Niels Meesschaert and Stefaan Vaes

摘要

oindent 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$_{|n/m|}$.