TOPOLOGICAL DYNAMICAL SYSTEMS AND REGULARITY PROPERTIES OF REDUCED CROSSED PRODUCT C∗-ALGEBRAS A Dissertation by XIN MA Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Chair of Committee, David Kerr Committee Members, Roger R. Smith Kenneth Dykema Mohsen Pourahmadi Head of Department, Emil J. Straube August 2019 Major Subject: Mathematics Copyright 2019 Xin Ma ABSTRACT 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 topolog- ically 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 ii 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 suggests a definition of comparison for dynamical systems in which neither necessarily the acting group is amenable nor the action is minimal. iii DEDICATION To Lan, whose love and support made everything of me possible. iv ACKNOWLEDGMENTS This dissertation contains my works as a Ph.D student in the department of mathematics at Texas A&M University. I should like to thank, first and foremost, my supervisor, Professor David Kerr, for his support in all ways. This work could not have progressed without his help, encourage- ment, and guidance. His wisdom and insight will always help me in my future career. So I express my deepest gratitude to him. I also should like to thank Professor Guoliang Yu for his teaching, help and forethought. I benefits a lot from invaluable discussions with him, which usually take me time to digest. I would like to thank Professor Roger Smith, Zhuang Niu, Hanfeng Li, Gilles Pisier, Christo- pher Philips, Gábor Szabó, Kenneth Dykema, Zhizhang Xie and Mohsen Pourahmadi for their teaching, discussion, help and time. Finally, I would like to thank Jianchao Wu, Kang Li and Jamali Saeid for all fruitful discussions. v CONTRIBUTORS AND FUNDING SOURCES Contributors This work was supervised by a dissertation committee consisting of Professor David Kerr, Roger Smith and Kenneth Dykema of the Department of Mathematics and Professor Mohsen Pourahmadi of the Department of Statistics. All work for the dissertation was completed by the student, under the advisement of Professor David Kerr of the Department of Mathematics. Funding Sources There are no outside funding contributions to acknowledge related to the research and compi- lation of this document. vi TABLE OF CONTENTS Page ABSTRACT ......................................................................................... ii DEDICATION....................................................................................... iv ACKNOWLEDGMENTS .......................................................................... v CONTRIBUTORS AND FUNDING SOURCES ................................................. vi TABLE OF CONTENTS ........................................................................... vii 1. INTRODUCTION............................................................................... 1 1.1 C∗-algebras ................................................................................ 1 1.1.1 Regularity Properties of C∗-algebras ............................................ 1 1.1.1.1 Finite Nuclear Dimension............................................. 1 1.1.1.2 Z-stability ............................................................. 1 1.1.1.3 Strict Comparison and the Cuntz Semigroup ........................ 2 1.1.2 Finiteness of C∗-algebras ........................................................ 3 1.1.3 Toms-Winter Conjecture and Elliott’s Program................................. 5 1.2 Topological Dynamical Systems ......................................................... 6 1.2.1 Basic notations.................................................................... 6 1.2.2 Crossed product C∗-algebras .................................................... 7 1.2.3 Dynamical Comparison .......................................................... 10 1.2.4 Almost Finiteness ................................................................ 12 1.2.5 The Small Boundary Property ................................................... 14 2. DYNAMICAL SYSTEMS OF AMENABLE GROUPS ..................................... 16 2.1 Decomposition of Ergodic Invariant Probability Measures ............................. 16 2.2 Dynamical Comparison and Almost Finiteness ......................................... 19 2.3 The Small Boundary Property Revisited ................................................. 25 2.4 Classification Results ..................................................................... 28 3. PARADOXICALITY IN DYNAMICAL SYSTEMS ......................................... 31 3.1 Dynamical Comparison and Paradoxical Phenomenon ................................. 31 3.2 Paradoxical Comparison for Non-minimal actions ...................................... 37 3.2.1 Paradoxical Comparison ......................................................... 38 3.2.2 Uniform Tower Property and Pure Infiniteness ................................. 43 3.3 Applications and Examples ............................................................... 48 vii 3.3.1 Finite Many Ideals Case.......................................................... 48 3.3.2 Product of Spaces Case .......................................................... 49 4. SEMIGROUPS OF DYNAMICAL SYSTEMS ............................................... 54 4.1 The Type Semigroup ...................................................................... 54 4.2 The Generalized Type Semigroup ........................................................ 59 4.3 A New Characterization of Dynamical Comparison .................................... 65 5. SUMMARY AND CONCLUDING REMARKS.............................................. 77 REFERENCES ...................................................................................... 79 viii 1. INTRODUCTION In this Chapter, we mainly recall some definitions and introduce our basic framework for our study on topological dynamical systems and their crossed product C∗-algebras. We begin with an introduction to regularity properties and the classification theory of C∗-algebras. 1.1 C∗-algebras 1.1.1 Regularity Properties of C∗-algebras 1.1.1.1 Finite Nuclear Dimension For a general introduction to C∗-algebras, we refer to [9]. The nuclear dimension of a C∗- algebra was introduced by Winter and Zacharias in [77] as a noncommutative analogue of covering dimension of a topological space. We recall the definition here. Let A be a C∗-algebra. (i) A completely positive map ' from a C∗-algebra B to A is said to be of order zero if '(a)'(b) = 0 whenever a; b are self-adjoint elements in B satisfying ab = 0. (ii) We say that a completely positive map ' from a finite-dimensional C∗-algebra B to A is n-decomposable if we can write B = B0 ⊕ · · · ⊕ Bn so that the restriction of ' to each Bi has order zero. Definition 1.1.1. The nuclear dimension of A, denoted by dimnuc(A) is the least integer n such that for every finite set F ⊂ A and > 0 there are a finite-dimensional C∗-algebra B, a