A note on the von Neumann algebra of a Baumslag-Solitar group

TL;DR

This paper investigates the properties of the von Neumann algebra associated with a Baumslag-Solitar group, revealing it is prime, not solid, and lacks Cartan subalgebras in certain cases.

Contribution

It provides new insights into the structure of the group von Neumann algebra of Baumslag-Solitar groups, especially in the non-amenable ICC case.

Findings

The ${ m II}_1$ factor is prime.

The algebra is not solid.

It has no Cartan subalgebra.

Abstract

We study qualitative properties of the group von Neumann algebra of a Baumslag-Solitar group. Namely, we prove that, in the non-amenable and {ICC} case, the associated ${\rm II}_1$ factor is prime, not solid, and does not have any Cartan subalgebra.