• Date: 8/28/20
  • Marcel Bischoff, Ohio University
  • Title: Compact Hypergroups from Discrete Subfactors
  • Abstract: We show that to any local braided discrete subfactor N \subset M of type III one can associate a ”compact hypergroup” acting by extremal ucp maps on M, such that N is given by the fixed point algebra under this action. If the subfactor is also of depth two, then the hypergroup is exactly a compact group G and N is the fixed point under a minimal action of G. The motivation is to obtain an invariant and understand discrete inclusions of conformal nets. Based on joint work with Simone Del Vecchio, Luca Giorgetti (arXiv:2007.12384)

• Date: 9/4/20
  • Roberto Hernandez Palomares, The Ohio State University
  • Title: Representations of rigid C*-tensor categories over GJS C*-algebras
  • Abstract: We will construct a fully faithful bi-involutive strong monoidal representation of an arbitrary countably generated RC*TC onto a subcategory of finitely generated projective Hilbert C*-bimodules over a simple separable unital monotracial C*-algebra, using the diagrammatic methods from Guionnet, Jones and Shlyakhtenko. Out of this category of bimodules we will construct a similar functor into the bifinite and spherical bimodules over an interpolated free group factor. The composite of these two functors recovers the representation constructed by Brothier, Hartglass and Penneys. Based on joint work with Hartglass arXiv:2005.09821.

• Date: 9/11/20
• Yi Wang, Vanderbilt University
• Title: The Arveson-Douglas Conjecture
• Abstract: The Arveson-Douglas Conjecture states that certain submodules and quotient modules of the Bergman/Hardy/Drury-Arveson module is essentially normal. The conjecture starts as an operator theoretic problem but recent research shows that it is deeply involved with complex geometry.  In this talk, I will introduce this conjecture, its background and applications. I will also talk about several approaches and recent developments.
Math Colloquium
• Date: 9/17/20 (Thursday, 4:10-5:00pm)
• Romain Tessera, Universite de Paris, IMJ-PRG, CNRS
• Title: Quantitative measure equivalence
• Abstract: Measure equivalence is an equivalence relation between countable groups that has been introduced by Gromov. A fundamental instance are lattices in a same locally compact group. According to a famous result of Ornstein Weiss, all countable amenable groups are measure equivalent, meaning that geometry is completely rubbed out by this equivalence relation. Recently a more restrictive notion has been investigated called integrable measure equivalence, where the associated cocycles are assumed to be integrable. By contrast, a lot of surprising rigidity results have been proved: for instance Bowen has shown that the volume growth is invariant under integrable measure equivalence, and Austin proved that nilpotent groups that are integrable measure equivalent have bi-Lipschitz asymptotic cones. I will present a work whose goal is to understand more systematically how the geometry survives through measure equivalence when some (possibly very weak) integrability condition is imposed on the cocycles. We shall put the emphasis on amenable groups, for which we will present new rigidity results, and the first flexibility results known in this context.
Date: 9/18/20
• Jesse Peterson, Vanderbilt University
• Title: Properly proximal von Neumann algebras
• Abstract: Proper proximality is a geometric property for groups, which has a number of applications to rigidity phenomena for von Neumann algebras. In 2019 the speaker, together with Ishan Ishan and Lauren Ruth, showed that proper proximality is a von Neumann algebra invariant, in the sense that if two groups have isomorphic group von Neumann algebras then one is properly proximal if and only if the other is. This suggests that proper proximality should be a notion that generalizes to all von Neumann algebras. In this talk we will show that this is indeed the case, and we will discuss how in this setting we are able to generalize previous decomposition results. This is based on joint work with Changying Ding and Srivatsav Kunnawalkam Elayavalli.
• Date: 9/25/20
  • Ionut Chifan, University of Iowa
  • Title: New examples of W* and C*-superrigid  groups 
  • Abstract: In the the mid thirties F. J. Murray and J. von Neumann found a natural way to associate a von Neumann algebra L(G) to every countable discrete group G. Classifying L(G) in terms of G emerged as a natural yet quite challenging problem as these algebras tend to have very limited “memory” of the underlying group. This is perhaps best illustrated by Connes’ famous result asserting that all icc amenable groups give rise to isomorphic von Neumann algebras; therefore, in this case, besides amenability, the algebra has no recollection of the usual group invariants like torsion, rank, or generators and relations. In the non-amenable case the situation is radically different; many examples where the von Neumann algebraic structure is sensitive to various algebraic group properties have been discovered via Popa’s deformation/rigidity theory.  

    In my talk I will focus on an extreme situation, namely when L(G) completely remembers the underlying group G; such groups G are called \emph{$W^*$-superrigid}. Currently there have been identified only two types of group theoretic constructions that lead to $W^*$-superrigid groups: some classes of generalized wreath products groups with abelian base (Ioana-Popa -Vaes '10, Berbec-Vaes ‘13) and amalgamated free products (Chifan-Ioana '16). After briefly surveying these results I will introduce several new constructions of $W^*$-superrigid groups which include direct product groups, semidirect products with non-amenable core, and tree groups (iterations of amalgams and HNN-extensions). In addition, I will present several applications of these results to the study of rigidity in the C*-setting. This is based on a very recent joint work with Alec Diaz-Arias and Daniel Drimbe. 

• Date: 10/2/20
  • Cain Edie-Michell, Vanderbilt University
  • Title: Symmetries of Modular Categories, and Quantum Subgroups
  • Abstract: Since the problem was introduced by Ocneanu in the late 2000's, it has been a long-standing open problem to completely classify the quantum subgroups of the simple Lie algebras. This classification problem has received considerable attention, due to the correspondence between these quantum subgroups, and the extensions of WZW models in physics. A rich source of quantum subgroups can be constructed via symmetries of certain modular tensor categories constructed from Lie algebras. In this talk I will describe the construction of a large class of these symmetries. Many exceptional examples are found, which give rise to infinite families of new exceptional quantum subgroups.

• Date: 10/9/20
  • Kate Juschenko, UT Austin
  • Title: On skew amenability of topological groups
  • Abstract: We study the concept of skew-amenability for topological groups. Among other things, we prove that Thompson's group F is skew-amenable with respect to the topology of point-wise convergence arising from its action on the set dyadic rationals; Aut(D, <) is not skew-amenable with respect to the corresponding topology of point-wise convergence; the amenability of the discrete group F is equivalent to skew-amenability of a certain topological group given by a semidirect product of F and abelian group. We also show that skew-amenability is not stable under semi-direct products of topological groups, which answers a question of Pestov. This is a joint Martin Schneider.

• Date: 10/16/20
  • Jason Crann, Carleton University
  • Title: Amenable dynamical systems over locally compact groups
  • Abstract: We establish several new characterizations of amenable W*- and C*-dynamical systems over arbitrary locally compact groups. In the W*-setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz-Schur multipliers of (M,G,alpha) converging point weak* to the identity of the von Neumann crossed product. In the C*-setting, we show that amenability of (A,G,alpha) is equivalent to an analogous Herz-Schur multiplier approximation of the identity of the reduced crossed product, as well as a particular case of the positive weak approximation property of Bédos and Conti. When Z(A**)=Z(A)**, it follows that amenability is equivalent to the 1-positive approximation property of Exel and Ng. In particular, when A=C_0(X) is commutative, amenability of (C_0(X),G,alpha) coincides with topological amenability the G-space X. This is joint work with Alex Bearden.

• Date: 10/23/20
  • Michael Brannan, Texas A&M University
  • Title: Quantum graphs and quantum Cuntz-Krieger algebras
  • Abstract: In this talk I will give a light introduction to the theory of quantum graphs. Quantum graphs are generalizations of directed graphs within the framework of non-commutative geometry, and they arise naturally in a surprising variety of areas including quantum information theory, representation theory, quantum symmetries of graphs, and in the theory of non-local games. I will give an overview of some of these connections and also explain how one can generalize the well-known construction of Cuntz-Krieger C*-algebras associated to ordinary graphs to the setting of quantum graphs.  Time permitting, I will also explain how quantum symmetries of quantum graphs can be used to shed a great deal of light on the structure of quantum Cuntz-Krieger algebras.  (This is joint work with Kari Eifler, Christian Voigt, and Moritz Weber.)  

• Date: 10/30/20
  • Rufus Willett, University of Hawai'i at Mānoa
  • Title:Two 'von Neumann algebra’ results for uniform Roe algebras
  • Abstract: Uniform Roe algebras are C*-algebras that encode the large scale geometry of a (discrete) metric space X such as the integers, or more generally a finitely generated group with word metric. They are von Neumann algebras only in the trivial case that the metric space is finite, but have a somewhat von Neumann algebraic flavor due to the presence of l^\infty(X) as a diagonal MASA.
    
    I’ll explain two theorems that hold for uniform Roe algebras: they are analogues of results that are well-studied for von Neumann algebras, but more typically fail for C*-algebras. The first is joint work from a couple of years ago with Stuart White, and shows that the diagonal MASA is (sometimes) unique up to unitary conjugacy. The second is more recent joint work with Matthew Lorentz, and shows that all bounded derivations are inner. In both cases a key ingredient is a Ramsey-theoretic idea of Braga and Farah that in some sense plays the role that weak-* compactness of the unit ball would play for a von Neumann algebra. I’ll try to explain all of this.

• Date: 11/6/20
  • James Tener, Australian National University
  • Title: Fusion, positivity, and finite-index subfactors 
  • Abstract: In the 90's, Wassermann showed that the subfactors of index less than 4 of type A were realized via conformal field theories corresponding to SU(2) at level N. This landmark work relied on an analysis of the "Connes' fusion" of bimodules via quantum fields and differential equations. In this talk we will discuss a generalization of these ideas in terms of a recent 'geometric realization' of subfactors arising in conformal field theory. In addition, we will discuss how to use this approach to obtain finite index subfactors from Lie groups of all types. 

• Date: 11/13/20
  • Pieter Spaas, UCLA
  • Title: Cohomological obstructions to the (local) lifting property for full C*-algebras of property (T) groups
  • Abstract: We will recall and discuss the lifting property (LP) and local lifting property (LLP) for full group C*-algebras. In particular, we will introduce a new method to refute these properties, based on non-vanishing of second cohomology groups. This will allow us to derive that many natural examples in the presence of (relative) property (T), like Z^2\rtimes SL_2(Z) and SL_n(Z), for n>2, fail the LLP, and further large classes of groups with property (T) fail the LP. (Moreover, even though these are results on the level of the group C*-algebras, we will see that they also have a strong von Neumann algebraic and approximate representation theoretic flavor.) This is based on joint work with Adrian Ioana and Matthew Wiersma.

• Date: 11/20/20
  • Luca Giorgetti, Vanderbilt University
  • Title: Distortion for II₁ multifactor inclusions
  • Abstract: We introduce an invariant, called distortion, for inclusions of II₁ von Neumann algebras with finite index and finite-dimensional centers, thus called multifactors. Distortion allows to determine when the inclusion is extremal, when it admits an infinite tunnel and, if not, how many times one can perform the downward basic construction. We extend Popa’s classification of finite index finite depth hyperfinite subfactors to multifactors by means of the standard invariant, i.e. the 2-shaded unitary planar algebra associated with the inclusion, and the distortion.

    Joint work with M. Bischoff, I. Charlesworth, S. Evington, D. Penneys, arXiv:2010.01067, supported by ERC MSCA-IF beyondRCFT n. 795151.

• Date: 11/27/17
  • No Meeting, Thanksgiving Break.

• Date: 12/4/20 Different time: 11am-12:30pm
  • Remi Boutonnet, CNRS, Universite de Bordeaux
  • Title: On stationary actions of higher rank semi-simple lattices on C*-algebras
  • Abstract: Over the past decade, extensive work on C*-simplicity of various groups has led to many breakthrough results. At the origin of this effervescence lies the study of group actions on (non-commutative) C*-algebras. In this talk, I will explain how to push this idea further to study arbitrary unitary representations of higher rank semi-simple lattices. The idea is to extend Margulis' approach of his normal subgroup theorem via ergodic theory to the non-commutative setting. This is partially inspired by recent work of Peterson, although we use a different strategy, based on stationary dynamics and work of Nevo and Zimmer. This talk is based on joint works with Uri Bader, Cyril Houdayer and Jesse Peterson.

• End of Fall Semester.

Past Subfactor and NCGOA seminars