Partial classification of the Baumslag-Solitar group von Neumann algebras

Abstract

We prove that the rational number $|n/m|$ is an invariant of the group von Neumann algebra of the Baumslag–Solitar group $\mathrm{BS}(n,m)$. More precisely, if $L(\mathrm{BS}(n,m))$ is isomorphic with $L(\mathrm{BS}(n^{\prime },m^{\prime }))$, then $|n^{\prime }/m^{\prime }|=|n/m{|}^{\pm 1}$. We obtain this result by associating to abelian, but not maximal abelian, subalgebras of a II${}_1$ factor, an equivalence relation that can be of type III. In particular, we associate to $L(\mathrm{BS}(n,m))$ a canonical equivalence relation of type III${}_{|n/m|}$.