# Kyoto Operator Algebra Seminar

Organizers: Benoit COLLINS, Masaki IZUMI, Narutaka OZAWA.
Time and Location: 15:00 - 16:30 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 Spring/Summer

 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) TBA June 11-19 SGU Lecture Sci 3-127 Vaughan F. R. Jones (Vanderbilt) 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) TBA July 07 15:00 - 16:00 RIMS 206 Pinhas Grossman (UNSW Australia) TBA 16:15 - 17:15 RIMS 206 David Penneys (UCLA) TBA July 21 15:00 - 16:30 RIMS 206 TBA TBA Aug. 19-21 RIMS 111 Recent developments in operator algebras (program) Sep. 08-11 Sci 3-110 Group Actions and Metric Embeddings Oct. 24-26 Myo-Ko Annual meeting on operator theory & operator algebra theory

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