## Coverings of foliations and associated $$C^*$$-algebras.(English)Zbl 0598.58032

Let $$(V,F)$$ and $$(V',F')$$ be $$C^{\infty}$$ foliations with the associated $$C^*$$-algebras $$C^*(V,F)$$ and $$C^*(V',F')$$. The author introduces the notion of homogeneous covering map of $$(V,F)$$ onto $$(V',F')$$ with the structure group $$Z$$. Then it is shown that such a map determines an action $$\beta$$ of $$Z$$ on $$C^*(V,F)$$ such that the reduced crossed product of $$C^*(V,F)$$ by $$\beta$$ is isomorphic to $$C^*(V',F')$$. This result is applied for studying two Anosov foliations.

### MSC:

 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 57R30 Foliations in differential topology; geometric theory 57S25 Groups acting on specific manifolds 46L35 Classifications of $$C^*$$-algebras 46L40 Automorphisms of selfadjoint operator algebras
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