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.

Keywords and Phrases:

Full text: dvi.gz 38 k, dvi 106 k, ps.gz 289 k, pdf 215 k.

Home Page of DOCUMENTA MATHEMATICA