#### DOCUMENTA MATHEMATICA, Vol. 19 (2014), 629-645

Niels Meesschaert and Stefaan Vaes

Partial Classification of the Baumslag-Solitar Group Von Neumann Algebras

\noindent 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 $|n'/m'| = |n/m|^{\pm 1}$. 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 $L(\BS(n,m))$ a canonical equivalence relation of type III$_{|n/m|}$.

2010 Mathematics Subject Classification: Primary 46L36; Secondary 20E06, 22D25.

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