# Kyoto Operator Algebra Seminar

## April 2013 - March 2014

 April 03 1:00 - 1:50 RIMS 204 Narutaka Ozawa (RIMS) Some useful but not well-documented facts about tracial states on $\mathrm{C}^*$-algebras I will present some useful but not well-documented facts about tracial states on $\mathrm{C}^*$-algebras, and mention related open problems. 2:00 - 2:50 RIMS 204 Hiroshi Ando (IHES) Ultraproducts, QWEP von Neumann algebras and Effros-Marechal Topology Haagerup and Winslow studied topological properties of the Polish space $\mathrm{vN}(H)$ of von Neumann algebras acting on the separable infinite-dimensional Hilbert space $H$. Motivated by the work of Effros, this topology was introduced by Marechal. Among other interesting results, they proved that Kirhchberg's QWEP conjecture is equivalent to the assertion that the set ${\cal F}_{\mathrm{inj}}$ of injective factors on $H$ is dense in $\mathrm{vN}(H)$, and moreover a $\mathrm{II}_1$ factor $M$ on $H$ is $R^{\omega}$-embeddable if and only if $M$ is a limit of a sequence of injective factors. Based on the work of Haagerup-Winslow and the recent work of the speaker and Haagerup on ultraproducts, we will give new characterizations of QWEP von Neumann algebras. This is a joint work with Uffe Haagerup and Carl Winslow (University of Copenhagen). April 17 Colloquium 4:30 - 5:30 MATH 110 Yasuhiko Sato (Kyoto) A characterization of classifiable nuclear $\mathrm{C}^*$-algebras April 24 2:40 - 5:30 RIMS 420 Enlarged Colloquium May 15 2:30 - 4:00 RIMS 204 Yuhei Suzuki (Tokyo/RIMS) On Quasidiagonal Representations of Nilpotent Groups (after Caleb Eckhardt) Recently, Eckhardt has shown the full group $\mathrm{C}^*$-algebras of discrete nilpotent groups are strongly quasidiagonal. In other words, any nilpotent subgroup of the unitary group on a Hilbert space is quasidiagonal. In this talk, I will give a slightly different proof from Eckhardt's one, which uses less knowledges about nilpotent groups. May 29 2:30 - 4:00 RIMS 204 Masato Mimura (Tohoku) 萔CGNXp_[ƗLP[[Ot Multi-way isoperimetries, expanders, and Cayley graphs LOt $G$ ́iʏ́j萔 $h_2(G)$ Ƃ́COt̒_W $V$ ́iłȂj $2$ $(A_1,A_2)$ 𓮂ƂC $|A_i$̕ӋE$|$ $|A_i|$ Ŋʂ $i=1,2$ ł̍ől̕ł̍ŏlƂ邱ƂŒD$2\le n\le |V|$ Ȃ鐮 $n$ ƂƂCOt̒_ẂiłȂj$n$ œl̂Ƃl邱ƂŁC$G$ $n$ 萔 $h_n(G)$ D$h_n(G)$ $G$ ̃vXpf̑ $n$ ŗLli$0$ $1$ ŗLlƂj$\lambda_n(G)$ Ɗ֘A邱ƂmĂCu[K[^̕svƌĂ΂Ăi$n=2$ ̂Ƃ Alon--V. Milman ̒Ȍ, ʂ̏ꍇ Lee--Gharan--Trevisan ɂŋ߂̌ʂɂjD $h_n(G)$ $n$ ɂĒP񌸏Cʂ $h_{n+1}(G)$ $h_n(G)$ ɔׂł傫Ȃ肤DḱCgOt $G$ AP[[Ot̏ꍇ $h_{n+1}$ $h_n$ ̒l̊Ԃɂ͔񎩖Ȋ֌Ŵł͂ȂhCƂND{uł́C̓ḱ̖Cu҂ɂʓIȉbDu҂̌ʂCCӂ $n\geq2$ ɑ΂CLAP[[Ot̗ $\lbrace G_m\rbrace_m$ ɂāu$\inf_m h_n(G_m)$ łvƂƁu$\inf_m h_2(G_m)$ łvƂ̓l]D܂CLȒ_ړIȃOt $G$ $n\geq2$ ɑ΂āCu$h_n(G)$ $h_{n+1}(G)$ ̊ԂɁiʓIȁjMbvƂɂ $G$ i$n$ Ɉ˂ŋLqłj̑Ώ̐vƂ炩ɂȂD For a finite regular graph $G$, the (usual) isoperimetric constant $h_2(G)$ is defined as the minimum among non-empty decompositions $(A_1,A_2)$ of the vertex set $V$ of the maximum among $i=1,2$ of the ratio $|$the edge boundary of $A_i|/|A_i|$. For $n$ between $2$ and $|V|$, the $n$-way isoperimetric constant $h_n(G)$ of $G$ is defined in terms of non-empty decomposiitons $(A_1,..., A_n)$ of $V$. Cheeger-type inequalities, which relate $h_n(G)$ to the $n$-th eigenvalue $\lambda_n(G)$ of the combinatorial Laplacian, are known: for $n=2$, this is a well-known result of Alon and V. Milman, and for general $n$ this is a recent result of Lee--Gharan--Trevisan. $h_n(G)$ is non-decreasing on $n$, and in general $h_{n+1}(G)$ can be arbitrarily bigger than $h_n(G)$. Koji Fujiwara asks whether there exists any non-trivial relation between the values of $h_{n+1}(G)$ and $h_n(G)$ for finite connected Cayley graphs $G$. In this talk, the answer to this question by the speaker shall be presented. This unversal inequality provides with a corollary that for $n>2$ and for a sequence of finite connected Cayley graphs $\lbrace G_m\rbrace_m$, "$\inf_m h_n(G_m)>0$" in fact implies "$\inf_m h_2(G_m)>0$." Furthermore, it is shown that for a finite vertex transitive graph $G$ and $n>1$, an (explicitly stated) numerical gap between $h_{n+1}(G)$ and $h_n(G)$ implies a certain symmmetry (subscribed in terms of $n$) of the graph $G$. June 08 Osaka Kansai Operator Algebra Seminar June 19 Colloquium 4:30 - 5:30 MATH 110 Yoshifumi Matsuda (Kyoto) Rotation number and actions of the modular group on the circle July 10 2:30 - 4:00 RIMS 204 Raphaël Ponge (Seoul) Noncommutative geometry and conformal geometry In this talk we shall report on a program of using the recent framework of twisted spectral triples to study conformal geometry from a noncommutative geometric perspective. One result is a local index formula in conformal geometry taking into account the action of the group of conformal diffeomorphisms. Another result is a version of Vafa-Witten's inequality for twisted spectral triples. Geometric applications include a version of Vafa-Witten's inequality in conformal geometry. There are also noncommutative versions for spectral triples over noncommutative tori and duals of discrete cocompact subgroups of semisimple Lie groups satisfying the Baum-Connes conjecture. (This is joint work with Hang Wang.) July 16-19 RIMS 420 Geometric Group Theory - Kyoto 2013 July 22 4:00 - 5:30 RIMS 109 Raman Srinivasan (Chennai) Non-cocycle-conjugate $E_0-$semigroups on non-type-I factors July 30 3:00 - 4:30 Sci 6-609 Andrzej Żuk (Paris) Expanders Aug. 02 1:00 - 2:30 MATH 305 Tullio G. Ceccherini-Silberstein (U. Sannio) Cellular automata and groups The purpose of this talk is to overview the theory of cellular automata on groups focusing on the Garden of Eden Theorem for cellular automata over amenable groups and Gottschalk's surjunctivity problem. This includes a presentation of a few examples of cellular automata (including Conway's Game of Life) and a discussion of the notions of amenability and soficity (due to Gromov--Weiss), and of the relation between surjunctivity and Kaplansky's conjecture on stable finiteness of group rings. The talk, completely self-contained, addresses to a wide audience, including graduate students, interested in dynamical systems, group theory, operator algebras and ring theory, and theoretical computer science. 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 A completely positive map is called order zero if it preserves orthogonality. Recent developments in the classification theorem of $\mathrm{C}^*$-algebras suggest that order zero c.p. maps are very compatible with projectionless $\mathrm{C}^*$-algebras. In this talk, we investigate the Murray--von Neumann equivalence for positive elements and see that plays a crucial role for the understanding of order zero c.p. maps and their conjugacy classes. As a consequence of this study, we obtain an affirmative answer to the Toms and Winter conjecture for $\mathrm{C}^*$-algebras with a unique tracial state. This talk is based on a joint work with Stuart White and Wilhelm Winter. Jan. 29-31 RIMS 111 Development of operator algebras and related topics (program) Mar. 24 Monday 10:30 - 12:00 RIMS 110 Detlev Buchholz (Göttingen) Quantum systems and resolvent algebras The standard $\mathrm{C}^*$-algebraic version of the Heisenberg algebra of canonical commutation relations, the Weyl algebra often causes difficulties since it does not admit physically interesting dynamical laws as automorphism groups. In this talk a $\mathrm{C}^*$-algebraic version of the canonical commutation relations is presented which circumvents such problems. It is based on the resolvents of the canonical operators and their algebraic relations. The resulting $\mathrm{C}^*$-algebras, the resolvent algebras, have many desirable analytic properties. In fact, they are of type I (postliminal) for finite quantum systems and nuclear in the infinite case. In either case they admit existence of an abundance of one--parameter automorphism groups corresponding to physically relevant dynamics. They are also useful in the discussion of supersymmetry and systems with constraints. Moreover, the resolvent algebras have a rich and interesting ideal structure which encodes specific information about the dimension of the underlying physical system. They thus provide an excellent framework for the rigorous analysis of finite and infinite quantum systems.

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