UC San Diego

The ultraproduct construction in the context of operator algebras (specifically tracial von Neumann algebras) was first implicitly introduced by Wright in 1954 and has since been used to great effects in both the theory of von Neumann algebras and C*-algebras. Its great success is owed to at least three reasons: 1. First, by considering an operator algebra as sitting within its ultrapower, one may consider various asymptotic phenomena that may not be available within the algebra itself. This includes, e.g., central sequence algebras and asymptotic freeness phenomena. These asymptotic phenomena allow one to obtain structural properties of the operator algebra in question by first passing to a larger algebra within the ultrapower witnessing certain asymptotic behaviors and then back to the original algebra in an approximate fashion; 2. Second, ultraproducts are intrinsically related to model theory. Model-theoretic results such as Łoś's theorem, Keisler-Shelah theorem, and downward Löwenheim-Skolem theorem pass to operator algebras via continuous model theory, allowing one to prove model-theoretic results by applying operator-algebraic techniques to ultraproducts or prove operator-algebraic results on ultraproducts by applying model-theoretic techniques; 3. Third, the definition of ultraproducts means they are closely related to the issue of approximations. As such, they may be used to study the approximation theory of both operator algebras themselves, as well as other structures that may be represented by operator algebras, such as groups. These approximations allow the development of theories such as various entropy theories and have applications across many different areas of mathematics. The results in this dissertation cover my works in all three aspects of the applications of ultraproduct methods in operator algebras. In Chapter 1, we follow the philosophy of studying an operator algebra sitting within its own ultrapower in order to obtain structural properties of the said algebra. Specifically, in Chapter 1.1, a natural ultrapower condition on II1 factors is obtained that guaranteed the said factors are singly generated. This result unifies the vast majority of known results on the single generation problem for II1 factors. While in Chapter 1.2, we apply the concept of selfless C*-algebras first introduced by Robert, which is an ultraproduct asymptotic freeness condition, and show that a large class of groups, including free groups, have their reduced group C*-algebras being selfless. This allows us to resolve the long-standing open problem of whether C*_r(F_2) has strict comparison. In Chapter 2, we turn to the question of model-theory. Specifically, in Chapter 2.1, we study the issue of elementary equivalence for full II1 factors, improving upon a result of Chifan, Ioana, and Kunnawalkam Elayavalli that shows there is a full II1 factor that is not elementarily equivalent to L(F_2). We greatly simplified their argument in Chapter 2.1.1, while in Chapter 2.1.2, we removed the need to appeal to property (T). In Chapter 2.2, we consider the question of elementary equivalence for tracial von Neumann algebras that are not factors. We show that elementary equivalence of such algebras nearly implies elementary equivalence of almost all fibers, thereby reducing the study of model theory of such algebras to that of factors. In Chapter 3, we consider the approximation theory aspects of ultraproducts, as applied to groups and group actions. Specifically, in Chapter 3.1, we study soficity of group actions, which is a condition that roughly means the actions can be approximated by finite permutations in an ultraproduct sense. While in Chapter 3.2, we study soficity of certain groups in Chapter 3.2.1 and matricial field properties of certain groups in Chapter 3.2.2. These are, respectively, finite permutation approximations of groups in a weak sense and finite-dimensional unitary approximations of groups in a strong sense. Both employ an ultraproduct framework.