## Operator Algebras Seminar Spring 2017

Organizers: Marius Dadarlat, Andrew Toms, and Thomas Sinclair

Thursday, 2:30-3:30pm, REC 317

• Date: 1/12
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• Date: 1/19
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• Date: 1/26
• Yasuhiko Sato, Kyoto University
• Title: Bogoliubov actions of amenable groups on the CAR algebra
• The canonical anti-commutation relation (CAR) is well-known to be based on the celebrated Pauli exclusion principle. For this reason, the CAR algebra is regarded as one of main examples in the study of C*-algebras. In a way parallel to the construction of the CAR algebra, the Bogoliubov actions can be defined. These actions are often applied to obtain unitary equivalence of quasi-free states and to diagonalize Hamiltonians. In this talk, starting with the basic study of the CAR algebra, I will explain about the classification theory on crossed products of Bogoliubov actions.

• Date: 2/2
• Sayan Das, Vanderbilt University
• Title: Poisson Boundaries of finite von Neumann algebras
• In my talk, I shall discuss the notion of noncommutative Poisson boundary, for finite von Neumann algebras (due to Izumi, Creutz and Peterson). I shall also talk about a noncommutative generalization of "Double Ergodicity of the boundary" (due to Kaimanovich), and provide some applications to the study bounded derivations on a finite von Neumann algebras. This is based on joint work with Jesse Peterson.

• Date: **COLLOQUIUM** 2/7
• Xiang Tang, Washington University, St. Louis
• Title: A Leray-Hirsch Principle for Gerbes on Orbifolds
• A G-gerbe is a principal BG bundle, where BG is the classifying space of G. In this talk, we will explain what they are, and introduce a Leray-Hirsch principle to study the geometry of G gerbes on orbifolds. The key ingredient in our development is the Mackey machine for the restriction of a representation. We will aim to make the talk accessible to graduate students.

• Date: 2/9
• Weihua Liu, Indiana University
• Title: Distributional symmetries in noncommutative probability
• I will be introducing the notion of distributional symmetries in noncommutative probability and some results related to these symmetries.

• Date: 2/16
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• Date: 2/23
• Martino Lupini, Caltech
• Title: Cocycle superrigidity and group actions on $\rm C^*$-algebras
• I will present the proof that any property (T) countable discrete group admits a continuum of pairwise non cocycle conjugate free actions on a UHF C*-algebra of infinite type. The main ingredient in the proof is Popa's cocycle superrigidity theory for noncommutative Bernoulli shifts. This is joint work with Eusebio Gardella.

• Date: **WABASH SEMINAR** 2/25
• Date: 3/2
• Scott Atkinson, Vanderbilt University
• Title: Minimal faces and Schur’s Lemma for embeddings into R^U
• As shown by N. Brown in 2011, for a separable II$_1$-factor N, the invariant Hom(N,R$^U$) given by unitary equivalence classes of embeddings of N into R$^U$--an ultrapower of the separable hyperfinite II$_1$-factor--takes on a convex structure. This provides a link between convex geometric notions and operator algebraic concepts; e.g. extreme points are precisely the embeddings with factorial relative commutant. The geometric nature of this invariant provides a familiar context in which natural curiosities become interesting new questions about the underlying operator algebras. For example, such a question is the following. "Can four extreme points have a planar convex hull?" The goal of this talk is to present a recent result generalizing the characterization of extreme points in this convex structure. After introducing this convex structure, we will see that the dimension of the minimal face containing an equivalence class $[\pi]$ is one less than the dimension of the center of the relative commutant of $\pi$. This result also establishes the "convex independence" of extreme points, providing a negative answer to the above question. Along the way we make use of a version of Schur's Lemma for this context. No prior knowledge of this convex structure will be assumed.

• Date: 3/9
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• Date: 3/16
• Spring Break

• Date: **COLLOQUIUM** 3/21
• Dietmar Bisch, Vanderbilt University
• Title: Subfactors with infinite representation theory
• Since the discovery of the Jones polynomial in the 1980's, it is well-known that subfactors of von Neumann factors are intimately related to quantum topology. A subfactor is said to have infinite representation theory, if its standard representation generates infinitely many non-equivalent irreducibles. Such subfactors are hard to construct, and very few methods are known to produce interesting examples. I will highlight one such procedure, due to Jones and myself. The construction yields new C$^*$-tensor categories and solutions of the quantum Yang-Baxter equation. I will try to make the talk accessible to non-experts.

• Date: 3/23
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• Date: 3/30
• Yunxiang Ren, Vanderbilt University
• Title: Classification of Thurston-relation planar algebra
• Planar algebras were introduced by Jones as a topological axiomatization of the standard invariants of subfactors. With this perspective, it is natural to understand subfactors through its skein theory, i.e, generators and relations (both algebraic and topological). In this talk, we will discuss the subfactor planar algebra generated by a 3-box with Thurston relation as a continuation of the classification program by skein theory proposed by Bisch and Jones. We give a full classification of such subfactors and explore these subfactors from its skein theory.

• Date: 4/6
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• Date: 4/13
• David Penneys, Ohio State
• Title: Realizations of operator algebra objects and discrete subfactors
• There has been recent success in the classification of small index ${\rm II}_1$ subfactors and developing invariants for irreducible discrete/quasi-regular ${\rm II}_1$ subfactors. In joint work with Corey Jones, we give a characterization of extremal irreducible discrete subfactors $(N\subseteq M, E)$ where $N$ is ${\rm II}_1$, $M$ is an arbitrary factor, and $E: M\to N$ is a faithful normal conditional expectation. We use our recently developed language of operator algebras internal to a rigid C*-tensor category to give an equivalence of categories between discrete inclusions and connected W*-algebra objects. We give an easily computable analog of Connes' modular spectrum for connected W*-algebra objects, and we produce many examples of discrete inclusions $N\subset M$ where $M$ is type ${\rm III}$ coming from non Kac-type discrete quantum groups and their module categories.

• Date: 4/20
• Bogdan Nica, McGill
• Title: Around RD
• The property of Rapid Decay originates from a seminal paper of Haagerup, in which he was interested in operator norms over free groups. I will discuss this property, giving a rapid overview and then focusing on the case of hyperbolic groups.

• Date: **WABASH SEMINAR** 4/22
• Date: 4/27
• Title: