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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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