Schedule for: 18w5155 - Model Theory and Operator Algebras

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will give an introduction to the theory of C*-algebras. Starting from the basics, we will treat spectral theory in some detail, culminating in the Gelfand-Naimark theorem. We will cover the GNS construction, with highlight being that every abstract C*-algebras can be realized as a C*-algebra of bounded operators on a Hilbert space. We will then discuss other constructions/examples such as certain universal C*-algebras or inductive limits. If there is time, I will give a rough outline of the Elliott classification program.

In the first lecture, I will review basic notions and constructions of von Neumann algebras. The second lecture will be devoted to property Gamma and McDuff’s property for II_1 factors. In the third lecture, I will discuss the isomorphism problem for ultrapowers of II_1 factors. ========== Lecture 1: Define vN algebras and state the bicommutant theorem. Introduce tracial vN algebras and the hyperfinite II_1 factor. Group and group measure space vN algebras. Lecture 2: Define property Gamma and discuss the connection with inner amenability of groups. Define McDuff’s property. Examples of II_1 factors that are Gamma but not McDuff. Lecture 3: The ultrapower construction for tracial vN algebras. Discuss dependence on the choice of the ultrafilter and examples of II_1 factors with non-isomorphic ultrapowers.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lecture 1: Structures, ultraproducts, and formulas I will introduce the fundamental notions of logic for metric structures, such as formulas and ultraproducts. I will then explain how C*-algebras and von Neumann algebras fit into this framework. Lecture 2: Axiomatizability and definability I will present the crucial model-theoretic concepts of axiomatizability and definability, and then provide many examples from the theory of operator algebras. Lecture 3: Nuclearity and omitting types I will discuss how nuclearity can be captured model-theoretically, and how this opens up the possibility to use constructions from model theory to produce interesting new examples of nuclear C*-algebras.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Abstract: In joint work with Goldbring and Sinclair, for tracial von Neumann algebras $M$ and $N$, we show how to capture the notion of an $M$-$N$ correspondence model theoretically. We use this correspondence framework to study $\sigma$-finite von Neumann algebras and the uniform 2-norm on a $C^*$-algebra with respect to a collection of states. The role of the ultraproduct will be highlighted for its guidance in determining the correct languages in these cases.

I will discuss some recent joint work with Jos\'e Carri\'on, Jamie Gabe, Aaron Tikuisis, and Stuart White, which provides a new abstract approach to the classification of simple, nuclear C*-algebras.

I will talk about recent joint work with Antoine, Perera, and Thiel. We have shown that the Cuntz semigroup of a separable C*-algebra of stable rank one is inf-semilattice ordered, i.e., has finite infima and addition is distributive over infima. We use this to gain new insights into the structure of these Cuntz semigroups and to answer a number of questions on C*-algebras of stable rank one. We are able to remove the assumption of separability in some of these applications using model theoretic tools.  

The Toms-Winter conjecture postulates that three very different-looking regularity-type conditions coincide for separable simple infinite-dimensional amenable unital C*-algebras. The different conditions are (i) finite nuclear dimension, (ii) Z-stability, and (iii) strict comparison of positive elements. In the first half of my talk, I will say some things about this conjecture and its important connections to the classification of C*-algebras. In the second half of the talk, I will discuss a key concept used in the proof that (ii) implies (i), called complemented partitions of unity (CPoU). As I will explain, this concept provides a method for gluing local witnesses of open types in tracial GNS representations to global witnesses (satisfying the type uniformly over all traces). I will explain complemented partitions of unity in the context of this local-to-global type satisfaction device. This is joint work with Jorge Castillejos, Sam Evington, Stuart White, and Wilhelm Winter.

The use of (central) sequence algebras in the theory  of operator algebras has a long history, dating back to McDuff’s  characterization of those factors which absorb the hyperfinite  II_1-factor. Applications in the context of C*-algebras are both  abundant and far-reaching, and they often appear in connection  with classification of C*-algebras. Central sequence algebras are fundamental tools in the study of strongly self-absorbing  C*-algebras, which themselves have tight connections with the Elliott classification programme. This has prompted a deeper study  of ultrapowers and (central) sequence algebras, where  model-theoretic methods have become predominant. Ultrapowers  and relative commutants have also been a crucial tool in the study  of group actions on operator algebras, dating back to the  classification of amenable group actions on the hyperfinite  II_1-factor. A more recent instance of their use in the equivariant  setting is the study of strongly self-absorbing actions. As such,  equivariant (central) sequence algebras are interesting objects  whose systematic study is justified by their wide application in the  literature. In this talk, we report on joint work with Lupini, where we  consider actions of a compact group on a C*-algebra as a structure  in the framework of continuous model theory. The realization that  the continuous part of the ultrapower of a G-C*-algebra is just its  ultrapower as a structure in the new equivariant language, allows  us to establish interesting properties, including saturation and Los'  theorem. We give various applications to C*-dynamics, including to  strongly self-absorbing actions as well as to Rokhlin dimension. 

We show that the convex set of factorizable quantum channels on a fixed matrix algebra of size at least 11 which factor through finite dimensional C*-algebras is non-closed, and that there exist factorizable quantum channels on matrix algebras that require an ancilla of type II_1. We also give a new and simplified proof of the result by Dykema, Paulsen and Prakash that the set of synchronous quantum correlations C_q^s(5,2) is non-closed. One can describe factorizable quantum channels on a given matrix algebra in terms of traces on the unital free product of that matrix algebra with itself. We give a description of which of these traces correspond to factorizable maps that can be approximated by ones with finite dimensional ancilla, and we relate this to the Connes Embedding Problem. This is a joint work with Magdalena Musat.

I present a joint work with Sorin Popa and Dimitri Shlyakhtenko. We prove that under a natural condition, the regular von Neumann subalgebras B of the hyperfinite II_1 factor R are completely classified (up to conjugacy by an automorphism of R) by the associated discrete measured groupoid. The key step in proving this result is the vanishing of the 2-cohomology for cocycle actions of amenable discrete measured groupoids and the approximate vanishing of the 1-cohomology. This leads us to a new notion of treeability for equivalence relations. I also discuss (non-)classification results for amenable discrete measured groupoids.

For each C*-algebra A, one can construct its opposite \(A^{op}\), which is the same as a Banach space, only with the order of multiplication is reversed. It is a long-standing and difficult open problem whether there exists a simple separable nuclear C*-algebra which is not isomorphic to its opposite. I will survey some of the known results and techniques, focusing on the nuclear case, and discuss a joint paper with Ilijas Farah from 2017 in which we construct a simple nuclear non-separable example.

Let \(p \in (1, \infty)\). We show that the Continuum Hypothesis implies that the \(l^p\)~Calkin algebra \(Q (\^p) = L (l^p ({\mathbb{Z}})) / K (l^p ({\mathbb{Z}}))\) has isometric automorphisms which are not given by conjugation by invertible isometries in \(Q (\^p)\). Depending on what is done between now and the time of the talk, we will describe progress towards proving that it is consistent with ZFC that there are no such isometric automorphisms. This is joint with with Andrey Blinov.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.