# Kyoto Operator Algebra Seminar

## April 2015 - March 2016

 Apr. 04 15:00 - 18:00 RIMS 209 Roadmap Meeting Apr. 07 14:00 - 15:30 RIMS 206 Yuhei Suzuki (Tokyo) Construction of minimal skew products of amenable minimal dynamical systems We give a generalization of a result of Glasner and Weiss. This provides many new examples of amenable minimal dynamical systems of exact groups. We also study the pure infiniteness of the crossed products of minimal dynamical systems arising from this result. For this purpose, we introduce and study a notion of the finite filling property for etale groupoids, which generalizes a result of Jolissaint and Robertson. As an application, we show that for any connected closed topological manifold M, every countable non-amenable exact group admits an amenable minimal free dynamical system on the product of M and the Cantor set whose crossed product is a Kirchberg algebra. This extends a result of Rørdam and Sierakowski. Apr. 14 DT Seminar 15:00 - 16:30 Sci 6-609 Masato Mimura (Tohoku) New algebraization of Kazhdan and fixed point properties Apr. 14 16:45 - 18:15 RIMS 206 Seung-Hyeok Kye (Seoul) Various notions of positivity for bi-linear maps and applications to tri-partite entanglement We consider bi-linear analogues of $s$-positivity for linear maps. The dual objects of these notions can be described in terms of Schimdt ranks for tri-tensor products and Schmidt numbers for tri-partite quantum states. These tri-partite versions of Schmidt numbers cover various kinds of bi-separability, and so we may interpret witnesses for those in terms of bi-linear maps. We give concrete examples of witnesses for various kinds of three qubit entanglement. This is a co-work with Kyung Hoon Han. May 12 15:00 - 16:30 RIMS 206 Igor Klep (Auckland) Commuting Dilations and Linear Positivstellensätze Given a tuple $A=(A_1,...,A_g)$ of real symmetric matrices of the same size, the affine linear matrix polynomial $L(x):=I-\sum A_j x_j$ is a monic linear pencil. The solution set $S_L$ of the corresponding linear matrix inequality, consisting of those $x$ in ${\mathbb R}^g$ for which $L(x)$ is positive semidefinite (PsD), is called a spectrahedron. It is a convex semialgebraic subset of ${\mathbb R}^g$. We study the question whether inclusion holds between two spectrahedra. We identify a tractable relaxation of this problem by considering the inclusion problem for the corresponding free spectrahedra $D_L$. Here $D_L$ is the set of tuples $X=(X_1,...,X_g)$ of symmetric matrices (of the same size) for which $L(X):=I-\sum A_j \otimes X_j$ is PsD. We explain that any tuple $X$ of symmetric matrices in a bounded free spectrahedron $D_L$ dilates, up to a scale factor, to a tuple $T$ of commuting self-adjoint operators with joint spectrum in the corresponding spectrahedron $S_L$. The scale factor measures the extent that a positive map can fail to be completely positive. In the case when $S_L$ is the hypercube $[-1,1]^g$, we derive an analytical formula for this scale factor, which as a by-product gives new probabilistic results for the binomial and beta distributions. The talk is based on joint work with Bill Helton, Scott McCullough and Markus Schweighofer. May 26 10:30 - 12:00 Sci 3-152 Nigel Higson (Penn State University) The Oka principle: commutative and noncommutative Kiyoshi Oka proved in 1938 that topological line bundles over closed, complex submanifolds of complex affine space admit unique holomorphic structures. Nearly twenty years later, Hans Grauert proved the same theorem for topological vector bundles of any rank. I will examine these results from the point of view of K-theory, and explain the proofs, which are strikingly similar to the proofs of some fundamental theorems in homology theory, for example the Jordan separation theorem. Okafs theorem is in some sense gcommutative,h since it concerns the abelian Lie group GL(1,C), whereas Grauertfs theorem concerns the non-abelian groups GL(n,C). But there are further extensions of both theorems into the realm of noncommutative geometry (in the sense of Alain Connes), and as I shall explain these extensions have interesting links to representation theory. May 26 DT Seminar 15:00 - 16:30 Sci 6-609 Ryunosuke Ozawa (Kyoto) xԂ̗̑]ڐ June 09 15:00 - 16:30 RIMS 206 Koichi Shimada (Tokyo) Approximate Unitary Equivalence of Finite Index Endomorphisms of the AFD Factors We characterize the condition for two finite index endomorphisms on an AFD factor 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. June 11-19 SGU Lecture Sci 3-127 Vaughan F. R. Jones (Vanderbilt/Kyoto) An introduction to subfactors in mathematics and physics. 11(T) 10:00 - 12:00,  12(F) 10:00 - 12:00,  15(M) 15:00 - 17:00,  17(W) 10:00 - 12:00,  19(F) 10:00 - 12:00. Short Abstract. We will introduce the theory of subfactors and their interactions with topology and physics. Goal. We intend to describe the current state of classification of subfactors and attack the question of whether all subfactors have something to do with quantum field theory, including an appearance of Richard Thompsonfs groups F and T. Plan of the lectures. Lecture 1. Introduction to von Neumann algebras. Definition and examples. Von Neumann density theorem. Factors of types I,II and III. Tomita-Takesaki theory and Connes decomposition and classification. Hyperfiniteness. Lecture 2. Subfactors and elementary examples. Index in the type II case. Bimodules. The tower of relative commutants and restrictions on index values. Construction of examples. Braid group representations and knot polynomials. Lecture 3. Kauffman diagrams for the Temperley-Lieb algebra. Planar algebras, lambda lattices - the standard invariant of a subfactor and reconstruction. Random matrices with a real number of matrices. Tensor categories, 2-categories. Endomorphisms and the type III approach. Lecture 4. Classification of small index subfactors. Izumifs Cuntz algebra approach. Skein theory presentations of subfactors, the exchange relation, the Yang-Baxter equation and the jellyfish algorithm. Lecture 5. Algebraic quantum field theory, conformal field theory and scaling limits of statistical mechanical models. Hilbert spaces from planar algebras-a toy algebraic QFT and the Thompson groups. Knots and links from the Thompson groups. Prerequisites: Some functional analysis-including the spectral theorem on Hilbert space. Some topology including the fundamental group and homology groups. Some quantum mechanics including the Schrodinger equation and the uncertainty principle. June 27-28 Sendai Takagi Lectures June 30 15:00 - 16:30 RIMS 206 Reiji Tomatsu (Hokkaido) Popa--VaesC*-tensor category̕\_̏ЉD July 03 SGU Seminar 14:45 - 16:15 Sci 3-127 Vaughan F. R. Jones (Vanderbilt/Kyoto) Knots and Braids (Introductory Lecture) July 07 15:00 - 16:00 RIMS 206 Pinhas Grossman (UNSW Australia) Some examples of fusion categories associated to finite groups and to subfactors A fusion category is a tensor category that glooks likeh the category of representations of a finite group. Fusion categories arise as categories of bimodules or endomorphisms associated to finite depth subfactors. In this talk we will explain some representation theory of fusion categories by looking at examples of fusion categories associated to some small groups, as well as fusion categories associated to certain subfactors which are related to those groups. 16:15 - 17:15 RIMS 206 David Penneys (UCLA) Bicommutant categories I'll discuss an ongoing joint project with Andre Henriques. Just as a tensor category is a categorification of a ring, and its Drinfel'd center is a categorification of the center of a ring, a bicommutant category is a categorification of a von Neumann algebra. I'll define the notion of the commutant C' of a tensor category C inside an ambient tensor category B. A bicommutant category is then a category which is equivalent to its own bicommutant inside B. Because we are interested in von Neumann algebras, we work in the ambient category B=Bim(R), the tensor category of bimodules over a hyperfinite von Neumann factor R, which can be thought of as a categorification of B(H). Given a unitary fusion category C inside Bim(R), we identify its bicommutant C'', which we show is, in fact, an example of a bicommutant category. Along the way, we provide machinery for constructing elements of C', and we see the Longo-Rehren subfactor appear naturally. July 21 15:00 - 16:30 RIMS 206 Mikael Pichot (McGill) Entropy invariants in analysis and algebra I will explain how to construct invariants in algebra and analysis using Boltzmann's S = K log W entropy formula. Aug. 05 Wed Sci 3-110 Workshop on interactions between commutative and noncommutative probability Aug. 19-21 RIMS 111 Recent developments in operator algebras (program) Aug. 28 Friday 15:00 - 16:30 RIMS 110 Gilles Godefroy (Paris) Lipschitz-free space and the approximation properties The predual of the space of Lipschitz functions on a metric space $M$ is called the Lipschitz-free space over $M$. The structure of this Banach space is naturally related with the behaviour of Lipschitz functions defined on $M$. In this talk, we will display some simple results which relate the validity of Grothendieck's bounded approximation property with the existence of linear extension operators for Lipschitz functions defined on subsets of $M$. We will also present some natural open problems. Sep. 08-11 Sci 3-110 Group Actions and Metric Embeddings Oct. 05-07 RIMS 111 Quantum fields and related topics Oct. 06 15:00 - 16:30 RIMS 206 Noriyoshi Sakuma (Aichi Kyoiku) Unimodality for free Lévy processes I will talk about recent developments of research of free infinitely divisible distributions based on joint work with T. Hasebe. Especially, some interesting distributional properties are founded comparing the classical probability. Oct. 20 15:00 - 16:30 RIMS 206 Narutaka Ozawa (RIMS) A functional analysis proof of Gromov's polynomial growth theorem The celebrated theorem of Gromov in 1980 asserts that any finitely generated group with polynomial growth contains a nilpotent subgroup of finite index. Alternative proofs have been given by Kleiner in 2007, Shalom--Tao in 2009, and Breuillard--Green--Tao in 2011. In this talk, I will give yet another proof of Gromov's theorem, along the lines of Shalom and Chifan--Sinclair, which is based on the analysis of reduced cohomology and Shalom's property $H_{\mathrm{FD}}$. Oct. 24-26 Myo-Ko Annual meeting on operator theory & operator algebra theory Nov. 09-11 RIMS 420 Research on structure of operators by order and geometry with related topics Nov. 10DT Seminar 15:00 - 16:30 Sci 6-609 Jeremie Brieussel (Montpellier) About the speed of random walk on solvable groups The speed of a random walk measures the average distance between the particle and its starting point. By Lee--Peres, for non-degenerate random walks on infinite groups, the speed is between $\sqrt{n}$ and $n$. By Amir--Virag, any regular function between $n^{3/4}$ and $n$ is the speed function of some random walk on some group. I will describe some solvable groups and some random walks on them with speed between $\sqrt{n}$ and $n^{3/4}$. Dec. 01 15:00 - 16:30 RIMS 206 Narutaka Ozawa (RIMS) Character Rigidity after Jesse Peterson Dec. 08DT Seminar 15:00 - 16:30 Sci 6-609 Narutaka Ozawa (RIMS) A functional analysis proof of Gromov's polynomial growth theorem Re-broadcasting. Dec. 12-13 Kinosaki Kansai Operator Algebra Seminar Jan. 12 15:00 - 16:30 RIMS 206 Gábor Szabó (Münster) Strongly self-absorbing C*-dynamical systems We discuss a generalization of the notion of strongly self-absorbing C*-algebras to the setting of C*-dynamical systems. The main result is an equivariant McDuff-type theorem that characterizes exactly when an action of a locally compact group on a separable C*-algebra absorbs a given strongly self-absorbing action tensorially up to cocycle conjugacy. This extends a similar folklore result that has been known for discrete group actions. I will present natural examples of strongly self-absorbing actions, and we then discuss what kind of (equivariant) permanence properties carry over in this context, similar to how D-stability is closed under various C*-algebraic operations. Jan. 25-27 RIMS 111 Some problems in operator algebras related to ergodic theory and group theory (program) Feb. 23 Sci 3-110 KTGU Mathematics Workshop for Young Researchers Feb. 23 15:35 - 17:05 RIMS 206 James Mingo (Queen's University, Kingston) Cumulants of Partially Transposed Random Matrices In quantum information theory the partial transpose of a positive matrix has been used to detect entanglement. I have recently investigated the effect of partial transposes on asymptotic freeness. The main tool for doing this is the use of free cumulants, both of first and second order. I will show how to use some simple geometric properties to find these cumulants. Mar. 08 15:00 - 16:30 RIMS 206 Motohisa Fukuda (Yamagata) Partially transposed random quantum states and meander problems In the first part of this talk, we study quantum states and their partial transpose. Then, through the next part, we make moment calculations of partially transposed random quantum states (some random matrices) and introduce the result by Aubrun, and Banica and Nechita, which have interesting meaning in terms of quantum information theory. For the last part, we see how those random matrices yield meander polynomials, and learn more about meander problems, along with our past and ''future" results. Mar. 28 - Apr. 04 RIMS 420 KTGU-IMU Mathematics Colloquia and Seminars

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