• Von Neumann Algebras
• NCGOA 2016

• Schedule
• Participants
• Poster
• Recorded Talks

Vaughan Jones and I classified subfactor planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of 3-boxes is at most 13 several years ago. In recent joint work with Jones and Liu, we settled the case of dimension 14. We find one depth 3 subfactor planar algebra coming from quantum SO(3), and a one-parameter family coming from quantum Sp(4). They are all BMW.

The aim of this series of lectures is to provide some background on von Neumann algebras. After giving the main definitions and examples, I will review some of the most fundamental concepts: basic construction and index; amenability and property (T); intertwining by bimodules. I will illustrate how one can use these concepts to study free product von Neumann algebras.

Video recording: Lecture 1  Lecture 2  Lecture 3

The modules of a rational vertex operator algebra form a modular tensor category. Could all modular tensor categories be associated to a rational vertex operator algebra in this way? The obvious place to check is the centre of the Haagerup subfactor: though surely no one any more can seriously believe the Haagerup is exotic, it has no direct connection to Lie theory or finite groups or indeed any known relation whatsoever to vertex operator algebras. In my talk I'll describe how to approach such a challenge and describe how far we've got.

I will talk about entropy for pmp actions of sofic groups (equivalently trace-preserving actions of sofic groups on abelian von Neumann algebras). We investigate structural properties of Pinsker algebras for sofic entropy. In the amenable case, the Pinsker algebra is the largest invariant subalgebra with entropy zero. In the nonamenable case, there is a modification of the Pinkser algebra called the Outer Pinsker algebra which is defined via sofic entropy in the presence (defined in an analogous manner to Voiculescu's free entropy in the presence). Entropy in the presence turns out to fix many "pathological" monotonicity properties of entropy (namely increase of entropy under restriction to invariant subalgebras) and so one expects the Outer Pinsker algebra to behave in the nonamenable case much as it does in the amenable case. We introduce the "independent microstates lifting property" for actions of sofic groups, (this is related to model surjectivity as defined by Austin) which is defined via an extension property of embeddings into a natural ultraproduct algebra. Under the assumption of the independent microstate lifting property, we prove that the Pinkser algebra of a tensor product is the tensor product of the Pinsker algebras. Time permitting, we will give some applications to the study of entropy for actions of a group on a compact group by automorphisms.

Slides

In 2007, Gaboriau and Lyons showed that any non-amenable group G has a free ergodic pmp action G ↷ X whose orbit equivalence relation ℜ(G ↷ X) contains ℜ(F₂ ↷ X) for some free ergodic pmp action of F₂. This talk will focus on joint work with Lewis Bowen and Adrian Ioana in which we extend this result, showing that given any ergodic non-amenable pmp equivalence relation ℜ, the Bernoulli extension ℜ^~ over a nonatomic base space must contain ℜ(F₂ ↷ X^~) for some free ergodic pmp action of F₂. We then show that any such ℜ admits uncountably many extensions {(ℜ^~)_a}_a∈A which are pairwise not stably von Neumann equivalent. From this we deduce that any non-amenable unimodular lcsc group G has uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent (in particular, pairwise not orbit equivalent).

Slides

I will present new results regarding the structure and the classification of Shlyakhtenko's free Araki-Woods factors. Firstly, I will show that all free Araki-Woods factors are strongly solid, meaning that the normalizer of any diffuse amenable subalgebra with normal expectation remains amenable. This provides the first class of nonamenable strongly solid type III factors (joint work with R. Boutonnet and S. Vaes). Secondly, I will present a complete classification of a large family of non-almost periodic free Araki-Woods factors up to *-isomorphism (joint work with D. Shlyakhtenko and S. Vaes).

Slides

Adrian Ioana: Applications of asymptotic properties of group actions and II₁ factors

In this course, I will present several applications of asymptotic properties to the study of group actions and II₁ factors. I will first explain the role of strong ergodicity in recent cocycle and orbit equivalence rigidity results for translation actions. In particular, I will discuss a cocycle superrigidity theorem for translation actions of product groups (joint with D. Gaboriau and R. Tucker-Drob). I will also explain how central sequences can be used to provide infinitely many II₁ factors with non-isomorphic ultrapowers (joint with R. Boutonnet and I. Chifan).

Video recording: Lecture 1  Lecture 2  Lecture 3

We give an overview of recent classification results of certain classes of fusion categories by using type III factors and the Cuntz algebra endomorphisms. We start with the basics of cohomological aspects of group actions on factors, and the tensor categories consisting of endomorphisms of type III factors. Then we show how we can use them to classify subfactors and fusion categories.

Slides (part 1) Slides (part 2) Slides (part 3)

Video recording: Lecture 1  Lecture 2  Lecture 3

Recently, Popa and Vaes introduced a framework for analytic properties of rigid C*-tensor categories, generalizing Popa's framework for standard invariants of subfactors. We will describe the tube algebra of a tensor category and its representation theory, which provides an alternate perspective on analytic properties. As an application, we will sketch a proof that quantum G₂ categories have property (T).

We formulate gapped domain walls between topological phases in terms of subfactors and tensor categories. Then we disprove a recent conjecture in condensed matter physics. We also study a certain fusion product of subfactors in this setting in connection to the alpha-induction subfactors.

Slides

Polynomial maps on groups (to be defined in the talk) can be related to a certain cohomology theory, referred to as polynomial cohomology, and in the case of nilpotent groups the information captured by its polynomials turns out to be very refined. In my talk I will explain the precise meaning of these statements and how polynomial cohomology can be used to prove a generalization of Shalom's result on quasi-isometry invariance of Betti numbers for nilpotent groups. This is all part of an ongoing joint project with Henrik Densing Petersen.

Slides

Although we do not yet have a complete description of the centre of the extended Haagerup subfactor, in joint work with Terry Gannon and with Kevin Walker we have now computed the modular data (that is, the S and T matrices). These can be derived using a combination of combinatorial, Galois theoretic, and representation theoretic methods. We have also found that there are "healthy looking" vector valued modular forms associated to these S and T matrices.

Slides

I will demonstrate a non-tracial analogue of Dabrowski's 2010 result that a tracial von Neumann algebra generated by elements with finite free Fisher information is a factor without property Γ. For the non-tracial context, we suppose M is a von Neumann algebra equipped with a faithful normal state φ, and is generated by a finite set G = G*, |G| ≥ 3. If G consists of eigenvectors of the modular operator Δ_φ and have finite free Fisher information, then the centralizer M^φ is a II₁ factor without property Γ and M is either a type II₁ factor or a type III_λ factor, 0 < λ ≤ 1, depending on the eigenvalues of G. Furthermore, M is full when it is a type III_λ factor for 0 < λ < 1.

Recently C. Houdayer and Y. Isono have proved among other things that every biexact group Γ has the property that for any non-singular strongly ergodic action Γ ↷ (X,μ) on a standard measure space the group measure space von Neumann algebra Γ ⋉ L^∞(X) is full. I will prove the same property for a wider class of groups, notably including SL(3,ℤ). Also, for any connected simple Lie group G with finite center, any lattice Γ ≤ G, and any closed non-amenable subgroup H ≤ G, the non-singular action Γ ↷ G/H is strongly ergodic and the von Neumann factor Γ ⋉ L^∞(G/H) is full.

arXiv:1602.02654

The question of classifying small subfactors (according to their principal graph or standard invariant) is almost as old as the field of subfactors. I will start by describing the classification of classical (index 4 or smaller) subfactors, talk about the techniques which extended the classification up to index 5, and finish by describing the current state of the art. I will also discuss methods of construction for the interesting examples of subfactors that pop up along the way.

Video recording: Lecture 1  Lecture 2  Lecture 3

A character on a group is a class function of positive type. Based on the rigidity results of Mostow, Margulis, and Zimmer, it was conjectured by Connes that for lattices in higher rank Lie groups the space of characters should be completely determined by the finite dimensional representations of the lattice. In this talk, I will discuss the solution of this conjecture, and I will also discuss the classification of characters on lattices in products of groups.

Slides

I will discuss a method for constructing Haar unitaries u in a subalgebra B of a von Neumann II₁ factor M that are "as independent as possible" (approximately) with respect to a given finite set of elements in M. This technique had most surprising applications over the years, e.g., to Kadison-Singer type problems, to proving vanishing cohomology results for II₁ factors (like compact valued derivations, or the Connes-Shlyakhtenko 1-cohomology), as well as to subfactor theory (e.g., unravelling the axiomatisation of standard invariants). After explaining this method, which I call incremental patching, I will comment on all these applications and its potential for future use.

Voiculescu’s free probability theory provides both a number of interesting von Neumann algebras (such as interpolated free group factors and free Araki-Woods factors) and a number of important tools for handling them. These tools range from the free product construction itself to Voiculescu’s free entropy theory and to non-commutative PDEs. Many times the motivation for the various constructions and theorems comes from the corresponding classical probability results. In the three lectures, we will cover: free products of C*- and W*-algebras; free group factors; Voiculescu’s microstates and non-microstates free entropy and free entropy dimension; selected applications to properties of free group factors; connections with the first L² Betti number; and free transport theory.

Video recording: Lecture 1  Lecture 2  Lecture 3

Recently Voiculescu introduced the notion of bi-free independence as a generalization of free independence in order to simultaneously study the left and right regular representations on free products of vector spaces. In this talk, we will provide a brief overview of the current state of bi-free probability. This overview will include basic definitions, bi-free cumulants, bi-free infinitely divisible distributions, operator-valued bi-free independence, bi-free matrix models, and bi-free partial transformations.

Slides

We define new noncommutative spheres with partial commutation relations for the coordinates. We investigate the quantum groups acting maximally on them, which yields new quantum versions of the orthogonal group. The corresponding version of a quantum permutation group, which turns out to be Bichon's quantum automorphism group of a graph, gives the quantum symmetry for Mlotkowski's Lamda-freeness (a mixture of classical and free independence), via Speicher and Wysoczanski's cumulant description of that concept.

This is joint work with Moritz Weber.

Slides

We show that a random d-regular bipartite graph has an optimally large spectral gap with nonzero probability. Notably, we use tools inspired by asymptotic (i.e., large n limit) random matrix theory to prove statements about finite-dimensional matrices. The mediating role is be played by the expected characteristic polynomials of the random matrices in question, exploiting in particular their real-rootedness, interlacing, and invariance properties. Our analysis of the roots of these polynomials is based on finite analogues of tools from Free Probability Theory, in particular a finite-dimensional free convolution and corresponding R-transform inequality.

Joint work with Adam Marcus and Daniel Spielman.

Slides

Frobenius-Schur indicators provide an important invariant for fusion categories, especially for application to classification problems. Their values can be obtained from the modular data of the Drinfel'd center. In several important cases of singly-generates fusion categories this modular data is given by quadratic forms on some associated groups. This leads to the expression of the indicators as quadratic Gauss sums, which yields examples of fusion categories that are completely determined by their indicators. We will discuss the indicators of near-groups and Haagerup-Izumi categories following from the conjectures of Evans and Gannon regarding the modular data for the centers of these categories.

Stefaan Vaes: II₁ factors with exactly two crossed product decompositions

In the first two lectures, I will introduce several of the key methods of Popa's deformation/rigidity theory and present a number of rigidity theorems for group von Neumann algebras and crossed product factors. In the third lecture, these results will be used to sketch the proof of a recent joint work with Anna Krogager, in which we construct II₁ factors that have exactly n group measure space decompositions, up to conjugacy by an automorphism.

Slides (part 1) Slides (part 2) Slides (part 3)

Video recording: Lecture 1  Lecture 2  Lecture 3

Expanders, especially those coming from box spaces of residually finite groups, have been used to test various forms of the coarse Baum-Connes conjecture. The first construction of a pair of expanders, one not coarsely embedding in the other, was provided by Mendel and Naor in 2012. This was extended by Hume in 2014 who constructed a continuum of expanders with unbounded girth, pairwise not coarsely equivalent. In joint work with A. Khukhro, we construct a continuum of expanders with geometric property (T) of Willett-Yu, as box spaces of SL(3,ℤ). We will discuss the following results: if box spaces of groups G, H are coarsely equivalent, then the groups G, H are quasi-isometric (Khukhro and myself), and moreover G and H are uniformly measure equivalent (K. Das).

I will provide a brief introduction to the extension of free probability for systems with two kinds of noncommutative random variables, a group of left and a group of right variables.

Slides (part 1)  Slides (part 2)

Video recording: Lecture 1  Lecture 2

I'll describe how to obtain quasidiagonality for faithful traces on nuclear C*-algebras with the UCT, how this fits in to recent work on the structure and classification of simple nuclear C*-algebras and draw some parallels with the structure of injective von Neumann algebras.

This is joint work with Aaron Tikuisis and Wilhelm Winter.

Slides