The \(C^*\)-algebras of codimension one foliations without holonomy. (English) Zbl 0581.46057

Natsume, Toshikazu

Math. Scand. 56, 96-104 (1985).

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

Keywords:

codimension one folitation without holonomy on a closed manifold; free Abelian group; Novikov transformation; \(C^*\)-algebra; stably isomorphic to the crossed product