Topological Dynamical Systems and Regularity Properties of Reduced Crossed Product C*-algebras

In this work, we will explore the relation between topological dynamical systems and their reduced crossed product C ∗ -algebras. More precisely, we mainly study some dynamical properties and how they imply various of regularity properties of C ∗ -algebras, say, stably finiteness, pure infiniteness, finite nuclear dimension and Z-stability. Let α : G y X be a minimal free continuous action of an infinite countable amenable group on an infinite compact metrizable space. Under the hypothesis that the invariant ergodic probability Borel measure space EG(X) is compact and zero-dimensional, we show that the action α has the small boundary property. This partially answers an open problem in dynamical systems that asks whether a minimal free action of an amenable group has the small boundary property if its space MG(X) of invariant Borel probability measures forms a Bauer simplex. In addition, under the same hypothesis, we show that dynamical comparison implies almost finiteness, which was shown by Kerr to imply that the crossed product is Z-stable. This also provides two classifiability results for crossed products, one of which is based on the work of Elliott and Niu. When the group G is not amenable it is possible for action α : G y X not to have a G-invariant probability measure, in which case we show that, under the hypothesis that the action α is topologically free, dynamical comparison implies that the reduced crossed product of α is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical comparison and the uniform tower property. Under the hypothesis that the action α is exact and essentially free, we show that paradoxical comparison together with the uniform tower property implies that the reduced crossed product of α is purely infinite. As applications, we provide new results on pure infiniteness of reduced crossed products in which the underlying spaces are not necessarily zero-dimensional. Finally, we study the type semigroups of actions on the Cantor set in order to establish the equivalence of almost unperforation of the type semigroup and comparison. This sheds a light to a question arising in the paper of Rørdam and Sierakowski. In addition, we construct a semigroup associated to an action of countable discrete group on a compact Hausdorff space, that can be regarded as a higher dimensional generalization of the type semigroup. Using this generalized type semigroup we obtain a new characterization of dynamical comparison. This answers a question of Kerr and Schafhauser. Furthermore, we suggest a definition of comparison for dynamical systems in which neither necessarily the acting group is amenable nor the action is minimal