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

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 ${\mathrm{II}}_1$ factor is prime, not solid, and does not have any Cartan subalgebra.