# Kyoto Operator Algebra Seminar

## April 2012 - March 2013

 April 11 2:30 - 4:00 RIMS 204 Rui Okayasu (Osaka Kyoiku University) Free group $\mathrm{C}^\ast$-algebras associated with $\ell_p$ For every $p\geq 2$, we give a characterization of positive definite functions on a free group, which can be extended to the positive linear functionals on the free group $\mathrm{C}^\ast$-algebra associated with $\ell_p$. As a consequence, the associated $\mathrm{C}^\ast$-algebras are mutually non-isomorphic, and they have a unique tracial state. April 18 2:30 - 4:00 RIMS 204 Hiroki Sako (RIMS) Property A and the operator norm localization property for discrete metric spaces We study property A defined by Yu and the operator norm localization property defined by Chen, Tessera, Wang, and Yu. These are coarse geometric properties for metric spaces, which have applications to operator K-theory. It is proved that these two properties are equivalent for discrete metric spaces with bounded geometry. Combining this theorem with a recent result by Brodzki, Niblo, Spakula, Willett, and Wright, we obtain several characterizations of property A. May 02 2:30 - 4:00 RIMS 204 Narutaka Ozawa (RIMS) Cantor minimal systems and finitely generated simple amenable groups (after Matui, Juschenko--Monod) Juschenko and Monod proved the Grigorchuk--Medynets conjecture that the topological full groups of any minimal transformation on the Cantor space is amenable. It was previously known that the commutator subgroup of such a group is simple (Bezuglyi--Medynets and Matui) and sometimes finitely generated (Matui). Thus, these groups give rise to the first examples of infinite finitely generated groups that are simple and amenable. I will discuss these results and their background. May 02 Colloquium 4:30 - 5:30 RIMS 110 Andrzej Żuk (Paris VI & RIMS) On a problem of Atiyah In 1976, Michael Atiyah defined $L^2$-Betti numbers for manifolds and asked a question about their rationality. These numbers arise as the von Neumann dimensions of kernels of certain operators acting on the $L^2$-space of the fundamental group of a manifold. The problem concerning their values is closely related to the Kaplansky zero-divisor question. We present constructions of closed manifolds with irrational $L^2$-Betti numbers. May 16 2:30 - 4:00 RIMS 204 Mamoru Tanaka (RIMS) Expander graphs and higher eigenvalues of the Laplacians on graphs In words, expander graphs are highly connected sparse graphs. They are used in theoretical computer science, combinatorics, functional analysis, and so on. It is known that expander graphs are characterized by the second eigenvalues of the Laplacians on the graphs. In this talk, we give relations between expander graphs and higher eigenvalues of the Laplacians on graphs. May 26 RIMS 420 Takagi Lectures May 28 -- June 01 RIMS 111 RIMS 420 Developments of the geometry of transformation groups Conference on Geometry June 04 - 08 RIMS 420 Geometric Group Theory - Kyoto 2012 June 06 Colloquium 4:30 - 5:30 RIMS 420 Mladen Bestvina (University of Utah) Group actions on quasi-trees and applications It is Serre who pointed out that SL_2(Z) acts on a tree. According to Bass-Serre theory, a group that acts on a tree can be decomposed as an amalgam of its subgroups. In this talk I will consider actions of groups on quasi-trees -- these are metric spaces (e.g. graphs) quasi-isometric to a tree. Having an action on a quasi-tree is a much more flexible condition than having an action on a tree. There is a general construction of such actions for "rank 1" groups. There are also many groups that do not admit nontrivial actions on quasi-trees, by the work of Manning, Burger-Mozes, N. Ozawa (SL_3(Z) is an example). Applications include a construction of quasi-morphisms and quasi-cocyces on various groups, characterization of elements in mapping class groups with zero stable commutator length, and finiteness of asymptotic dimension of mapping class groups. This is joint work with Ken Bromberg and Koji Fujiwara. June 20 2:30 - 4:00 RIMS 204 Hirokazu Maruhashi (Kyoto) Parameter rigidity of actions of nilpotent Lie groups A locally free smooth action $\rho$ of a connected Lie group on a closed manifold is said to be parameter rigid if each action which has the same orbits as $\rho$ is conjugate to $\rho$. There are not so many known parameter rigid actions of noncommutative groups. In this talk we give a criterion for parameter rigidity of nilpotent group actions and construct parameter rigid actions of nilpotent groups. June 20 Colloquium 4:30 - 5:30 RIMS 110 Igor Mineyev (UIUC & RIMS) An introduction to the Hanna Neumann Conjecture The Hanna Neumann Conjecture is an easy-to-state question about subgroups of free groups that has been open since 1956-1957. I will present examples, ways to restate the conjecture, and some ideas of its recent proof. June 27 Enlarged Colloquium 2:40 - 3:40       & 4:30 - 5:30 RIMS 420 Koji Fujiwara (Kyoto University) Hyperbolicity of groups and its application Kaoru Ono (RIMS) Lagrangian Floer theory on compact toric manifolds July 04 2:30 - 4:00 RIMS 204 Yusuke Isono (Tokyo) Weak Exactness for $\mathrm{C}^\ast$-algebras and Application to Condition (AO) July 04 Colloquium 4:30 - 5:30 MATH 110 Yoshikata Kida (Kyoto) Ergodic theory for type $\mathrm{III}$ actions and Baumslag-Solitar groups July 18 2:30 - 4:00 RIMS 204 Hiroshi Ando (RIMS) Ultraproducts of von Neumann algebras The ultraproduct of $\mathrm{II}_1$ factors has been one of the most important tools in von Neumann algebra theory. However, due to the lack of a trace, it is not so clear how one should define the ultraproduct of general von Neumann algebras. There are at least two approaches, one given by A. Ocneanu using a normal state in stead of a trace, and the one given by U. Groh and Y. Raynaud based on the Banach space ultraproduct of the predual of a target von Neumann algerba. We will study how these two algebras are related, and show some properties of the modular operator of a ultrapower state. (Joint work with Uffe Haagerup.) Sep 24 - 26 RIMS 420 Recent developments in operator algebras and related topics (program) Oct 10 2:30 - 4:00 Sci 6-809 Hiroki Sako (RIMS) Amenability of Thompson's group ? (after Justin Moore) I will introduce a recent work of Justin Moore (arXiv:1209.2063). He proved that Richard Thompson's group F is amenable. I will explain the outline of the proof. The problem is reduced to "existence of an idempotent mean" on rooted binary trees. NB: Moore has retracted his claim. Oct 17 2:30 - 4:00 Sci 6-809 Hiroki Sako (RIMS) A generalization of expander graphs and local reflexivity of uniform Roe algebras Topics of my talk are discrete metric spaces and their operator algebraic properties. I will introduce a generalization of expander graphs, which is called a weak expander sequence. It is proved that a uniform Roe algebra of a weak expander sequence is not locally reflexive. It follows that uniform Roe algebras of expander graphs are not exact. Key tools for the proof are amenable traces and measured groupoids associated to discrete spaces. Nov 07 2:30 - 4:00 Sci 6-809 Takahiro Hasebe (Kyoto) Stochastic independence and cumulants based on spreadability Many notions of stochastic independence have been proposed to understand noncommutative random variables. Cumulants are quite useful quantities to study independent random variables. The main purpose of this talk is to explain a unifying approach to cumulants and independence. This talk is based on a joint work with Franz Lehner. Nov 26 2:30 - 4:00 RIMS 111 Masato Mimura (Tohoku) p-Kazhdan constants and non-expanders Fundamental idea to study graphs and finitely generated groups (as Cayley graphs) with the path metrics is to "linearize" them. More precisely, to consider good embeddings into well-behaved Banach spaces. Specially embeddings into the Hilbert space or the $\ell^p$ spaces are intensively studied. A well-known result states that a sequence of "expanders" does not coarsely embed into the Hilbert space. Here a sequence of "expanders" denotes a family of finite graphs of uniformly bounded degree whose spectral gaps are uniformly bounded below by zero. In this talk, we make an estimate of a decay rate of $\ell^p$-version of spectral gaps of a family of "non"-expanders. We will then have some information of such family in terms of embedding into the $\ell^p$ spaces. Main exapmle is a family of the Cayley graphs of $\mathrm{SL}_n({\mathbf Z}/k_n{\mathbf Z})$ for $n$ at least $3$ with repsect to standard generating sets of cardinal $4$, where $(k_n)$ is a sequence of integers bigger than two. Dec 12 2:30 - 4:00 Sci 6-809 Reiji Tomatsu (Hokkaido) Some remarks on non-commutative Poisson boundaries and product type actions of compact quantum groups I will begin with an overview of theory of non-commutative Poisson boundary that was introduced by Izumi and then study the following topics: (1) A conjecture about the classical parts of Poisson boundaries; (2) Product type actions and Poisson boundaries. I will prove the conjecture affirmatively for q-deformations of any compact Lie groups and give some observations on their product type actions. Jan 07 - 10 Komaba Rigidity School Jan 11 Komaba Miniworkshop on Operator Algebras I Jan 16 Komaba Miniworkshop on Operator Algebras II Jan 22 - 25 Sci 3-110 Discrete Geometry and Dynamical Systems Jan 30 Komaba Miniworkshop on Operator Algebras III Feb 02 - 03 Awaji Kansai Operator Algebras Seminar Feb 04 - 06 RIMS 420 Quantization and operator algebras (program) Feb 13 2:30 - 4:00 RIMS 204 Takumi Enomoto (Kyoto) Classification theorem of characters of the unitary group of the CAR-algebra via asymptotic method Boyer studied the representation theory of some very big groups. He claimed the factorial characters of the unitary group of the CAR-algebra are only tensor products of the canonical representation and its complex conjugate. However there are some gaps in his proof. I will talk about a new proof by Vershik-Kerov's "ergodic method". Feb 12 - 15 RIMS 420 Markov Chains on Graphs and Related Topics Feb 20 1:00 - 6:00 RIMS 109 Kyoto Operator Algebra Day Semester End Party Mar 11 Komaba Miniworkshop on Operator Algebras IV Mar 20 - 23 Kyoto MSJ Annual Meeting

Back to the main Index.