Operator Algebras and Abstract Classification

This dissertation is dedicated to the study of operator spaces, operator algebras, and their automorphisms using methods from logic, particularly descriptive set theory and model theory. The material is divided into three main themes. The first one concerns the notion of Polish groupoids and functorial complexity. Such a study is motivated by the fact that the categories of Elliott-classifiable algebras, Elliott invariants, abelian separable C*-algebras, and arbitrary separable C*-algebras have the same complexity according to the usual notion of Borel complexity. The goal is to provide a functorial refinement of Borel complexity, able to capture the complexity of classifying the objects in a functorial way. Our main result is that functorial Borel complexity provides a finer distinction of the complexity of functorial classification problems.

The second main theme concerns the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory. Our results show that, for any non-elementary simple separable C*-algebra, the problem of classifying its automorphisms up to unitary equivalence transcends countable structures. Furthermore we prove that in the unital case the relation of unitary equivalence obeys the following dichotomy: it is either smooth, when the algebra has continuous trace, or not classifiable by countable structures.

The last theme concerns applications of model theory to the study and construction of interesting operator spaces and operator systems. Specifically we show that the Gurarij operator space introduced by Oikhberg can be characterized as the Fraisse limit of the class of finite-dimensional 1-exact operator spaces. This proves that the Gurarij operator space is unique, homogeneous, and universal among separable 1-exact operator spaces. Moreover we prove that, while being 1-exact, the Gurarij operator space does not embed into any exact C*-algebra. Furthermore the ternary ring of operators generated by the Gurarij operator space is canonical, and does not depend on the concrete representation chosen. We also construct the operator system analog of the Gurarij operator space, and prove that it has analogous properties.