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I have a uniquely ergodic dynamical system preserving a finite ergodic measure (specifically, I have a nice aperiodic tiling space with an action of $\mathbb{R}^d$). Thus the transformation group von Neumann algebra it generates is a Type $II_\infty$ factor. If I now choose a transversal to the action and look at the groupoid of the transversal, will the corresponding groupoid VN algebra be a type $II_1$ factor, even if the $\mathbb{R}^d$ action does not reduce to a $\mathbb{Z}^d$ action on the transversal? I know that the $C^*$-algebras are Morita equivalent, but I don't know if the VN algebras are in some sense equivalent. Perhaps the transversal VN algebra is a full corner of the tiling VN algebra?

Edit: For reference, this paper http://jot.theta.ro/jot/archive/1987-017-001/1987-017-001-001.pdf by Muhly, Renault, and Williams shows that the $C^*$-algebras are Morita equivalent.

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I am not familiar with Von Neumann algebra of ergodic action so I am not sure that the following help, but if two C* algebra C and C' are Morita equivalence, then the Morita equivalence induce an equivalence between representation of C and C'. If R is a representation of C, and R' the corresponding representation of C' then the double comutant of C in B(R) is Morita equivalent (in the sense of VN algebra) to the double comutant of C' in B(R'). –  Simon Henry Jan 9 at 11:52
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