb{R}^n$ and factoriality of group von Neumann algebras

TL;DR

This paper identifies conditions under which group actions on Lie-type abelian groups lead to factorial group von Neumann algebras, expanding the class of known factors.

Contribution

It provides new natural conditions on group actions that ensure factoriality of the associated group von Neumann algebras, with numerous examples.

Findings

Conditions for factoriality of group von Neumann algebras

Large class of groups with factorial von Neumann algebras

Examples of factors from semidirect product groups

Abstract

We give some natural conditions on actions of discrete countable groups on abelian locally compact groups of Lie type that imply factoriality of the group von Neumann algebras of their semidirect products. This allows us to give a fairly large class of examples of locally compact groups whose group von Neumann algebras are factors.