# Kyoto Operator Algebra Seminar

Organizers: Masaki IZUMI, Narutaka OZAWA, Yoshikata KIDA
Time and Location: 14:30 - 16:00 on Wednesday (followed by Colloquium), at RIMS 204
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.
Previous Years: 2011  2012  2013 Spring/Summer

## 2013-2014 Fall/Winter

 Sept. 11-13 RIMS 420 Recent Progress in Operator Algebras (program) Oct. 08 Tuesday 16:30 - 18:00 RIMS 206 Yuhei Suzuki (Tokyo/RIMS) Amenable minimal Cantor systems of free groups arising from diagonal actions G. A. Elliott and A. Sierakowski have constructed an amenable minimal Cantor system of free groups whose $K_0$-group vanishes. In particular this is distinguished from the boundary action by $K$-theory. In this talk we show every (f.g., noncommutative) free group admits continuum many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras with UCT by a different approach from that of Elliott--Sierakowski. The $K$-theory of these Cantor systems are also determined explicitly. Oct. 21-28 MATH 127 Intensive Course by Hideki Kosaki (Kyushu) Oct. 21-28 MATH 108 Intensive Course by Shin-ichi Oguni (Ehime) Oct. 23 Colloquium MATH 110 Shin-ichi Oguni (Ehime) The coarse Baum-Connes conjecture for relatively hyperbolic groups Oct. 29 Tuesday 16:30 - 18:00 RIMS 206 Pierre Fima (Paris 7/Tokyo) Amenable, transitive and faithful actions of groups acting on trees We study under which condition an amalgamated free product or an HNN-extension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees. Nov. 06-08 RIMS 110 Operator monotone functions and related topics Nov. 12 Tuesday 16:30 - 18:00 RIMS 206 Yusuke Isono (Tokyo) Some prime factorization results for free quantum group factors Ozawa and Popa proved some unique factorization properties for tensor products of free group factors. Roughly speaking, it means such a tensor product "remembers" each tensor component. In this seminar, we study similar factorization results for free quantum group factors. In the $\mathrm{II}_1$ factor case, we can follow Ozawa--Popa's method for free group factors. In the general case, we use continuous cores. More precisely, we observe a condition (AO) type phenomena for cores of the tensor products and then deduce some weak factorization properties. Nov. 13 Enlarged Colloquium 14:40 - 15:40       & 16:30 - 17:30 MATH 110 Benoit Collins (Ottawa/Tohoku) Free probability and quantum information theory Kenji Fukaya (Stony Brook) z`FC̕@̌ Nov. 16-17 RIMS Takagi Lectures Dec. 04 13:30 - 14:40       & 14:50 - 16:00 RIMS 204 Issan Patri (Chennai) Automorphisms of Compact Quantum Groups We will study automorphisms of Compact Quantum Groups. We will be define a notion of "inner" automorphism in the group-theoretic sense and see the behaviour of normal quantum subgroups under these automorphisms. It will turn out that automorphisms are, in a sense, strongly not "inner", give ergodic actions on the the Compact Quantum Group. We will end with a question on finding a suitable automorphism of a Tarski Monster Group, which will show how the quantum case can be very different from the classical one. Makoto Yamashita (Ochanomizu) Categorical dual of quantum homogeneous spaces The notion of compact quantum group actions on operator algebras can be dualized into that of $\mathrm{Rep}(G)$-module $\mathrm{C}^*$-categories, in the spirit of Woronowicz's Tannaka--Krein duality. This allows us to give several classification results. First, ergodic actions of $\mathrm{SU}_q(2)$ can be classified in terms of certain weighted graphs, extending the classical McKay correspondence. Second, recasting the noncommutative Poisson boundary in this framework, we obtain an explicit list of the non-Kac quantum groups with the same fusion rule and classical dimension of representations as $\mathrm{SU}(n)$. This talk is based on joint works with Kenny De Commer and Sergey Neshveyev. Dec. 09-13 MATH 110 Metric geometry and analysis Dec. 15 Sunday 10:00 - 12:00 MATH 110 Zhizhang Xie (Texas A&M University) Finitely embeddable groups and strongly finitely embeddable groups The notion of groups finitely embeddable into Hilbert space was introduced by Weinberger and Yu. It is a more flexible notion than coarse embeddability. For finitely embeddable groups, Weinberger and Yu obtained a lower bound of the free-rank of the finite part of K-theory of the maximal group $\mathrm{C}^*$-algebra, which in turn was used to give a lower bound of the size of the structure group of an oriented manifold and to give a lower bound of the size of the space of positive scalar curvature metrics on a given manifold. In order to detect finer information of the structure group and the space of positive scalar curvature metrics, Yu and I were led to the notion of groups with strongly finite embeddability into Hilbert space. It is an open question whether every group is (strongly) finitely embeddable. The talk is based on joint work with Guoliang Yu. Dec. 16-20 MATH 110 Further development of Atiyah-Singer index theorem and K-theory Dec. 21-22 Kinosaki Kansai Operator Algebra Seminar Jan. 07 Tuesday 16:30 - 18:00 RIMS 206 Narutaka Ozawa (RIMS) A nonseparable amenable operator algebra which is not isomorphic to a $\mathrm{C}^*$-algebra The notion of amenability for Banach algebras was introduced by B. E. Johnson in 1970s and has been studied intensively since then. For several natural classes of Banach algebras, the amenability property is known to single out the ggoodh members of those classes. For example, B. E. Johnsonfs fundamental observation is that the Banach algebra $L_1(G)$ of a locally compact group $G$ is amenable if and only if the group $G$ is amenable. Another example is the celebrated result of Connes and Haagerup which states that a $\mathrm{C}^*$-algebra is amenable as a Banach algebra if and only if it is nuclear. In this talk, I will talk about recent progress on the longstanding problem whether every amenable operator algebra is isomorphic to a (necessarily nuclear) $\mathrm{C}^*$-algebra. This problem was recently solved in the negative for general nonseparable operator algebras by Choi, Farah and me; and in the affirmative for commutative operator algebras by Marcoux and Popov. The existence of a separable counterexample---the most interesting case---remains an open problem. Jan. 14 Tuesday 16:30 - 18:00 RIMS 206 Yasuhiko Sato (Kyoto) Murray--von Neumann equivalence for positive elements and order zero c.p. maps tba Jan. 29-31 RIMS 111 Development of operator algebras and related topics (program)

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