Partial classification of the Baumslag-Solitar group von Neumann algebras
Summary: We prove that the rational number \(|n/m|\) is an invariant of the group von Neumann algebra of the Baumslag-Solitar group \(\text{BS}(n,m)\). More precisely, if \(L(\text{BS}(n,m))\) is isomorphic with \(L(\text{BS}(n',m'))\), then \(|n'/m'| = |n/m|^{\pm 1}\). 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 \(L(\text{BS}(n,m))\) a canonical equivalence relation of type III\(_{|n/m|}\).
scientific article; zbMATH DE number 6358805