zbMATH — the first resource for mathematics

The $$C^*$$-algebras of codimension one foliations without holonomy. (English) Zbl 0581.46057
Let $${\mathcal F}$$ be a codimension one folitation without holonomy on a closed manifold M. To (M,$${\mathcal F})$$ is associated an action $$\alpha$$ of a free Abelian group $${\mathbb{Z}}^ k$$ on the circle $$S^ 1$$. This action is called the Novikov transformation of (M,$${\mathcal F})$$. The main result is that the $$C^*$$-algebra $$C^*(M,{\mathcal F})$$ of (M,$${\mathcal F})$$ is stably isomorphic to the crossed product $$C(S^ 1)\rtimes_{\alpha}{\mathbb{Z}}^ k$$.

MSC:
 46L55 Noncommutative dynamical systems 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 57R30 Foliations in differential topology; geometric theory 46L30 States of selfadjoint operator algebras
Full Text: