  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.
