An Introduction to Abstract Classification Theory in the Operator Algebraic Setting

Abstract

In the setting of modern mathematical logic and model theory, classification theory has been one of the landmark achievements of the field. Likewise, the classification of UHF-algebras and AF-algebras were substantial contributions to the field of operator algebra theory. These seemingly disparate topics of study in mathematics, model theory and operator algebras, have in recent years become closely related in many respects. I here attempt to bridge the gap between these two topics by discussing how operator algebraic classifications may be understood in terms of model-theoretic classification theory. This introductory article assumes basic familiarity with model theory and linear operator, but higher-level concepts are introduced when necessary. The focus of this introduction is conceptual and informal, and as such, many results are stated without proof, but relevant sources are cited for completeness. The reader should take this not as a detailed review, but rather as an overview of a general narrative thread connecting these two branches of modern mathematics.