Correspondences, Ultraproducts and Model Theory

TL;DR

This paper explores the model-theoretic properties of correspondences between tracial von Neumann algebras, introducing ultraproducts and proving elementary class and stability results, with applications to property (T) and C*-algebras.

Contribution

It introduces an ultraproduct of correspondences, proves elementary class and stability results, and applies these to property (T) and C*-algebra ultraproducts.

Findings

Correspondences form an elementary class.

The theory of correspondences is classifiable and stable.

A simpler proof that certain structures are elementary classes.

Abstract

We study correspondences of tracial von Neumann algebras from the model-theoretic point of view. We introduce and study an ultraproduct of correspondences and use this ultraproduct to prove, for a fixed pair of tracial von Neumann algebras M and N, that the class of M-N correspondences forms an elementary class. We prove that the corresponding theory is classifiable, all of its completions are stable, that these completions have quantifier elimination in an appropriate language, and that one of these completions is in fact the model companion. We also show that the class of triples (M, H, N), where M and N are tracial von Neumann algebras and H is an M-N correspondence, form an elementary class. As an application of our framework, we show that a II_1 factor M has property (T) precisely when the set of central vectors form a definable set relative to the theory of M-M correspondences. We then use our approach to give a simpler proof that the class of structures (M, Phi), where M is a sigma-finite von Neumann algebra and Phi is a faithful normal state, forms an elementary class. Finally, we initiate the study of a family of Connes-type ultraproducts on C*-algebras.