# Schedule of the Workshop "Von Neumann Algebras"

## Monday, July 4

10:30 - 11:00 |
Registration & Welcome coffee |

11:00 - 11:10 |
Opening remarks |

11:10 - 12:00 |
Ken Dykema: Haagerup-Schultz projections and upper triangular forms in II_{1}-factors |

12:00 - 14:00 |
Lunch break |

14:00 - 14:50 |
Hari Bercovici: Conjectures related to the Connes embedding problem |

15:00 - 15:50 |
Thomas Sinclair: Product rigidity for group von Neumann Algebras |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:00 |
Tim de Laat: Howe-Moore type theorems for quantum groups and rigid C*-tensor categories |

17:10 - 17:40 |
Yusuke Isono: On fundamental groups of tensor product II_{1} factors |

17:50 - 18:20 |
Tucker-Drob: Ergodic hyperfinite subgraphs and primitive subrelations |

afterwards |
Reception |

## Tuesday, July 5

9:10 - 10:00 |
Sergey Neshveyev: Drinfeld center, tube algebra, and representation theory of monoidal categories |

10:10 - 10:40 |
Sven Raum: Operator algebras of locally compact groups acting on trees |

10:40 - 11:10 |
Group photo and coffee break |

11:10 - 12:00 |
Yoshimichi Ueda: Structural results of free product von Neumann algebras |

12:00 - 14:00 |
Lunch break |

14:00 - 14:50 |
Makoto Yamashita: Subfactor and operator system theoretic aspects of categorical Poisson boundary |

15:00 - 15:50 |
Wilhelm Winter: Von Neumann algebra techniques in C*-algebra classification |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:20 |
Yuki Arano: Unitary representations of Drinfeld doubles |

## Wednesday, July 6

9:10 - 10:00 |
Vaughan F. R. Jones: On the semicontinuous limit of quantum spin chains |

10:10 - 10:40 |
James Tener: Finite index subfactors from vertex operator algebras |

10:40 - 11:10 |
Coffee break |

11:10 - 12:00 |
Tobias J. Osborne: Towards effective conformal field theories for tensor network states |

12:00 - 14:00 |
Lunch break |

14:00 - 14:50 |
Roberto Longo: KMS states in Conformal Field Theory |

15:00 - 15:50 |
Arthur Jaffe: Reflection Positivity: Something Old, Something New, Something Still to Do |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:00 |
Marcel Bischoff: Generalized fixed points of conformal nets and quantum double subfactors |

## Thursday, July 7

9:10 - 10:00 |
Zhengwei Liu: Planar anionic algebras |

10:10 - 10:40 |
Arnaud Brothier: Analytical properties for subfactors |

10:40 - 11:10 |
Coffee break |

11:10 - 12:00 |
Pinhas Grossman: Brauer-Picard groups of certain small-index subfactors |

12:00 - 14:00 |
Lunch break |

14:00 - 14:50 |
David Penneys: Bicommutant categories from (multi)fusion categories |

15:00 - 15:50 |
Noah Snyder: Where do subfactors come from? |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:20 |
Shamindra Kumar Ghosh: Annular representations of C*-tensor categories |

## Friday, July 8

9:10 - 10:00 |
Alice Guionnet: Random tilings and Schwinger-Dyson equations |

10:10 - 10:40 |
Michael Hartglass: Graphs (and standard invariants) as compact quantum metric spaces |

10:40 - 11:10 |
Coffee break |

11:10 - 12:00 |
Yoann Dabrowski: Free entropy and a Laplace Principle for Hermitian Brownian Motion. |

12:00 - 14:00 |
Lunch break |

14:00 - 14:50 |
Yoshikata Kida: Stable actions and central extensions |

15:00 - 15:50 |
Marius Junge: Operator valued q-gaussian algebras |

16:00 - 16:30 |
Tea and cake, end of workshop |

# Abstracts

(Underlined titles can be clicked for the video recording)

## Yuki Arano: Unitary representations of Drinfeld doubles

As has been observed by many authors, the Drinfeld double of the q-deformation of a compact Lie group can be regarded as a quantization of the complexification of the original Lie group. Using this point of view, I will discuss irreducible unitary representations of these Drinfeld doubles and compare with the classical case. As an application, we prove the dual of a q-deformation of compact Lie group of rank equal or more than 2 has central property (T).

It is known that one can associate with each pair of selfadjoint elements in II_{1} factor an essentially combinatorial object known as a hive. Making this association more constructive offers a possible approach to the embedding problem for such factors. We will discuss one approach to an "extremal" part of the collection of hives.

Conformal field theory can be axiomatized using von Neumann algebras, namely by so-called conformal nets. I will review the structure of finite index subnets of completely rational conformal nets, their representation theory and relations to quantum double subfactors.

Then I will introduce a notion of fixed points of conformal nets by actions of finite hypergroups which are generalizing fixed points by actions of finite groups.

Finally, I will discuss some structural results for hypergroup actions on conformal nets. For example, if a net has trivial representation category, the representation category of the fixed point net is the Drinfeld center of a categorification of the hypergroup.

We will discuss about analytic properties for groups and their generalizations to subfactors, standard invariants, and certain tensor categories. We will present examples of subfactors with prescribed properties such as weak amenability or the Haagerup property.

I will explain the solution of old problems concerning Voiculescu's free entropy. I will survey several ideas behind these results ranging from ultraproducts techniques and continuous model theory to convex analysis and optimization related to free stochastic differential equations. Finally, I will suggest how this renews the free probability approach to Connes embedding conjecture.

## Ken Dykema: Haagerup-Schultz projections and upper triangular forms in II_{1}-factors

Brown measure is a sort of spectral distribution for arbitrary operators (including non-selfadjoint ones) in II_{1}-factors. Haagerup and Schultz proved existence of hyperinvariant projections for operators in II_{1}-factors, that decompose the Brown measure. With Sukochev and Zanin, we used these to prove a sort of upper trianguler decomposition result for such elements, analogous to Schur's famous result for matrices. More recently, we have partially extended these results to certain unbounded operators affiliated with II_{1}-factors. One application is to show that every trace is spectral (i.e., the value of the trace depends only on the Brown measure of the operator) for traces on certain bimodules of affiliated operators (these are often called Dixmier traces). Time permitting, we will mention some results about the nature of the spectrally trivial parts in these upper triangular forms.

Annular representations of finite index subfactors / planar algebras were introduced by Vaughan Jones. This turned out to be one of the most important tool to construct new subfactors. On the other hand, (along with Paramita Das and Ved Gupta) we showed that the annular representation category is a very nice braided tensor category (which in finite depth case, is equivalent to the Drinfeld center of the bimodule categories of the subfactor). Further, along with Corey Jones, we relate this with representations of Ocneanu's tube algebra, and also show that the annular representation category of a rigid C*-tensor category C, captures various analytic properties of C in the sense of Popa and Vaes. Neshveyev and Yamashita studied the center of Ind C which turns out to be equivalent to the annular representation category.

In my talk, I will describe the above theory and discuss some examples.

## Pinhas Grossman: Brauer-Picard groups of certain small-index subfactors

The Brauer-Picard group of a fusion category is its group of Morita autoequivalences. It is isomorphic to the group of braided autoequivalences of the Drinfeld center. The Brauer-Picard group plays a central role in the theory of extensions of fusion categories by finite groups, due to Etingof, Nikshych, and Ostrik.

In this talk we will discuss some recent progress in describing the Brauer-Picard groups of some small-index subfactors.

Given a weighted graph, I will construct a "loop algebra" associated to the graph, and will discuss how it forms a compact quantum metric space in the sense of Rieffel. I will then present an application to planar algebras. This is joint work with Dave Penneys.

## Yusuke Isono: On fundamental groups of tensor product II_{1} factors

We introduce another notion of primeness for II_{1} factors, and give some examples. As a corollary, we prove that if G and H are groups realized as fundamental groups of II_{1} factors (with separable predual), then so is the group generated by G and H.

## Arthur Jaffe: Reflection Positivity: Something Old, Something New, Something Still to Do

What is RP? Why is RP important? Some new insight into RP.

In an attempt to produce a conformal field theory associated with any subfactor/fusion category we have constructed a Hilbert space corresponding to a scale invariant quantum spin chain on the dyadic rationals. This Hilbert space carries a natural action of the dyadic rationals on the circle but we can prove that this action is highly discontinuous for the topology of the circle. The proof uses the transfer matrix and we go on to construct a scale invariant transfer matrix depending on a “spectral parameter” which has interesting properties, some pathological as we would expect from the nogo theorem and some rather striking like commuting transfer matrices for different values of the spectral parameter. We work in some specific models - the simplest trivalent fusion categories in the sense of Morrison-Peters-Snyder - which give rise to the R-matrices of Izergin-Korepin. One may ask whether such nice R-matrices exist for other trivalent categories.

We consider a particular example of algebras given by generators and relations. Indeed, q-deformations of the classical gaussian variables have obtained a fair amount of attention through recent work of Darbrowski, Guionnet, Nelson, and Shlyahtenko. We consider operator valued generalization of these results, given by interesting combinatorial multiplications rules and show strong solidity results under suitable assumptions.

Joint work with Bogdan Udrea.

## Yoshikata Kida: Stable actions and central extensions

A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite equivalence relation of type II_{1} under direct product. Some conditions for groups to have a stable action involve inner amenability and property (T). Among other things, I will discuss a characterization of a central group-extension having a stable action.

## Zhengwei Liu: Planar anionic algebras

We recall some fundamental concepts, results and questions in subfactor theory. Motivated by them, we introduce planar anionic algebras and answer a couple of questions. We give new methods to construct subfactors using planar anionic algebras. Furthermore, we talk about parafermion algebras which are planar anionic algebras satisfying additional axioms, namely planar para algebras. These examples lead to holographic software for quantum information and surprising applications from subfactor theory to quantum information and backwards.

## Tim de Laat: Howe-Moore type theorems for quantum groups and rigid C*-tensor categories

A locally compact group has the Howe-Moore property if for every unitary representation without invariant vectors, the matrix coefficients of the representation vanish at infinity. It is well known that this property holds for connected non-compact simple Lie groups with finite center and certain classes of other groups. I will explain a recent joint work with Yuki Arano and Jonas Wahl, in which we proved Howe-Moore type results in the setting of quantum groups and in the setting of rigid C*-tensor categories. Most importantly, we proved that the representation categories of q-deformations of connected simply connected compact simple Lie groups satisfy the Howe-Moore property for rigid C*-tensor categories.

A survey on the structure of KMS states on a local conformal nets with respect to translations (joint work with P. Camassa, Y. Tanimoto and M. Wiener) and with respect to rotations (joint work with Y. Tanimoto)

I will review and clarify the connections between several constructions in category theory, subfactor theory and quantum groups, such as Drinfeld center, Drinfeld double, Ocneanu's tube algebra and annular representation theory. Some of these connections have been known for a long time, for other a progress has been made over the last couple of years. One outcome of this progress is the possibility to define a C*-completion of the fusion algebra of a tensor category analogous to the construction of the full group C*-algebra of a discrete group. This, in turn, allows one to introduce approximation properties of tensor categories and, in particular, unify such notions as central property (T) for discrete quantum groups and Popa's property (T) for subfactors. (Joint work with Makoto Yamashita)

Recently, building on new ideas in quantum information theory emerging from the study of quantum entanglement, expressive variational classes of quantum states known as tensor network states (TNS) have been developed to model quantum lattice systems approaching quantum phase transitions. These classes have many similarities to representations of planar algebras, and includethe projected entangled-pair states (PEPS) and the multiscale entanglement renormalisation ansatz (MERA). After several years of intensive investigation in the physical literature it has been comprehensively established that PEPS and MERA states provide faithful and parsimonious descriptions of quantum critical phenomena in low dimensions. Tensor network states such as the MERA class enjoy many fascinating properties, including: (1) all n-pt functions may be calculated efficiently; (2) they are not Gaussian states; and (3) they capture conformal data for critical lattice systems. In this talk I will review some of the state of the of the TNS literature. Then I will explain how to build a continuum limit for these states known as the semicontinuous limit. The construction crucially builds on an idea of Jones, who found unitary representations of a discrete analogue of the conformal group known as Thompson's group T. Here I'll describe how to identify lattice observables corresponding to the (primary, secondary, …) quantum fields with fluctuation operators, which obey a lie algebra given by an Inönü-Wigner group contraction of a loop algebra giving rise, in some cases, to representations of Kac-Moody algebras. Ultimately this should give rise to conformal nets for new CFTs. This is work in progress.

I'll discuss an ongoing joint project with Andre Henriques. Just as a tensor category is a categorification of a ring, and its Drinfeld center is a categorification of the center of a ring, a bicommutant category is a categorification of a von Neumann algebra. I'll define the notion of the commutant C' of a tensor category C inside an ambient tensor category B. Since there is a functor C' to B, we can then take the bicommutant C'' inside B, and C naturally sits inside C''. Note, however, that C'' is not always equivalent to its bicommutant!

Because we are interested in von Neumann algebras, we work in the ambient category B=Bim(R), the tensor category of bimodules over a hyperfinite von Neumann factor R, which can be thought of as a categorification of B(H). Given a unitary fusion category C inside Bim(R), we identify its bicommutant C'', and we show that C'' is a bicommutant category. This categorifies the theorem by which a finite dimensional *-algebra that can be faithfully represented on a Hilbert space is actually a von Neumann algebra.

We will also discuss applications, including machinery to construct elements of C', the quantum double subfactor, and commutants of multifusion categories.

I will present my work on C*-simplicity of locally compact groups, focusing on its relevance for studying locally compact groups acting on trees. First, I will summarising results that I could obtain in 2015 on simplicity and non-simplicity of reduced group C*-algebras of locally compact groups. After describing some examples of C*-simple groups acting on trees, I will describe the type I conjecture for closed subgroups of Aut(T) and how different C*-algebraic and von Neumann algebraic results can contribute to its clarification.

I will discuss some recent results around rigidity results of group von Neumann algebras of product groups and applications to group factors of irreducible lattices.

## Noah Snyder: Where do subfactors come from?

The first subfactors studied came from actions of finite groups. The subject took off with the Jones-Temperley-Lieb subfactors, which are closely related to the quantum group su(2). These two worlds, finite groups and quantum groups, remain the two main sources of subfactors. One can use various constructions (such as reduced subfactors, orbifolds, higher Morita equivalence, graded extensions, free products) to find create new subfactors out of finite groups and quantum groups. But are there any subfactors that don't come from groups or quantum groups? A third major source of subfactors are Izumi's quadratic subfactors. Although these are still not completely understood, there appear to be infinitely many such subfactors based on computations of Izumi and Evans-Gannon. Are there any other subfactors which do not fit into any of these three worlds? The main way to find new subfactors is through a classification program, for example the small index classification or the Bisch-Jones skein theoretic classification. I'll summarize these classification programs and explain which fit into which of these three worlds, and which subfactors remain a mystery.

## James Tener: Finite index subfactors from vertex operator algebras

A recent theorem of Carpi, Kawahigashi, Longo and Weiner produces a conformal net from a suitable vertex operator algebra. Representations of the conformal net produce subfactors, which should correspond to representations of the VOA. In this talk, I will discuss a method for showing that these subfactors have finite index in a large family of examples.

## Robin Tucker-Drop: Ergodic hyperfinite subgraphs and primitive subrelations

We show that every ergodic p.m.p. graph has an ergodic hyperfinite subgraph. We use this to show that every p.m.p. ergodic treeable equivalence relation has an ergodic hyperfinite primitive subrelation, and that every free p.m.p. ergodic action of the free group F_n is orbit equivalent to an action in which one of the generators acts ergodically. This also leads to a new proof of Hjorth's Lemma on cost attained.

## Yoshimichi Ueda: Structural results of free product von Neumann algebras

I will report on several structural results on general free product von Neumann algebras. Part of the contents are taken from my joint works with Cyril Houdayer and Reiji Tomatsu.

## Wilhelm Winter: Von Neumann algebra techniques in C*-algebra classification

I will survey some recent breakthroughs in the structure and classification theory of nuclear C*-algebras. These run parallel to (and were originally motivated by) Connes’ classification of injective factors - but only in the last few years it has become clear just how solid the analogy really is. I will highlight some of the key steps involving ideas from von Neumann algebras and speculate about future developments.

The study of categorical Poisson boundary involves in a crucial way intricate techniques from the theories of subfactors and operator systems. These include a generalized Pimsner-Popa type estimate for relative entropy for finite dimensional algebras, and the use of Jensen’s inequality to show the multiplicativity of Poisson integral. I will highlight some of these results which would be of interest even outside of categorical framework. Based on joint work with S. Neshveyev.