# Kyoto Operator Algebra Seminar

Organizers: Benoit COLLINS, Masaki IZUMI, Narutaka OZAWA.
Time and Location: 14:00 - 15:30 on Tuesday at RIMS 206
Seminar Description: This seminar features both research and introductory talks on topics in Operator Algebra, Noncommutative Geometry, Ergodic Theory, and Group Theory of various kinds (geometric, measure theoretic, functional analytic, etc.). The talks are informal and take between an hour and an hour and a half.
Useful Tips: How to Give a Good Colloquium. Advice on Giving Talks (Upgrade). Myths.
Memento: 2011  2012  2013  2014  2015

## 2016 Spring/Summer

 Mar. 28 - Apr. 04 RIMS 420 KTGU-IMU Mathematics Colloquia and Seminars May 17 14:00 - 15:30 RIMS 206 Hiroshi Ando (Chiba) Unitarizability, Maurey-Nikishin factorization and Polish groups of finite type In the seminal work of cocylcle superrigidity theorem, Sorin Popa introduced the class of finite type Polish groups. A Polish group $G$ is of finite type, if it is embeddable into the unitary group of a separable II$_1$ factor equipped with the strong operator topology. Popa proposed a problem of finding abstract characterization of finite type Polish groups. As Popa pointed out, there are two conditions which are clearly necessary for a Polish group $G$ to be of finite type, namely that (a) $G$ is unitarily representable (i.e., $G$ is embeddable into the full unitary group of $\ell^2$) and (b) $G$ is SIN, i.e., $G$ admits a two-sided invariant metric compatible with the topology. Popa asked whether these two conditions are actually sufficient. In 2011, Yasumichi Matsuzawa and I obtained several partial positive answers for some classes of Polish groups. In this talk, we show that there exists a unitarily representable SIN Polish group which is not of finite type, answering the above question. Our analysis is based on the Maurey-Nikishin Theorem on bounded maps from a Banach space of Rademacher type 2 to the space of all measurable maps on a probability space. This is joint work with Yasumichi Matsuzawa, Andreas Thom, and Asger Törnquist. May 30 - June 03 RIMS 420 Geometric Analysis on Discrete Groups June 07 14:00 - 15:30 RIMS 206 Yusuke Isono (RIMS) Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence 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 examples of group measure space type III factors with no central sequences. This is joint work with C. Houdayer. July 06&08 11:00 - 12:30 Sci 3-305 Saeid Molladavoudi (uOttawa) Symmetry Reduction in Quantum Information Theory Symmetries are ubiquitous in natural phenomena and so in their mathematical descriptions and according to a general principle in Mathematics, one should exploit a symmetry to simplify a problem whenever possible. In these mini-lecture series, we focus on elimination of symmetries from multi-particle quantum systems and discuss that the existing methods equip us with a powerful set of tools to compute geometrical and topological invariants of the resulting reduced spaces. We first introduce some of the group-theoretical and geometrical settings that are required to study multi-particle quantum systems and present a mathematical framework for symmetry reduction purposes, namely characterizing the space of entanglement classes of multi-particle quantum states provided the local information encoded in their local density matrices. Then, as an intermediate step, we consider the maximal torus subgroup $T$ of the compact Lie group of Local Unitary operations $K$ and elaborate on the symmetry reduction procedure and use methods from symplectic geometry and algebraic topology to obtain some of the topological invariants of these relatively well-behaved quotients for multi-particle systems containing $r$ qubits. More precisely, by fixing the relative phases of isolated $r$ qubits and then varying them inside their domain in an $r$-dimensional polytope, we utilize recursive wall-crossing procedures to obtain some of the topological invariants of the corresponding reduced spaces, e.g. Poincaré polynomials and Euler characteristics. July 10 Kyoto Kyoto Prize Symposium (For a general audience. Registration required.) July 12 13:30 - 15:00 RIMS 206 Richard Cleve (Waterloo) Entangled strategies for linear system games Binary linear system games provide a simple framework for investigating the non-local effects that can occur from entangled quantum states. In particular, perfect strategies for such games are characterized by operator solutions to certain non-commutative equations: finite-dimensional solutions for tensor-product entanglement; and possibly-infinite-dimensional solutions for commuting-operator entanglement. I will explain some context and motivation for considering these games and then prove the characterizations. The proofs are based around a finitely-presented group associated with the linear system. This is joint work with Rajat Mittal, Li Liu and William Slofstra. July 12 15:15 - 16:45 RIMS 206 Henry Tucker (USC) Frobenius-Schur indicators and modular data for singly-generated fusion categories Frobenius--Schur indicators provide an important invariant for fusion categories, especially for application to classification problems. Their values can be obtained from the modular data of the Drinfel'd center. In several important cases of singly-generates fusion categories this modular data is given by quadratic forms on some associated groups. This leads to the expression of the indicators as quadratic Gauss sums, which yields examples of fusion categories that are completely determined by their indicators. We will discuss the indicators of near-groups and Haagerup-Izumi categories following from the conjectures of Evans and Gannon regarding the modular data for the centers of these categories. July 19 14:00 - 15:30 RIMS 206 Takuya Takeishi (Kyoto) Primitive ideals and K-theoretic approach to Bost-Connes systems We would like to deal with the classification problem of Bost-Connes systems. For a number field $K$, there is a $\mathrm{C}^*$-dynamical system ($\mathrm{C}^*$-algebra equipped with an $\mathbf{R}$-action) so called the Bost-Connes system for $K$. By KMS-classification theorem of Laca-Larsen-Neshveyev, the Dedekind zeta function is an invariant of Bost-Connes systems. However, this invariant turned out to be an invariant of Bost-Connes $\mathrm{C}^*$-algebras (without $\mathbf{R}$-acitons),. In this talk, we will introduce definitions and some basic facts about Bost-Connes systems, and give an outline of the proof of this theorem. Aug. 01-12 Sendai MSJ-SI: Operator Algebras and Mathematical Physics Sep. 12-14 RIMS 420 Recent developments in operator algebras (program)

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