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 BS(n,m). More precisely, if L(BS(n,m)) is isomorphic with L(BS(n′,m′)), then |n′/m′| = |n/m|±1. We obtain this result by as-sociating to abelian, but not maximal abelian, subalgebras of a II1 factor, an equivalence relation that can be of type III. In particular, we associate to L(BS(n,m)) a canonical equivalence relation of type III|n/m|.