On the Computability Theoretic and Continuous Model Theoretic Structure of General von Neumann Algebras

Abstract

In this thesis, we introduce and develop the model theory of general von Neumann algebras with a faithful normal semifinite weight. Our framework admits various computability theoretic properties that align it with recent work on uncomputable universal theories in the tracial von Neumann algebra setting. We study the ultraproduct that our framework suggests and we prove analogues of known theorems about the Ocneanu ultraproduct for this new ultraproduct, ultimately providing 3 new operator algebraic characterizations of our ultraproduct. We show that our framework captures the Connes-Takesaki decomposition which is central to the classification of injective factors. We capture the Connes-Takesaki decomposition via definable groupoids, examining other aspects of definability in continuous logic along the way. Finally, we study the uncomputability of various classes of operator algebras and, as an application, give consequences about ultraproduct embeddings.