Groups and Operator Algebras Seminar

Speaker: David Jekel (University of Copenhagen)

Title: Logical reasons why von Neumann algebras are hard

Abstract: I will introduce and motivate model theory for tracial von Neumann algebras and present some related recent work. Many properties (such as property Gamma and various embedding properties) can be described by variational formulas composed by taking iterated suprema and infima over the unit ball. These formulas can be viewed as continuum analogs of logical predicates where the quantifiers are sup and inf rather than for all and there exists, allowing the formulation of model theory for von Neumann algebras. For a diffuse commutative von Neumann algebra (i.e. a probability space), any such formula with suprema and infima is approximately equivalent to one without any quantifiers, meaning such spaces have quantifier elimination. Farah showed that no II1 factors have quantifier elimination, and my joint work with Farah and Pi showed that (a large class including all Connes-embeddable) II1 factors are not model complete, meaning that it is also impossible to reduce every formula to one quantifier. We also know, thanks to Alekseev and Thom's result on universal sofic groups, that formulas for the n x n matrices together with the distinguished subset of diagonal matrices do not always have a limit as n goes to infinity, which raises deep questions about the structure of matrix algebras (and random matrix theory) in the large-n limit.

GOA seminar webpage: https://sites.google.com/view/copenhagen-goa-seminar/home