Let $X$ be a non vanishing vector field on a compact manifold $M$ so we have a one dimensional foliation $F$ of $M$ with orbits of $X$. This foliation defines a $C^{*}$ algebra $C^{*}(F)$. On the other hand the flow of $X$ define an action of $\mathbb{R}$ on $C(M)$. So we have the $C^{*}$ algebra $C(M)\rtimes \mathbb{R}$. Are there some relations between these two $C^{*}$ algebras?
Comparision of two $C^{*}$ algebras associated to a non vanishing vector field on a compact manifold
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asked Apr 14, 2015 at 18:06
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It depends if you use the monodromy groupoid or the holonomy groupoid (Is there a more canonical choice for this construction ? ).
Basically, the monodromy groupoid is exactly the same as the action groupoid (whose convolution algebra the cross-product $C^*$-algebra) hence if you use this one the two $C^*$-algebras are going to be the same.
There is a canonical map from the monodromy groupoid to the holonomy groupoid which induces a comparaison morphisms between the two $C^*$-algebras (maybe just a bi-module if this morphism is not well behaved I haven't thought about this yet).
If your flow does not have any circular orbit then there is no distinction to be done.
Take a look to the case where $M =S^1$ to figure out what is the groupoid you are using/want to use.
answered Apr 14, 2015 at 19:28
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$\begingroup$ @SimonHenrey By $A\rtimes R$ I mean the completion of all compactly support continuos functions from R to A, with a natural convolution. By $C^{*}(F)$ I mean the standard C* algebra associated to the holonomy groupoid of a foliation, the C* algebra associated to (non Hausdorff manifold) Graph of a foliation. Now could you please more explain about such comparison? $\endgroup$ Commented Apr 15, 2015 at 8:08