# Kyoto Operator Algebra Seminar

## April 2014 - September 2014

 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

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