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For $\MvN$ a separable, purely infinite von Neumann algebra with almostperiodic weight $\phi$, a decomposition of $\MvN$ as a crossed product of asemifinite von Neumann algebra by a trace--scaling action of a countableabelian group is given. Then Takasaki's continuous decomposition of the samealgebra is related to the above discrete decomposition via Takesaki's notion ofinduced action, but here one induces up from a dense subgroup. The aboveresults are used to give a model for the one--parameter trace--scaling actionof $\Real_+$ on the injective II$_\infty$ factor. Finally, another model of thesame action, due to work of Aubert and explained by Jones, is described.

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