On the regular representation of solvable Lie groups with open coadjoint quasi-orbits

We obtain a Lie theoretic intrinsic characterization of the connected and simply connected solvable Lie groups whose regular representation is a factor representation. When this is the case, the corresponding von Neumann algebras are isomorphic to the hyperfinite \(\textrm{II}_\infty \) ${\text{II}}_{\infty }$ factor, and every Casimir function is constant. We thus obtain a family of geometric models for the standard representation of that factor. Finally, we show that the regular representation of any connected and simply connected solvable Lie group with open coadjoint orbits is always of type  \(\textrm{I}\) $\text{I}$ , though the group needs not be of type  \(\textrm{I}\) $\text{I}$ , and include some relevant examples.