Subfactor Seminar
Fall 2015

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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: 8/28/15
  • Ben Hayes, Vanderbilt University
  • Title: 1-bounded entropy and regularity problems in von Neumann algebras
  • Abstract: We introduce and investigate the singular subspace of an inclusion of a tracial von Neumann algebra N into another tracial von Neumann algebra M. The singular subspace is a canonical N-N subbimodule of L^2(M) containing the normalizer, the quasi-normalizer (introduced by Izumi-Longo-Popa), the one-sided quasi-normalizer (introudced by Fang-Gao-Smith), and the wq-normalizer (introduced by Galatan-Popa). By abstracting Voiculescu’s original proof of absence of Cartan subalgebras, we show that the von Neumann algebra generated by the singular subspace of a diffuse, hyperfinite subalgebra is not L(F_2). We rely on the notion of being strongly 1-bounded, due to Jung, and the 1-bounded entropy, a quantity which measures “how” strongly 1-bounded an algebra is. Our methods are robust enough to repeat this process by transfinite induction and we use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of II_1-factors. We also present applications to nonisomoprhism problems for Free-Araki woods factors, as well as crossed products by Free Bogoliubov automorphisms in the spirit of Houdayer-Shlyakhtenko.

• Date: 9/4/15
  • Sandeepan Parekh, Vanderbilt University
  • Title: Uncountably many II_1 factors with non-isomorphic ultrapowers [After Boutonnet-Chifan-Ioana]
  • Abstract: McDuff proved the existence of uncountably many non-isomorphic separable II_1 factors in 1969. In a recent paper, Boutonnet-Chifan-Ioana have shown the same family has non-isomorphic ultrapowers, thus showing the existence of a continuum of non-elementarily equivalent II_1 factors. In this seminar talk I shall present their proof.

• Date: 9/11/15
  • Koichi Shimada, University of Tokyo
  • Title: Approximate Unitary Equivalence of Finite Index Endomorphisms of the AFD Factors
  • Abstract: We characterize the condition for two finite index endomorphisms on AFD factors to be mutually approximately unitarily equivalent. The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of the AFD factors, obtained by Kawahiashi–Sutherland–Takesaki and Masuda–Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.

• Date: 9/18/15
  • Arnaud Brothier, Vanderbilt University
  • Title: Subfactors, Hecke pairs, and approximation properties
  • Abstract: We consider subfactor planar algebras that are constructed with a group acting on a bipartite graph. There is a Hecke pair of countable discrete groups associated with this construction. We show that if this Hecke pair is amenable, has the Haagerup property, or is weakly amenable, then the subfactor planar algebra is amenable, has the Haagerup property, or is weakly amenable respectively. We exhibit an infinite family of subfactor planar algebras with non-integer index that are non-amenable, have the Haagerup property and have the complete metric approximation property.

• Date: 9/25/15
  • Corey Jones, Vanderbilt University
  • Title: Quantum G_2 categories have property (T)
  • Abstract: We discuss the notion of property (T) for rigid C*-tensor categories, and sketch a proof that the categories arising as unitary representations of the quantum groups U_q(G_2) corresponding to the exceptional Lie group G_2 have property (T) for positive q \ne 1.

• Date: Wednesday, 9/30/15, 3:10pm-4:00pm, in SC 1432
  • Robert Coquereaux, Centre de Physique Théorique. CNRS.
  • Title: Immmersions of the circle in the sphere and in higher Riemann surfaces.
  • Abstract: We encode circle immersions with n crossings in terms of orbits of the centraliser of a special element of the symmetric groups S(2n) or S(4n) acting by conjugation on particular subsets, or in terms of appropriate double cosets. The details depend on the various orientability hypothesis made on the source (the circle) and on the target (a surface of genus g), and also on a possible constraint of bi-colariability that one can furthermore impose.
    
    We count and tabulate non-equivalent images of spherical immersions up to 10 crossings therefore recovering and extending results by Arnold (5 crossings) and followers (7 crossings), we also obtain the corresponding classifications for genus higher than 0. In the latter case we introduce the notion of bicolourability and determine the first terms (up to 9 crossings) of the corresponding sequences.
    
    This presentation summarizes recent work done in collaboration with with J.-B. Zuber.

• Date: 10/3/15 – 10/4/15
  • East Coast Operator Algebras Symposium, at University of Iowa

• Date: 10/10/15 – 10/11/15
  • West Coast Operator Algebras Seminar, at UCSD

• Date: Wednesday, 10/14/15, 3:10pm-4:00pm, in SC 1432
  • Jean Renault, Université d’Orléans
  • Title: Random walks on Bratteli diagrams.
  • Abstract: Let X be the path space of a Bratteli diagram. The Markov measures are the measures \mu on X which are quasi-invariant under the tail equivalence relation and which admit a quasi-product cocycle as Radon-Nikodym derivative. This observation provides a classical identification of the ergodic decomposition of \mu. This will be applied to the study of random walks on groups or on groupoids. This study is a joint ongoing project with T. Giordano

• Date: 10/16/15
  • No Meeting, Fall Break.

• Date: 10/23/15
  • Zhengwei Liu, Harvard University
  • Title: Parasymmetry for subfactors
  • Abstract: The universal skein theory provides a general framework to study subfactors and planar algebra. I will talk about different generalizations based on some examples. Motivated by the example from parafermions, I introduce the parasymmetry for subfactors and generalize some concepcts in subfactor theory, such as standard invariants, paragroups, $\lambda$ lattices, and planar algebras. As an application, infinitely many finite/infinite depth subfactors are constructed.

• Date: 10/30/15
  • Yunxiang Ren, Vanderbilt University
  • Title:
  • Abstract:

• Date: 11/6/15
  • Marcel Bischoff, Vanderbilt University
  • Title: Generalized Orbifold Construction for Conformal Nets
  • Abstract: A conformal net A describes a chiral conformal field theory on the circle using von Neumann algebras. A G-orbifold of A is the fixed point by a finite group G acting by (gauge) automorphisms of A. We generalize this construction to certain actions of finite hypergroups and show that every finite index inclusion of rational conformal nets is such a generalized orbifold. These inclusions can be seen as generalizations of GHJ subfactors. We also get a generalization of the folk theorem saying that the representation category of a G-orbifold of a holomorphic conformal net is equivalent to the representation category of a possibly twisted quantum double of G.

• Date: 11/7/15
  • VOTCAM 2015, at The University of Virginia

• Date: 11/13/15
  • Matthew Kennedy, University of Waterloo
  • Title: C*-simplicity for discrete groups.
  • Abstract: A discrete group is said to be C*-simple if its reduced C*-algebra is simple. It is not difficult to see that a group with this property does not have any non-trivial normal amenable subgroups, however it was an open question for many years to determine whether the converse holds. Recent examples constructed by Le Boudec show that the answer to this question is negative, but raise the question of whether there is an intrinsic algebraic characterization of C*-simplicity. In this talk, after a brief review of some background material, I will discuss recent work providing such a characterization.

• Date: 11/20/15
  • Rolando de Santiago, University of Iowa
  • Title: Product rigidity for the von Neumann algebras of hyperbolic groups.
  • Abstract: Suppose $\Gamma_1,\ldots, \Gamma_n $ are each hyperbolic i.c.c. groups and $L(\Gamma_1\times\cdots\times \Gamma_n) \cong L(\Lambda) $ for an arbitrary group $\Lambda $. Then we show $\Lambda $ decomposes as an $n $-fold product, $\Lambda= \Lambda_1\times\cdots\times \Lambda_n $ such that for each $i=1,\ldots, n $, we have \$L(\Gamma_i)\cong L(\Lambda_i) $ up to amplification. This strengthens Ozawa and Popa’s unique prime decomposition results.

• Date: 11/27/15
  • No Meeting, Thanksgiving Break.

• Date: 12/4/15
  • Daniel Hoff, UC San Diego
  • Title: Unique Prime Factorization for ${\rm II}_1$ Factors with Cartan Subalgebras
  • Abstract: A tracial von Neumann algebra $M$ is called prime if it cannot be decomposed as the tensor product of two nontrivial (not type ${\rm I}$) subalgebras. Naturally, if $M$ is not prime, one asks if $M$ can be uniquely factored as a tensor product of prime subalgebras. The first result in this direction is due to Ozawa and Popa in 2003, who gave a large class of groups $\mathcal{C}$ such that for any $\Gamma_1, \dots, \Gamma_n \in \mathcal{C}$, the associated von Neumann algebra $L(\Gamma_1) \,\overline{\otimes}\, \cdots \,\overline{\otimes}\, L(\Gamma_n)$ is uniquely factored. This talk will focus on unique prime factorization in the setting of von Neumann algebras which contain Cartan subalgebras. Here we encounter an interesting general obstruction to unique prime factorization in the sense of Ozawa and Popa. We will describe this obstruction and how their techniques can be adapted to avoid it.

• Date: Wednesday, 12/9/15, 3:10-4:00pm in SC 1320.
  • Yusuke Isono, RIMS
  • Title: Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence
  • Abstract: We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras arising from arbitrary actions of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We particularly prove a spectral gap rigidity result for the crossed products and, using recent results of Boutonnet-Ioana-Salehi Golsefidy, we provide the first example of group measure space type III factors with no central sequences. This is joint work with C. Houdayer.

• End of Fall Semester.