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

O’Uchi, Moto

Math. Scand. 58, 69-76 (1986).

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.

Reviewer: Dumitru Motreanu (Perpignan)

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

Keywords:

smooth manifold; transverse submanifold; holonomy groupoid; action of a group; \(C^*\)-algebras; homogeneous covering map; crossed product; Anosov foliations