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Simplicity and Minimality in Crossed Products

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Simplicity and Minimality in Crossed Products Simplicity and Minimality in Crossed Products Jackson Morris A Thesis Presented to the Honors Tutorial College In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics. 2018 April Contents 1 Introduction 2 2 General Background7 2.1 Hilbert Spaces.....................................7 2.2 Analysis......................................... 13 2.2.1 Measure Theory................................ 13 2.2.2 Functional Analysis.............................. 15 2.3 Group Theory...................................... 20 2.3.1 Fundamentals of Group Theory........................ 20 2.3.2 Topological Groups............................... 23 2.4 Dynamical Systems................................... 25 3 C*-Algebras 33 3.1 Definitions and Examples............................... 33 3.2 Positivity........................................ 40 3.3 Representations..................................... 45 3.4 Completely Positive Maps............................... 50 4 Crossed Products 56 4.1 Amenability....................................... 56 4.2 Crossed Product Fundamentals............................ 62 4.3 Minimality and Simplicity............................... 73 4.4 Minimality in Abelian Groups............................. 81 1 Chapter 1 Introduction Our objective in this document is to study a construction known as the crossed product of a dynamical system. This construction allows us to take a dynamical system and create a C*-algebra which encodes information about the original dynamical system. Constructing such operator algebras out of changing spaces is an established technique, explored initially by Murray and von Neumann in [25], [26], and [24], and later by Zeller-Meier [27], Effros, and Hahn [7]. From this baseline, many, such as Elliot [8], Kishimoto [12], Power [18], Archbold and Spielberg [1], and Quigg [19] began to explore the relationship between properties of the crossed product and the properties of the base system. For the purposes of this document, we will mostly be following Power [18] as adapted in Davidson [5], and Archbold-Spielberg [1]. But what is a dynamical system? What is a C*-algebra? How does this crossed product even work? Let us answer each of these questions in turn, beginning with dynamical systems. In broad terms, dynamical systems is the study of iterated maps on spaces. To give the classical example from Brin [3], imagine a taking a circle and rotating it by an angle, doing so again and again repeatedly. The field then asks questions about this system-how do points travel over time? Do points initially close together stay close together? How frequently does a point come back to where it started? These questions are easy if not trivial in this case, but give an idea as to the kind of things that are done in the study of dynamical systems. As for what a C*-algebra is, that is a bit more difficult to pin down. The best way to illustrate the concept is to give some examples and break down why these examples are C*-algebras. For our first example, consider M2(R), the set of two-by-two matrices with real coefficients. Recall 2 from linear algebra that such a matrix defines a linear map on R2, the real plane. This collection of matrices will be the first example of a C*-algebra. The first important property this has is that it is closed algebraically. You can multiply a matrix by a number to get another matrix, and you can take the sum or product of two matrices to get another matrix. It is also important that we can define a \norm" for each matrix. The word \norm" has a precise mathematical definition, but we needn't worry about it. We need only to understand that a norm is generalization of the idea of length. For our real matrices, the norm is the measure of how much their linear operation stretches vectors, without paying heed to how much it might rotate them. Two more important, but technical properties this space satisfies is that it is closed and complete with respect to the norm. These properties can collectively be thought of as there not being any holes in our set, or there not being anything missing that should be there. The most important condition for a C*-algebra, however, is that of the adjoint, or \*" operation. It is this * that distinguishes C*-algebras from other kinds of algebraic constructions and in fact, is part of the name. For our matrices, our adjoint is the conjugate, the operation that flips the matrix along the diagonal. The adjoint needs to satisfy a small collection of basic algebraic properties that the transpose does satisfy in this case. It is this collection of norm, closure, completeness, and adjoint which characterize a C*-algebra. In general, C*-algebras are defined over complex numbers, not just the reals. Furthermore, we can generalize this to matrices of any dimension, where it acts analogously on Cn. For the details on the matrix example in this case, please see Example3 :1:3. A second example which is illustrative of the breadth of these concepts is that of the algebra of continuous functions on a space, such as the circle. For a visualization of this, imagine the collection of various ways to stretch a short, circular length of string. The details are a bit more technical than matrices, but it's also fairly easy to see that they are a C*-algebra in their own right. A natural question at this point is what either of these concepts have to do with each other at all. And this is where the crossed product construction and the function algebra example come in. This is because, if we have a dynamical system, the set of continuous functions on that space is again a C*-algebra, and one that the dynamical system acts upon. For a visualization, return to the example of the rotating circle and a continuous function on it. As we rotate the \base" circle 3 of the string, we rotate the string itself. And this is where much of operator algebra theory comes in, giving us a vector to consolidate this system into one C*-algebra that encodes information about the original system. The details of this are highly technical, but in broad strokes, we begin by simply smashing the structures together, which creates an algebraic structure that needs only a norm to be a C*-algebra. To do this, we represent the entire interwoven system onto an appropriate type of space, and use the \largest" of these representations as the norm. After completing the \proto"-algebra with respect to this norm, we get a C*-algebra that we call the crossed product. Using this, we will explore connections between the original dynamical system and this crossed product. In particular, we want to find equivalent conditions for the \simplicity" of the crossed product. Simplicity is an algebraic property that has to do with a lack of a certain kind of substructure, one that is invariant under multiplication. And as it turns out, there is also a property based on lacking of invariant substructures for dynamical systems, known as minimality. A dynamical system is minimal if it does not have a smaller part that the iterated map doesn't change. And in fact, it turns out that the minimality of a dynamical system is equivalent to the simplicity of its crossed product. Our primary objective in this text will be to rigorously show this equivalence. We begin with a basic overview of various background material. First, we discuss Hilbert spaces, which can be though of as infinite-dimensional generalizations of Cn. This will be pri- marily utilized for the aforementioned representations of C*-algebras. Next, we cover measure theory, which is a broadly applicable generalization of the idea of area or volume. After some useful but technical results, we cover groups, a generalized structure to which the integers belong to. Finally, we cover many basic ideas in dynamical systems, providing formal definitions for the concepts we've discussed in this introduction and several other highly useful results. In the next chapter, we begin to dig in to the basics of C*-algebras. We naturally start with formalized definitions and examples that for the most part have been discussed in this introduction. The most important addition here is that the bounded operators on Hilbert space are a C*-algebra in Example3 :1:7. The next major topic is that of the spectrum (Definition 3:2:1). This is a little technical, but we can see it as a mutual generalization of the idea of a function's range and a matrix's eigenvalues. And, in the vein of a positive function, we can call an element of a C*-algebra positive if its entire spectrum is positive. A big result from this 4 section is the Continuous Functional Calculus (Theorem3 :2:18), which lets us take continuous functions on the spectrum of an element and apply them to the element in the C*-algebra. We can then use this in Lemma3 :2:20 to show the extremely useful characterization of positive elements as those of the form b∗b. Next, we deal with those aforementioned representations of C*-algebras. Bounded operators on Hilbert space are actually really easy to work with, so our representations will be maps from C*-algebras to these operator spaces. From our examples, this is also pretty easy to see- matrices act on bounded operators on Cn, and Hilbert space is a generalization of Cn. It is also in this section that we define ideals, which are those invariant structures we talked about earlier with simplicity. The most important result from this section is the GNS construction (Example3 :3:17, Theorem3 :3:18), which gives us that for any C*-algebra, such a representation onto the bounded operators of a Hilbert space exists. The final section of this chapter deals with completely positive maps, which will come in for later results.
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