# Kyoto Operator Algebra Seminar

## April 2016 - March 2017

 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 September 05-09 KTGU Lecture Sci 3-127 Sorin Popa (UCLA/Kyoto) Paving over arbitrary MASAs in von Neumann algebras 05(M) 15:00 - 17:00,  06(T) 15:00 - 17:00,  07(W) 15:00 - 17:00,  08(T) 15:00 - 17:00,  09(F) 15:00 - 17:00. Motivated by an intriguing claim in Dirac's 1947 book on "Quantum Mechanics", Kadison and Singer have asked the question of whether any pure state on the diagonal maximal abelian subalgebra (MASA) $D$ of $B(H)$ extends to a unique state on $B(H)$. They also showed that this unique pure state extension property is equivalent to norm paving over $D$ for operators in $B(H)$. The Kadison-Singer paving problem has been recently solved in the affirmative by Marcus--Spielman--Srivastava. In these lectures, we will introduce a general paving property for a MASA $A$ in a von Neumann factor $M$, called so-paving, involving approximation in the so-topology, rather than in norm, but which coincides with norm-paving in the case $D\subset B(H)$. We conjecture that so-paving holds true for any MASA in any factor. We check the conjecture in many cases, including singular and regular MASAs in hyperfine factors. Related problems will be discussed. Sep. 12-14 RIMS 420 Recent developments in operator algebras (program) Oct. 09-11 Maebashi Annual meeting on operator theory & operator algebra theory Oct 18 15:00 - 16:30 RIMS 206 Yusuke Isono (RIMS) Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors Let $M$ be a type $\mathrm{III}$ factor associated with a free (unitary or orthogonal) quantum group. We prove that for any factor $B$, the tensor product of $M$ and $B$ has no Cartan subalgebras. The main ingredient of the proof is a generalization of Ozawa--Popa and Popa--Vaes's weakly compact action at the level of the continuous core. We study it by using an operator valued weight to $B$ and the central weak amenability of $M$. Oct 25 15:00 - 16:30 RIMS 206 Koichi Shimada (Kyoto) Maximal amenability of the generator subalgebra in q-Gaussian von Neumann algebras We give explicit examples of maximal amenable von Neumann subalgebras of the $q$-Gaussian von Neumann algebras. More precisely, the generator subalgebra is maximal amenable inside the $q$-Gaussian algebras for real numbers $q$ with $|q|<1/9$. We would like to show this based on Popa's theory. In order to achieve this, we construct a Riesz basis in the spirit of Radulescu. This is a joint work with Sandeepan Parekh and Chenxu Wen. Nov 01 15:00 - 16:30 RIMS 206 Ivan Ip (Kyoto) Positive representations: a bridge between Drinfeld-Jimbo quantum groups and C*-algebra The finite dimensional representation theory of Drinfeld-Jimbo quantum group is well-known for representation theorist, and many applications have been discovered in the last 30 years. However, the non-compact case is a lot more complicated and much less is known. The notion of "positive representations" was introduced in a joint work with I. Frenkel to study the representation theory of split real quantum groups, which involves representations by unbounded operators. In this talk, I will give some motivations for such representation theory, and explain how the techniques from C*-algebra allow us to study the harmonic analysis and braiding structure of split real quantum groups. Nov 11 RT Seminar 16:30 - 18:00 RIMS 402 Benoit Collins (Kyoto) Positivity for the dual of the Temperley-Lieb basis A problem raised by Vaughan Jones is to consider the basis dual to the canonical basis of the Temperley-Lieb algebra for non-degenerate loop values, and investigate the coefficients of this basis element in the original basis. For example, the dual of the identity element is a multiple of the Jones Wenzl projection, and computing it is an important problem for which some formulas have been given recently (e.g. by Morrisson). The goal of this talk is to describe a new combinatorial formula for all of these coefficients. As a byproduct, we solve one question of Jones and prove that all these coefficients are never zero for real parameters $\ge 2$, and we compute their sign. Our strategy relies on identifying these coefficients with the Weingarten function of the free orthogonal quantum group, and on developing quantum integration techniques. I will spend some time on recalling definitions and properties of some objects that are less well-known, such as Weingarten functions and free orthogonal quantum groups. This talk is based on joint work with Mike Brannan, arXiv:1608.03885. Nov 29 15:00 - 16:30 RIMS 206 Yuki Arano (Tokyo) Torsion-freeness for fusion rings and tensor C*-categories Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum--Connes conjecture for discrete quantum groups. In this talk, we define torsion-freeness for abstract rigid tensor C*-categories and abstract fusion rings and show that the free unitary quantum group is torsion-free. This is joint work with Kenny De Commer. Dec 06 15:00 - 16:30 RIMS 206 Zhigang Bao (Hong Kong) Local law of addition of random matrices The question of how to describe the possible eigenvalues of the sum of two general Hermitian matrices dates back to Weyl. A randomized version of this question can give us a "deterministic" answer. Specifically, when two large-dimensional matrices are in general position in the sense that one of them is conjugated by a random Haar unitary matrix, the eigenvalue distribution of their sum is asymptotically given by the free convolution of the respective eigenvalue distributions. This result was obtained by Voiculescu on the macroscopic scale. In this talk, we show that this law also holds in a microscopic scale. This allows us to get an optional convergence rate for Voiculescu's result. Dec. 10 Sat Osaka-Kyoiku U Kansai Operator Algebra Seminar Dec 13 15:00 - 16:30 RIMS 206 Raphael Ponge (Seoul) The Cyclic Homology of Crossed-Product Algebras This talk will be a preliminary report on an explicit computation of the cyclic homology and periodic cyclic homology of crossed-product algebras over commutative rings. By explicit computation it is meant the construction of explicit quasi-isomorphisms. This enables us to recover and somewhat simplify various known results by Baum-Connes, Brylinski-Nistor, Crainic, Feigin-Tsygan, Getzler-Jones, Nistor, among others. In particular, we recover the spectral sequences of Feigin-Tsygan and Getzler-Jones, and obtain several other spectral sequences as well. The approach is purely algebraic. It grew out of an attempt to extend to crossed-product algebras the algebraic approach of Marciniak to the computation of the cyclic homology of group rings by Burghelea. At the conceptual level, we introduce a generalization of the cyclindrical complexes of Getzler-Jones. This provides us with the rele- vant homological tool to understand the cyclic homology of crossed-product algebras, especially in the non-torsion case. The results extend to the cyclic homology of crossed-products associated with locally convex algebras. In particular, in the case of group actions on manifolds we obtain an explicit construction of cyclic cycles in terms of equivariant characteristic classes. Dec 20 14:30 - 16:00 RIMS 206 Yosuke Kubota (Tokyo) Compact Lie group actions on C*-algebras with the continuous Rokhlin property The Rokhlin property is a dynamical analogue of freeness of group actions on C*-algebras. In this talk, we deal with its variation, the continuous Rokhlin property, which is compatible with KK-theory. We give a complete classification of equivariant Kirchberg algebras with the continuous Rokhlin property when G is a compact Lie group with torsion-free fundamental group. For the proof, the Baum--Connes isomorphism for the Pontrjagin dual quantum group plays an essential role. This is joint work with Yuki Arano. Jan 10 14:00 - 15:00 RIMS 206 Takuya Kawabe (RIMS) On the ideal structure of reduced crossed products Let $X$ be a compact Hausdorff space with an action of a countable discrete group $\Gamma$. We say that $X$ has the intersection property if every non-zero ideals in its reduced crossed product has a non-zero intersection with $C(X)$. We characterize the intersection property of $X$ by a certain property for amenable subgroups of its stabilizer subgroups in terms of the Chabouty space of $\Gamma$. This generalizes Kennedy's algebraic characterization of the simplicity for a reduced group $\mathrm{C}^{*}$-algebra of a countable discrete group. Jan 10 15:30 - 16:30 RIMS 206 Yul Otani (Tokyo) Entanglement entropy in algebraic quantum field theory We consider the problem of defining the entanglement entropy for chiral nets in the framework of algebraic quantum field theory. Considering a Möbius covariant local net with the split property, we give a sensible definition for the entropy of a state restricted to a local algebra of an open connected non-dense interval $I$, with a given conformal energy cutoff $E$. Considering the vacuum state restricted to any such interval, we prove that the latter is finite, and we give some upper bound estimates in terms of the dimensions of eigenspaces of the conformal Hamiltonian. This talk is based on a joint work with Yoh Tanimoto. Jan 24 15:00 - 16:30 RIMS 206 Kei Hasegawa (Kyushu) Bass--Serre trees of amalgamated free product C*-algebras For a given reduced amalgamated free product of C$^*$-algebras, we introduce a C$^*$-algebra including it as a unital subalgebra. This C$^*$-algebra is an analogue of the crossed product of the compactification of the Bass--Serre tree associated with an amalgamated free product group, and naturally identified with a Cuntz--Pimsner algebra. As applications we give simple proofs of several known results about exactness, nuclearity and KK-theory for reduced amalgamated free products. Feb. 01-03 RIMS 111 Operator Algebras and Quantum Information Theory (program) Feb 09 Thursday 15:00 - 16:30 RIMS 110 Amine Marrakchi (Paris-Sud) Spectral gap for full type III factors We present new spectral gap characterizations for full type III factors, generalizing a theorem of Connes in the tracial case, and we give applications to the fullness of tensor products factors and crossed product factors. We will also discuss some related open problems. This is partly joint work with Cyril Houdayer and Peter Verraedt. Mar 08 Wednesday 15:00 - 16:30 Sci 3-305 Takeshi Katsura (Keio) On magic square C*-algebras In '98, Wang defined a series of quantum groups $A_s(n)$ ($n=1, 2, \ldots$) which are free versions of the symmetric groups $\mathfrak{S}_n$. Wang remarked that for $n=1, 2, 3$, $A_s(n)$ becomes the classical one, i.e. $C(\mathfrak{S}_n)$, and that for $n \geq 4$, $A_s(n)$ is non-commutative. In '08, Banica and Collins showed that $A_s(4)$ can be embedded into the homogeneous C*-algebra $C(SU_2, M_4(\mathbb{C}))$ by using integration techniques. In this talk, we focus on the underlying C*-algebras of the quantum groups $A_s(n)$ which we call the magic square C*-algebras. After general discussion on the magic square C*-algebras $A_s(n)$, we give another proof of the embedding of $A_s(4)$ into $C(SU_2,M_4(\mathbb{C}))$ using crossed products. As a byproduct of our proof, we give an explicit computation of the image of the embedding. The talk is based on the master thesis of my student Masahito Ogawa.

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