# Kyoto Operator Algebra Seminar

Organizers: Benoit COLLINS, Masaki IZUMI, Narutaka OZAWA.
Time and Location: 15:00 - 16:30 on Tuesday at RIMS 006
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

 Mar. 09&10 Sci 3-127 SGU Mathematics Kickoff Meeting Mar. 12-18 SGU Lecture Sci 3-110 Gilles Pisier (TAMU) Grothendieck Inequality, Random matrices and Quantum Expanders Thu 15:30 - 17:00,  Fri 15:30 - 17:00,  Mon 10:30 - 12:00,  Tue 10:30 - 12:00,  Wed 10:30 - 12:00. Lecture 1: Grothendieck's inequality in the XXIst century In a famous 1956 paper, Grothendieck proved a fundamental inequality involving the scalar products of sets of unit vectors in Hilbert space, for which the value of the best constant $K_G$ (called the Grothendieck constant) is still not known. Surprisingly, there has been recently a surge of interest on this inequality in Computer Science, Quantum physics and Operator Algebra Theory. The first talk will survey some of these recent developments. Lecture 2: Non-commutative Grothendieck inequality Grothendieck conjectured a non-commutative version of his Fundamental theorem on the metric theory of tensor products", which was established by the author (1978) and Haagerup (1984). This gives a factorization of bounded bilinear forms on $C^*$-algebras. More recently, a new version was found describing a factorization for completely bounded bilinear forms on $C^*$-algebras, or on a special class of operator spaces called exact". We will review these results, due to the author and Shlyakhtenko (2002) and also Junge, Haagerup-Musat (2012) , Regev-Vidick (2014). Lecture 3: The importance of being exact The notion of an exact operator space (generalizing Kirchberg's notion for $C^*$-algebras) will be discussed in connection with versions of Grothendieck's inequality in Operator Space Theory. Random matrices (Gaussian or unitary) play an important role in this topic. Lectures 4 and 5: Quantum expanders Quantum expanders will be discussed with several recent applications to Operator Space Theory. They can be related to smooth" points on the analogue of the Euclidean unit sphere when scalars are replaced by $N\times N$-matrices. The exponential growth of quantum expanders generalizes a classical geometric fact on $n$-dimensional Hilbert space (corresponding to $N=1$). Miscellaneous applications will be presented: --to the growth of the number of irreducible components of certain group representations in the presence of a spectral gap, --to the metric entropy of the metric space of all $n$-dimensional normed spaces for the Banach-Mazur distance or its non-commutative (matricial) analogues, --to tensor products of $C^*$-algebras. Mar. 13 13:30 - 15:00 Sci 3-110 Kei Hasegawa (Kyushu) Relative nuclearity and its applications We prove a relative analogue of equivalence between nuclearity and the CPAP. In its proof, the notion of weak containment for C$^*$-correspondences plays an important role. As an application we prove $KK$-equivalence between full and reduced amalgamated free products of C$^*$-algebras under a strengthened variant of `relative nuclearity'. Mar. 16 13:30 - 15:00 Sci 3-110 Yuki Arano (Tokyo) Unitary spherical representations of Drinfeld doubles Motivated by the work by De Commer-Freslon-Yamashita, we introduce central property (T) for discrete quantum groups and discuss on some operator algebraic applications of this property. We also show that the dual of SUq(2n+1) has central property (T). Mar. 17 13:30 - 15:00 Sci 3-110 Takuya Takeishi (Tokyo) Irreducible representations of Bost-Connes systems The classification problem of Bost-Connes systems was studied by Cornellissen and Marcolli partially, but still remains unsolved. In this talk, we will give a representation-theoretic approach to this problem. We will generalize the result of Laca and Raeburn, which concerns with the primitive ideal space on Bost-Connes system for the rational field. As a consequence, Bost-Connes $C^*$-algebra for a number field $K$ has $h^1_K$-dimensional irreducible representations and doesn't have finite dimensional irreducible representations for other dimensions, where $h^1_K$ is the narrow class number of K. Mar. 31 15:00 - 16:30 RIMS 006 El-kaïoum M. Moutuou (Southampton) Categorification in functional analysis The term "categorification", invented from Louis Crane, roughly refers to the process of developing a category theoretic approach to theories phrased in a set theoretic language. In functional analysis, the idea would be to replace normed vector spaces by categories equipped with some "topologies", bounded linear operators by continuous functors, equations by bounded natural transformations, duality by limits and colimits, etc. In my talk, I will address such a goal by discussing Banach categories, introducing their big spectrums, and outlining some of their properties generalising basic ideas from functional analysis. Apr. 04 15:00 - 18:00 RIMS 209 Roadmap Meeting Apr. 07 15:00 - 16:30 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 15:00 - 16:30 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) TBA

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