The primitive ideal space of Deaconu-Renault groupoid ∗ Nk C -algebras coming from 0-actions Lars Andreas van Woensel Kooy Tveiten Master’s Thesis, Spring 2017 This master’s thesis is submitted under the master’s programme Mathematics, with programme option Mathematics, at the Department of Mathematics, University of Oslo. The scope of the thesis is 60 credits. The front page depicts a section of the root system of the exceptional Lie group E8, projected into the plane. Lie groups were invented by the Norwegian mathematician Sophus Lie (1842–1899) to express symmetries in differential equations and today they play a central role in various parts of mathematics. Abstract The main goal of this thesis is to reproduce the article [SW16] by Sims and Williams, categorising the primitive ideal space of the class of Deaconu- ∗ k Renault groupoid C -algebras generated by N0 -actions. We go through the required groupoid prerequisites, from the definition of a groupoid to the construction of covariance C∗-algebras. We apply what we learn from [SW16] to study simplicity of the covariance C∗-algebras, and give an explicit application to the rotation algebra. Contents Abstract i Contents ii Acknowledgements iii 1 Introduction 1 2 Preliminaries 3 3 Introduction to Groupoids 7 3.1 Definition and Basic Properties . 7 3.2 Locally Compact Groupoids . 11 3.3 Representation Theory for Locally Compact Groupoids . 19 4 Covariance C∗-algebras 29 4.1 Inverse Semigroups . 29 4.2 Localisations and Covariance C∗-algebras . 31 5 Deaconu-Renault Groupoids 43 5.1 Groupoid Actions and Isotropy . 43 5.2 Deaconu-Renault Groupoids . 53 5.3 The Primitive Ideals of the C∗-algebra of an Irreducible Deaconu- Renault Groupoid . 64 5.4 The Primitive Ideals of the C∗-algebra of a Deaconu-Renault Groupoid . 81 5.5 An Application to Simplicity . 88 A The Growth Rate of Periodic Points for Hyperbolic Toral Automorphisms 95 A.1 Introduction . 95 A.2 The case where d = 3, k = 2 .................... 96 A.3 Generalisations to Zk-actions . 100 Bibliography 105 Acknowledgements I would like to thank my supervisor, Adam Sørensen, for all his advice and guidance. A thank is also in order to my partners in crime, Sissel and Paul (we finally made it!) – and all the friends and family who have kept my spirits up. CHAPTER 1 Introduction A dynamical system is, intuitively, a space in which the position of each point changes over time, and this change can be described by a function. There are numerous mathematical formulations of this umbrella term, and their applications extend widely. Chaos theory, fluid mechanics and statistical mechanics are a few examples of fields that have benefited from, or indeed have their foundations built on, the theory of dynamical systems. There are also real-life applications to engineering, biology and medicine (among others). Needless to say, dynamical systems are important. In this thesis, we will study dynamical systems by looking at C∗-algebras associated to groupoids. k Explicitly, dynamical systems coming from N0 -actions of commuting local homeomorphisms on a locally compact Hausdorff space. Let (X, T ) denote such a dynamical system, where T is a function describing the position of points in the space X over time (as in our intuitive definition above). Then one can associate to (X, T ) a groupoid GT , called a Deaconu-Renault groupoid, and ∗ ∗ ∗ ∗ thereafter a C -algebra C (GT ). The C -algebra C (GT ) is a much larger object than the original dynamical system. Indeed, the groupoid GT itself can be viewed as mashing the space X and the function (or action) T together to form a single object containing all information of both X and T . Afterwards, we ∗ form C (GT ), which contains among other things the set of certain continuous ∗ functions on GT . The fact that C (GT ) is larger than the original dynamical system has both positive and negative aspects. On one hand, one loses some intuition and “hands-on” properties of the system. On the other hand, the added structure gives us more to work with, and in particular, it enables us to apply the machinery that is developed in both groupoid and C∗-algebra theory. To get some hands-on experience with dynamical systems, we go through part of the paper [Pol12] in Appendix A. That article is about the growth rate of fixed points of hyperbolic toral automorphisms. The dynamical systems built out of commuting toral automorphisms are closely connected to Deaconu-Renault groupoids. We build the theory of groupoid C∗-algebras by following the first few chapters of [Pat99]. The construction is very similar to the construction of group C∗-algebras; their difference lies in generalising known notions of the group case (such as locally compactness, the Haar measure, representations and so on) to the groupoid case. After these basics are completed, we move on to the so-called covariance C∗-algebras, which are basically a way to characterise groupoid C∗-algebras. The construction of these are more similar to that of crossed products, of which the groupoid C∗-algebras are also a generalisation. About 1. Introduction half the thesis is devoted to reproducing [SW16]. The goal is to characterise the primitive ideal space of Deaconu-Renault groupoid C∗-algebras. We first do this in the easier case of when T is an irreducible action, and then generalise to non-irreducible actions. Finally, we apply the result from [SW16] to say something about simplicity for Deaconu-Renault C∗-algebras. As an application of what we arrive at, we give an alternative proof that the rotation algebra Aθ is simple if and only if θ is irrational. The theory in [Pat99] is very general: Nice properties such as étale and Hausdorff are mostly not assumed. Deaconu-Renault groupoids are both Haus- dorff and étale, so I have spent some time dechiphering the general results of [Pat99] to the easier case. This has not always been successful, so some of the proofs may be more technical than needed. We also go quite a lot into the details in our proofs. In particular, [SW16] sketches (or skips entirely) parts of the proofs that are clear to someone who is already well versed with this field of research. I have spent quite a lot of time writing those arguments out fully. 2 CHAPTER 2 Preliminaries General Topology As one might expect, we will need various results from general topology through- out the thesis. This subsection will be the point of reference for those results. Definition 2.0.1. A function f : X → Y between topological spaces is called proper if, for every compact set K ⊆ Y , the preimage f −1(K) is compact. Lemma 2.0.2. Let f : X → Y be a bijective function between topological spaces X and Y . If for all S ⊆ X we have x ∈ S ⇐⇒ f(x) ∈ f(S), then f is a homeomorphism. Proof. If S ⊆ X is closed, then x ∈ S if and only if f(x) ∈ f(S), so f(S) is closed. If S is not closed, then f(S) =6 f(S) so f(S) is not closed. Hence S is closed if and only if f(S) is closed, and f is a homeomorphism. Lemma 2.0.3. Let (aλ)λ∈Λ be a net in a topological space A, and let a ∈ A. Then limλ→∞ aλ = a if every subnet of (aλ)λ has a subnet converging to a. Proof. Suppose (contrapositively) that aλ 6→ a. Then for every neighbourhood U of b and every λ ∈ Λ, there is some βU,λ ≥ λ with aβU,λ 6∈ U. Let U vary over all neighbourhoods of b, and let λ vary over Λ. Define a partial order by 0 0 0 0 setting βU,α βU ,α if α ≤ α and U ⊆ U. Then (aβU,λ ) is a subnet of (aλ) with no subnet converging to b. Proposition 2.0.4. If f : K → B is a continuous bijection, where K is compact and B is Hausdorff, then f is a homeomorphism. Lemma 2.0.5. Let X be a topological space, Y ⊆ X a subspace and χY : X → {0, 1} the indicator function of Y . Then χY is continuous if and only if Y is closed in X as well as open. Definition 2.0.6. A topological space X is called a Baire space if, given a T countable collection of open dense subsets {Un}n∈ , their intersection Un N n∈N is also dense. If every locally closed subset of X has the Baire property, X is called totally Baire. (A locally closed subset of X is any intersection C ∩ U where C is closed and U is open.) 2. Preliminaries Proposition 2.0.7. If X is a totally Baire space, Y is a topological space and p: X → Y is an open continuous map, then p(X) is totally Baire. Theorem 2.0.8 (Baire category theorem). Every locally compact Hausdorff space is Baire. Remark 2.0.9. If X is locally compact Hausdorff, then every closed subspace C ⊆ Xis also locally compact Hausdorff, and therefore Baire. If U ⊆ X is open, then C ∩ U is open in U. Since open subsets of Baire spaces are Baire, C ∩ U is Baire. Hence X is totally Baire. Group Theory and Algebra In this thesis, we generalise the notion of a group C∗-algebra to a groupoid C∗-algebra. It is therefore appropriate to recall some basic definitions in group theory. Definition 2.0.10. A group is a set G together with an associative binary operation G × G → G, denoted (g, h) 7→ gh for g, h ∈ G, such that the following hold: (i) G has a unit e, such that ge = eg = g for all g ∈ G, and (ii) every element g has an inverse g−1 such that gg−1 = g−1g = e.