Subfactor Seminar
Fall 2014

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Organizers: Dietmar Bisch, Vaughan Jones, and Jesse Peterson

Fridays, 4:10-5:30pm in SC 1432

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• Date: 8/22/14
  • Zhengwei Liu, Vanderbilt University
  • Title: A universal skein theory for subfactor planar algebras
  • Abstract: Given the principle graphs (with dual datas) of a subfactor planar algebra, we provide a universal method to construct its generators and relations and derive its evalutation algorthms. Without the presumed subfactor planar algebra, we can show that the relations are consistent under certain local and global conditions. If the conditions are satisfied, then we will obtain a subfactor planar algebra with the given principle graphs. This method reduces the construction of a subfactor planar algebra to solving two families of equations. As one of its applications, we recover Izumi’s construction of near group subfactors. We also discuss how this universal skein theory is related to Ocneanu’s paragroups and Popa’s $lambda$-lattices.

• Date: 8/29/14
  • Ben Hayes, Vanderbilt University
  • Title: Fuglede-Kadison Determinats and Sofic Entropy
  • Abstract: Let G be a countable discrete group. An algebraic action of G is an action of G by automorphisms on a compact, metrizable, abelian group X. We will typically ignore the algebraic structure of X and think of the action of G on X as either an action on a compact metrizable space by homeomorphisms, or as a probablity measure-preserving action (giving X the Haar measure). When G is sofic, (to be defined in the talk) Lewis Bowen, with an extension by David Kerr and Hanfeng Li, defined the entropy of an action of G on a compact metrizable space or probability space. We will discuss results on the entropy of an algebraic action in the case that X is the Pontryagin dual of a finitely presented Z(G)-module. It turns out the answer is related to Fuglede-Kadison determinants (defined via the group von Neumann algebra) of finite matrices over Z(G). Our work extends results many others: Li-Thom,Kerr-Li,Bowen-Li etc, and in particular is a generalization of recent results of Li-Thom from amenable groups to sofic groups. Moreover, the techniques are the first in the subject to avoid a nontrivial determinant approximation. Time permitting, I will sketch a proof of the main result.

• Date: 9/5/14
  • Marcel Bischoff, Vanderbilt University
  • Title: A center construction for braided subfactors and defects of conformal nets
  • Abstract: Braided subfactors are subfactors whose even part sits inside a given (non-degenerately) braided tensor category C. Using the braiding of C, there is a notion of commutativity for such subfactors. Maximal commutative subfactors in Z(C) can all be obtained by a subfactor in C via a double construction, the generalized Longo-Rehren construction.
    
    Given two commutative subfactors there is the notion of a defect between them, which is basically a commuting square containing both subfactors. We introduce a fusion product between defects and discuss how these defects describe topological defects in algebraic conformal quantum field theory.
    
    We give a classification of defects for subfactors coming from the double construction in terms of bimodules of the original subfactors. This classification will also show that the double construction is basically a special case of the “Functoriality of the center of an algebra” by Kong and Runkel.

• Date: 9/12/14
  • Uffe Haagerup, Copenhagen University
  • Title: The Thompson groups F, T, V and their group C*-algebras and group von Neumann algebras.
  • Abstract: In the talk, I will give an introduction to the three Thompson groups F, T and V, and discuss some recent result obtained in collaboration with Søren Haagerup, Maria Ramirez-Solano and Kristian Knudsen Olesen.
    
    It is a long standing open problem whether the Thompson group F is amenable, or equivalently whether its group von Neumann algebra L(F) isomorphic to the hyperfinite II-1 factor R? Paul Jolissaint has proved that F is inner amenable and L(F) has property Gamma. In a recent work with Kristian Knudsen Olesen, we prove that T and V are not inner amenable and L(T) and L(V) does not have property Gamma. We also prove that if the reduced C*-algebra C*_r(T) of T is simple, then F is non-amenable.
    
    In collaboration with Maria Ramirez-Solano and Søren Haagerup we use extensive numerical computations to test the amenability problem for F by estimating the norms of certain elements of C*_r(F). Numerical computions alone cannot detect whether or not F is amenable, but the results we have obtained suggest that the most likely outcome is that F is non-amenable.

• Date: 9/19/14
  • François Le Maître, École Normale Supérieure de Lyon
  • Title: More Polish full groups
  • Abstract: Full groups were introduced in Dye’s visionary paper of 1959 as subgroups of Aut(X,m) stable under cutting and gluing their elements along a countable partition of the probability space (X,m). However, the focus has since then been rather on full groups generated by countable groups, which are Polish for the uniform topology. This situation is justified by the very nice interplay between these full groups and von Neumann algebras of countable measure preserving equivalence relations. On the other hand, note that the group Aut(X,m) itself is a full group, Polish for the weak topology, and its topological properties are still an important subject of investigation.
    
    In a work in progress with A. Carderi, we investigate Polish full groups of a new kind, whose topology is intermediate between the uniform and the weak topology. They arise as full groups of equivalence relations generated by the Borel action of a Polish group on (X,m), and we will discuss some of their topological properties such as topological rank or amenability.

• Date: 9/22/14 – 9/26/14
  • Von Neumann algebras and ergodic theory, at UCLA

• Date: 9/27/14 – 9/28/14
  • Annual Wabash Miniconference, at IUPUI

• Date: 10/3/14
  • Brent Nelson, UCLA
  • Title: Free transport for finite depth subfactor planar algebras
  • Abstract: Given a finite depth subfactor planar algebra $\mathcal{P}$ endowed with the graded $*$-algebra structures $\{Gr_k \mathcal{P}\}_{k\in\mathbb{N}}$ of Guionnet, Jones, and Shlyakhtenko, there is a sequence of canonical traces $Tr_{k}$ on $Gr_k\mathcal{P}$ induced by the Temperley-Lieb diagrams. Moreover, with trace-preserving embeddings $c_k: Gr_k\mathcal{P} \to \mathcal{B}(H)$ into the bounded operators on a Hilbert space one can generate a tower of von Neumann algebras $\{M_{k}\}_{k\in\mathbb{N}}$ whose inclusions recover $\mathcal{P}$ as its standard invariant. In this talk we discuss how one can use free transport to construct traces $Tr_{k}^{(v)}$ induced by certain small perturbations of the Temperley-Lieb diagrams and trace-preserving embeddings of the $Gr_k\mathcal{P}$ that generate the same tower of von Neumann algebras. Moreover, we obtain a criterion for when the GNS representations of a sequence of traces $\{\tau_k\}_{k\in \mathbb{N}}$ on $\{Gr_k\mathcal{P}\}_{k\in\mathbb{N}}$ generate a tower of von Neumann algebras isomorphic to $\{M_{k}\}_{k\in\mathbb{N}}$.

• Date: 10/10/14
  • Claire Levaillant, UCSB
  • Title: Universal single qubit and qutrit gates in the Kauffman-Jones version of SU(2) Chern-Simons theory at level 4.
  • Abstract: this is a recent development regarding universal topological quantum computation in a specific anyonic system, as appearing in the title, and joint work with Michael Freedman and Station Q. The anyonic system we use is hoped to become physically realizable.
    
    Our starting point are two Jones unitary representations of the braid group on four strands. One representation arises from braiding four anyons of respective topological charges 1,2,2,1 and the second representation occurs when braiding four anyons of identical topological charge 2. Both representations have a finite image and this image yields a finite subgroup of SU(2) and SU(3) respectively whose elements are called quantum gates. By protocols involving both braids and measurements, we show how to make in each case an additional quantum gate. In the qubit case, this new gate generates an infinite subgroup of SU(2) and in the qutrit case, the new gate enlarges the size of the finite SU(3) group issued from braiding only. Our method uses ancilla preparation with adequate norms and interesting relative phases and fusion of the ancilla into the input in order to form the gate.

• Date: 10/11/14 – 10/12/14
  • The 12th East Coast Operator Algebras Symposium, at The Fields Institute

• Date: 10/17/14
  • No Meeting, Fall Break.

• Date: 10/24/14
  • Andre Kornell, UC Berkeley
  • Title: Operator algebras when every set is Lebesgue measurable
  • Abstract: The assumption that every set of real numbers is Lebesgue measurable is particularly convenient in functional analysis. I will describe the theory of operator algebras in one model of set theory satisfying this assumption. I will then discuss the V*-algebras, which are $*$-algebras of bounded operators closed in a continuum analog of the ultraweak topology. Every unital separable C*-algebra has an enveloping V*-algebra, which may be identified with the space of strongly affine functions on the state space.

• Date: 10/31/14
  • Marcel Bischoff, Vanderbilt University
  • Title: Realizing representation categories of compact groups in higher dimensional quantum field theories (after Doplicher, Piacitelli).
  • Abstract: In low-dimensional quantum field theory symmetries can in some sense be described by subfactors and it is an interesting open question if all subfactors arise from quantum field theory.
    
    Doplicher, Haag and Roberts have shown that in 4 dimensions the symmetries of quantum field theory are described by a compact groups. In opposite to low dimensions it is known by a work of Doplicher and Piacitelli, that every metrizable compact group arises as a symmetry group of a quantum field theory model and the goal of this talk is to show how this was achieved: Namely, given any metrizable compact group G, a suitable generating subset K of G and a mass function on K one can construct a net of von Neumann algebras, whose superselection theory is exactly the representation category of G. The construction is based on free fields (CCR and CAR algebras). An important analytical tool to show that G is the full symmetry group is the split property, which states that between the algebras of compactly embedded regions there exists an intermediate type I factor.

• Date: 11/7/14
  • Rémi Boutonnet, UC San Diego
  • Title: Maximal amenable subalgebras arising from maximal amenable subgroups.
  • Abstract: In this talk, based on a joint work with A. Carderi, I will present a new elementary method to prove maximal amenability of certain subalgebras in group von Neumann algebras. The subalgebras in question are coming from maximal amenable subgroups and the method is largely inspired from the group situation: it relies on certain dynamical properties of the groups involved. I shall present various examples where this method applies. I shall also discuss some links with Ozawa’s solidity and strong solidity.

• Date: 11/14/14
  • Vaughan Jones, Vanderbilt University
  • Title:
  • Abstract:

• Date: 11/21/14
  • Akram Aldroubi, Vanderbilt University
  • Title: Dynamical Sampling
  • Abstract: Let $Y=\{f(i), Af(i), \dots, A^{\li}f(i): i \in \Omega\}$, where $A$ is a bounded operator on $\ell^2(I)$. The problem under consideration is to find necessary and sufficient conditions on $A, \Omega, \{l_i:i\in\Omega\}$ in order to recover any $ f \in \ell^2(I)$ from the measurements $Y$. This is the so called dynamical sampling problem in which we seek to recover a function $f$ by combining coarse samples of $f$ and its futures states $A^lf$. For self adjoint operators in infinite dimensional spaces, the M\”untz-Sz\’asz Theorem combined with the Kadison-Singer/Feichtinger Theorem allows us to show that $Y$ can never be a Riesz basis when $\Omega$ is finite. Moreover, when $\Omega$ is finite, $Y=\{f(i), Af(i), \dots, A^{\li}f(i): i \in \Omega\}$ is not a frame except for some very special cases. The existence of these special cases is derived from Carleson’s Theorem for interpolating sequences in the Hardy space $H(D)$.

• Date: 11/28/14
  • No Meeting, Thanksgiving Break.

• Date: 12/5/14
  • Jesse Peterson, Vanderbilt University
  • Title: Character rigidity for lattices in higher-rank groups
  • Abstract: We show that lattices in a higher rank center-free simple Lie groups are operator algebraic superrigid, i.e., any unitary representation of the lattice which generates a II_1 factor extends to a homomorphism of its group von Neumann algebra. This generalizes results of Margulis, and Stuck and Zimmer, and answers in the affirmative a conjecture of Connes.

• End of Fall Semester.