# Kyoto Operator Algebra Seminar

## April 2014 - March 2015

 Apr. 22 15:00 - 16:30 RIMS 006 Benoit Collins (Kyoto) Describing exchangeable SU(n) invariant separable states In quantum information theory, a state is a positive matrix in $M_n({\mathbb C})$ of trace $1$. In a tensor product (multipartite system), a state is called separable iff it is the convex combination of tensor product states. In $M_n({\mathbb C})^{\otimes k}$, we are interested in the problem of describing the convex body of separable states who are $\mathrm{SU}(n)$ invariant (with respect to the diagonal action), and exchangeable (i.e. invariant under the canonical action of the permutation group on k points). This problem arises from quantum information theory, where the notion of separable state is seen as the negation of the crucial notion of entangled state. It turns out that this problem admits a nice solution through the study of Martin boundary of random walks on Bratelli diagrams. Joint work in preparation with M. Al Nuwairan and T. Giordano. May 13 15:00 - 16:30 RIMS 006 Norio Nawata (Osaka Kyoiku) Rohlin actions on $\mathcal{W}_2$ In this talk, we shall consider finite group actions on $\mathcal{W}_2$ with the Rohlin property. Let $\alpha$ be an infinite tensor product type action on a UHF algebra $B$. We shall determine when the action $\alpha\otimes\mathrm{id}$ on $B\otimes\mathcal{W}_2 \cong \mathcal{W}_2$ has the Rohlin property. May 20 15:00 - 16:30 RIMS 006 Masato Mimura (Tohoku) Group approximation in Cayley topology and coarse geometry, part II: fibered coarse embeddings This is part II of a project with Hiroki Sako (Tokai University). We study coarse disjoint unions of finite groups. Our main strategy is to employ the Cayley topology (and "the space of marked groups"), which are introduced by Grigorchuk. The topology allows us to regard a group as a point in a compact metrizable space. The subject of this talk is generalized embeddability of metric spaces, which is called the "fibered coarse embeddability." We observe that fibered coarse embeddability of a sequence of finite groups can be characterized by its "Cayley boundary." As a byproduct, we construct a first example of an expander family that does not admit fibered coarse embeddings into any non-singular CAT(0) space, but that has a biLipschitz embedding into some singular CAT(0) space. May 27 15:00 - 16:30 RIMS 006 Ion Nechita (Toulouse) Positive and completely positive maps via free additive powers of probability measures We give examples of maps between matrix algebras with different "degrees" of positivity using ideas from free probability. We discuss applications to entanglement detection in quantum information theory, and compare the new methods with existing ones. June 03 15:00 - 16:30 RIMS 006 Rui Okayasu (Osaka Kyoiku) Haagerup approximation property and positive cones associated with a von Neumann algebra We introduce the notion of the $\alpha$-Haagerup approximation property for $\alpha\in[0,1/2]$ using a one-parameter family of positive cones studied by Araki and show that the $\alpha$-Haagerup approximation property actually does not depend on a choice of $\alpha$. This enables us to prove the fact that the Haagerup approximation properties introduced in two ways are actually equivalent, one in terms of the standard form and the other in terms of completely positive maps. June 17 15:00 - 16:30 RIMS 006 Yuhei Suzuki (Kyoto) Realization of hyperbolic group $\mathrm{C}^*$-algebras as decreasing intersection of Cuntz algebras ${\mathcal O}_2$ We will see that for every ICC group which is embeddable into a hyperbolic group, the reduced group $\mathrm{C}^\ast$-algebra is realized as the intersection of a decreasing sequence of isomorphs of the Cuntz algebra $\mathcal{O}_2$. In this talk, we will give a proof for finitely generated free groups. June 24 15:00 - 16:30 RIMS 006 Ping Zhong (Indiana) Superconvergence to freely infinitely divisible distributions This talk is based on a joint work with H. Bercovici and J.C. Wang. Given an infinitely divisible distribution $\nu$ relative to free independence in the sense of Voiculescu, let $\mu_n$ be a sequence of probability measures and let $k_n$ be an increasing sequence of integers such that $(\mu_n)^{\boxplus k_n}$ converges weakly to $\nu$. We show that the density $d(\mu_n)^{\boxplus k_n}/dx$ converges uniformly to the density of $d\nu/dx$ except possibly in the neighborhood of one point. This phenomenon is called super convergence. The special case when the limit distribution $\nu$ is the semicircle law was proved by Bercovici and Voiculescu. June 28 Nara Kansai Operator Algebra Seminar July 01 DT Seminar 15:00 - 16:30 Sci 6-609 Brian Bowditch (Warwick/TITECH) Rigidity properties of mapping class groups and related spaces July 08 15:00 - 16:30 RIMS 006 Yusuke Isono (Kyoto) On fundamental groups of tensor product $\mathrm{II}_1$ factors We study a notion of strong primeness for $\mathrm{II}_1$ factors, which was introduced in my previous work. As a result, we give examples of $\mathrm{II}_1$ factors $M$ which satisfies $\mathcal{F}(B\otimes M)=\mathcal{F}(B)$ for arbitrary $\mathrm{II}_1$ factor $B$. July 15 15:00 - 16:30 RIMS 006 Cyril Houdayer (Lyon) Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity We introduce an algebraic invariant for aperiodic inclusions of probability measure preserving equivalence relations. We use this invariant to prove that every stable orbit equivalence between free pmp actions of direct products of non-amenable Baumslag-Solitar groups whose canonical subgroup acts aperiodically forces the number of factors of the products to be the same and the factors to be isomorphic after permutation. This generalises some of the results obtained by Kida and moreover provides new measure equivalence rigidity phenomena for Baumslag-Solitar groups. We also obtain a complete classification of direct products of relative profinite completions of Baumslag-Solitar groups, continuing recent work of Elder and Willis.This is a joint work with Sven Raum. July 16 Colloquium 16:30 - 17:30 Sci 3-110 Narutaka Ozawa (Kyoto) Noncommutative real algebraic geometry of Kazhdan's property (T) July 19-22 Otaru Summer Camp on Operator Algebras Aug. 21 Math Phys 14:00 - 15:30 RIMS 402 Gábor Etesi (UIUC) Gravity as a four dimensional algebraic quantum field theory Based on an indefinite unitary representation of the diffeomorphism group of an oriented 4-manifold an algebraic quantum field theory formulation of gravity is exhibited. More precisely the representation space is a Krein space therefore as a vector space it admits a family of direct sum decompositions into orthogonal pairs of maximal definite Hilbert subspaces coming from the Krein space structure. It is observed that the C*-algebra of bounded linear operators associated to this representation space contains algebraic curvature tensors. Classical vacuum gravitational fields i.e., Einstein manifolds correspond to quantum observables obeying at least one of the above decompositions of the space. In this way classical general relativity exactly in 4 dimensions naturally embeds into an algebraic quantum field theory whose net of local C*-algebras is generated by local algebraic curvature tensors and vector fields. This theory is constructed out of the structures provided by an oriented 4-manifold only, and hence possesses a diffeomorphism group symmetry. Motivated by the Gelfand-Naimark-Segal construction and the Dougan-Mason construction of quasi-local energy-momentum we construct certain representations of the limiting global C*-algebra what we call the "positive mass representations". Finally we observe that the bunch of these representations give rise to a 2 dimensional conformal field theory in the sense of G. Segal. Sep. 08-10 RIMS 420 Recent developments in operator algebras (program) Sep. 30 15:00 - 16:30 RIMS 006 Mike Brannan (UIUC) Probabilistic aspects of (non-tracial) free orthogonal quantum groups In this talk, I will discuss the problem of computing moments (with respect to the Haar state) of the generators of Van Daele and Wang's $F$-deformed free orthogonal quantum groups $O^+_F$. The main technique here is a combinatorial Weingarten formula due to Banica and Collins. Using this technique, we show that as the quantum dimension'' of the quantum group tends to infinity, the rescaled generators converge in distribution to canonical generators of almost-periodic free Araki-Woods factors. Time-permitting, we will also discuss how the free orthogonal quantum groups $O^+_F$ can be realized as distributional symmetries of almost periodic free Araki-Woods factors. This talk is based on joint work with Kay Kirkpatrick. Oct. 06-10 Sci 3-127 Shigeru Yamagami (Nagoya) Intensive lecture course on quantum algebras Mon 15:00 - 17:00,  Tue 15:00 - 17:00,  Wed 10:00 - 12:00,  Thu 15:00 - 17:00,  Fri 10:00 - 12:00. Colloquium @ 3-110: Wed 15:00 - 16:00 Oct. 14 DT Seminar 15:00 - 16:30 Sci 6-609 Moon Duchin (Tufts University) Large-scale geometry of nilpotent groups Oct. 17 SGU Lecture 13:00 - 15:00 Sci 3-108 (Sci 3-305) Gennadi Kasparov (Vanderbilt University) Introduction to Index theory and KK-theory Every Friday from October 17 to November 28 from 13:00 to 15:00 at 3-108, except on October 24 and November 14 at 3-305. The course will contain the operator $K$-theory approach to the Atiyah-Singer index theorem. We will start with some classical examples of elliptic differential operators on compact smooth manifolds without boundary. This will naturally lead us to two special cases of $KK$-theory: $K$-cohomology and $K$-homology. $KK$-theory will be introduced gradually, as much as it is needed for index theory. Most examples will come from differential and pseudo-differential operators. Large part of technical results related with $KK$-theory will be given without proof: because time is limited, and also because we need $KK$-theory for this course only as a tool. Other technical tools include Clifford algebras and Dirac operators. Although all definitions will be given in the course, I advise the listeners to consult the book Spin geometry'' by H. B. Lawson and M.-L. Michelsohn on these issues. The main part of the course will contain a proof of the $K$-theoretic version of the Atiyah-Singer index theorem. The cohomological Atiyah-Singer index formula for compact manifolds will be obtained as a corollary. Various applications will be discussed as much as time allows. Recommended literature: 1. M. F. Atiyah, I. M. Singer: The index of elliptic operators'', I, III, Annals of Math., 87 (1968), 484-530, 546-604. 2. H. B. Lawson, M.-L. Michelsohn: Spin geometry'', Princeton Univ. Press, 1989. Oct. 21 15:00 - 16:30 RIMS 006 Yusuke Isono (Kyoto) Free independence in ultraproduct von Neumann algebras and applications We generalize Popa's free independence result for ultra products of $\mathrm{II}_1$ factors to the framework of type $\mathrm{III}$ factors with large centralizer algebras. Then we give two applications. First, we give a direct proof of stability under free product of QWEP for von Neumann algebras. Second, we give a new class of inclusions of von Neumann algebras with relative Dixmier property. This is a joint work with C. Houdayer. Oct. 28 15:00 - 17:00 RIMS 006 Narutaka Ozawa (Kyoto) The Furstenberg boundary and $\mathrm{C}^*$-simplicity I A (discrete) group $G$ is said to be $\mathrm{C}^*$-simple if the reduced group $\mathrm{C}^*$-algebra of it is simple. I will first explain Kalantar and Kennedy's characterization of $\mathrm{C}^*$-simplicity for a group $G$ in terms of its action on the maximal Furstenberg boundary. Then I will talk about my result with Breuillard, Kalantar, and Kennedy about examples and stable properties of $\mathrm{C}^*$-simple groups. Nov. 04 13:15 - 14:45 RIMS 006 Narutaka Ozawa (Kyoto) The Furstenberg boundary and $\mathrm{C}^*$-simplicity II 15:00 - 16:30 RIMS 006 Hiroshi Ando (Copenhagen) On the noncommutativity of the central sequence $\mathrm{C}^*$-algebra $F(A)$ We show that the central sequence $\mathrm{C}^*$-algebra of the free group $C_r^*(\mathbb{F}_n)\ (n\ge 2)$ is noncommutative, answering a question of Kirchberg in 2004. This is in contrast to the fact that the $\mathrm{W}^*$-central sequence algebra of the group von Neumann algebra $L(\mathbb{F}_n)$ is trivial. This is joint work with Eberhard Kirchberg. Nov. 11 Kyoto Prize 13:00 - 16:30 ICC Kyoto Edward Witten (IAS) Adventures in Physics and Math Nov. 18 15:00 - 16:30 RIMS 006 Takahiro Hasebe (Hokkaido) Extension of $q$-Fock space in terms of Coxeter group of type B A $q$-Fock space was introduced by Bozejko and Speicher in 1991. Its inner product contains a parameter $q$ and it interpolates the boson ($q=1$), fermion ($q=-1$) and full ($q=0$) Fock spaces. In this talk I will explain a further deformation by two parameters $(a,q)$ which come from Coxeter groups of type B. This is a joint work with Marek Bozejko and Wiktor Ejsmont. Nov. 29&30 Shirahama Kansai Operator Algebra Seminar Dec. 09 15:00 - 16:30 RIMS 006 Yoshikata Kida (Kyoto) OE and $\mathrm{W}^*$-superrigidity results for actions by surface braid groups We show that some natural subgroups of the mapping class group has rigidity in the title. Particularly I explain strategy for OE rigidity of strong type. This is based on my old work on mapping class groups. I will first briefly review mapping class groups and how their rigidity is obtained. This is a joint work with Ionut Chifan. Dec. 16 15:00 - 16:30 RIMS 006 Reiji Tomatsu (Hokkaido) Fullness of the core von Neumann algebra of free product factors I will talk about the fullness of the core von Neumann algebras of free product factors of type III$_1$ and present a characterization in terms of Connes' $\tau$-invariant. This is a joint work with Yoshimichi Ueda. Dec. 24-26 Hakusan Annual meeting on operator theory & operator algebra theory Jan. 06 15:00 - 16:30 RIMS 006 Sven Raum (RIMS/Münster) Character rigidity for lattices in higher rank groups I (after Creutz-Peterson, Peterson) A discrete group $G$ is called character rigid if every representation of $G$ on a Hilbert space generating a finite factor is either the left regular representation or generates a finite dimensional factor. Creutz-Peterson and Peterson proved that many lattices in Lie groups and their products with totally disconnected groups are character rigid. In this talk we review results of Creutz-Peterson proving character rigidity for lattices in products of certain property (T) Lie groups with totally disconnected Howe-Moore groups. We introduce necessary terminology and explain the strategy of proof. This will be set the precursor to understand Petersons result on character rigidity for lattices in higher rank semi-simple Lie groups. Jan. 13 15:00 - 16:30 RIMS 006 Sven Raum (RIMS/Münster) Character rigidity for lattices in higher rank groups II (after Creutz-Peterson, Peterson) Jan. 16 Prob Seminar 15:30 - 17:00 Sci 3-552 J L (kC) R蕪z̐ьÓT蕪zƂ̊֌W Jan. 20 15:00 - 16:30 RIMS 006 Rui Okayasu (Osaka Kyoiku) Haagerup approximation property and bimodules I try giving a characterization of Haagerup approximation property for arbitrary von Neumann algebra in terms of bimodules. Jan. 27 15:00 - 16:30 RIMS 006 Narutaka Ozawa (RIMS) Maximal amenable von Neumann algebras (after Boutonnet and Carderi arXiv:1411.4093) Feb. 02-04 RIMS 111 Classification of operator algebras and related topics (program) 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.

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