Crossed product decompositions of a purely infinite von Neumann algebra with faithful, almost periodic weight

For $\text{MvN}$ a separable, purely infinite von Neumann algebra with almost periodic weight $\varphi$, a decomposition of $\text{MvN}$ as a crossed product of a semifinite von Neumann algebra by a trace–scaling action of a countable abelian group is given. Then Takasaki’s continuous decomposition of the same algebra is related to the above discrete decomposition via Takesaki’s notion of induced action, but here one induces up from a dense subgroup. The above results are used to give a model for the one–parameter trace–scaling action of ${\text{Real}}_{+}$ on the injective II${}_{\mathrm{\infty }}$ factor. Finally, another model of the same action, due to work of Aubert and explained by Jones, is described.