  For $\MvN$ a separable, purely infinite von Neumann algebra with almost
periodic weight $\phi$, a decomposition of $\MvN$ 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 $\Real_+$ on the injective II$_\infty$ factor. Finally, another model of the
same action, due to work of Aubert and explained by Jones, is described.
