An Introduction to Crossed Products by Minimal Homeomorphisms

Abstract

In this section, we discuss free and essentially free minimal actions of countable discrete groups on compact metric spaces, with emphasis on minimal homeomorphisms (actions of ℤ). We give two simplicity proofs, using very different methods. One works for free minimal actions, and the method gives further information, as well as some information when the action is not minimal; see Theorems 11.1.20 and 11.1.22. The second proof is a special case of a more general simplicity theorem; the case we prove allows some simplification of the argument. Our theorem is Theorem 11.1.10, and its proof is given before Theorem 11.1.25. The full theorem is stated as Theorem 11.1.25. Both proofs end with an argument related to the proof that Kishimoto’s condition (see Definition 10.4.20) implies simplicity of the crossed product (see Theorem 10.4.22), but the two proofs use quite different routes to get there.