# 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

 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 Sci 3-110 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 19 2:30 - 4:00 RIMS 204 TBA TBA July 10 2:30 - 4:00 RIMS 204 Raphaël Ponge (Seoul) TBA

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