Existence of optimal ultrafilters and the fundamental complexity of simple theories

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such characterization of an unstable class. In the present paper, we first establish the existence of so-called optimal ultrafilters on Boolean algebras, which are to simple theories as Keisler's good ultrafilters are to all theories. Then, assuming a supercompact cardinal, we characterize the simple theories in terms of saturation of ultrapowers. To do so, we lay the groundwork for analyzing the global structure of simple theories, in ZFC, via complexity of certain amalgamation patterns. This brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability.Comment: The revisions aim to separate the set theoretic and model theoretic aspects of the paper to make it accessible to readers interested primarily in one side. We thank the anonymous referee for many thoughtful comment

An invitation to model theory and C*-algebras

We present an introductory survey to first order logic for metric structures and its applications to C*-algebras

An invitation to model theory and C*-algebras

We present an introductory survey to first order logic for metric structures and its applications to C*-algebras

Rigidity of Corona Algebras

In this thesis we use techniques from set theory and model theory to study the isomorphisms between certain classes of C*-algebras. In particular we look at the isomorphisms between corona algebras of the form $\prod\mathbb{M}_{k(n)}(\mathbb{C})/\bigoplus \mathbb{M}_{k(n)}(\mathbb{C})$ for sequences of natural numbers $\{k(n): n\in\mathbb{N}\}$. We will show that the question ``whether any isomorphism between these C*-algebras is trivial", is independent from the usual axioms of set theory (ZFC). We extend the classical Feferman-Vaught theorem to reduced products of metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent. We also use this to find examples of corona algebras of the form $\prod\mathbb{M}_{k(n)}(\mathbb{C}) / \bigoplus \mathbb{M}_{k(n)}(\mathbb{C})$ which are non-trivially isomorphic under the Continuum Hypothesis. This gives the first example of genuinely non-commutative structures with this property. In chapter 6 we show that $SAW^{*}$-algebras are not isomorphic to $\nu$-tensor products of two infinite dimensional C*-algebras, for any C*-norm $\nu$. This answers a question of S. Wassermann who asked whether the Calkin algebra has this property