KOAS

有限正則グラフ $G$ の（通常の）等周定数 $h_2(G)$ とは，グラフの頂点集合 $V$ の（空でない） $2$ 分割 $(A_1,A_2)$ を動かすとき， $|A_i$の辺境界$|$ を $|A_i|$ で割った量の $i=1,2$ での最大値の分割での最小値をとることで定義される．$2\le n\le |V|$ なる整数 $n$ をとるとき，グラフの頂点集合の（空でない）$n$ 分割で同様のことを考えることで，$G$ の $n$ 分割等周定数 $h_n(G)$ が定義される．$h_n(G)$ は $G$ のラプラス作用素の第 $n$ 固有値（$0$ を第 $1$ 固有値とする）$\lambda_n(G)$ と関連することが知られており，「チーガー型の不等式」と呼ばれている（$n=2$ のときは Alon--V. Milman の著名な結果, 一般の場合は Lee--Gharan--Trevisan による最近の結果による）．
$h_n(G)$ は $n$ について単調非減少だが，一般に $h_{n+1}(G)$ は $h_n(G)$ に比べいくらでも大きくなりうる．藤原耕二は，“グラフ $G$ が連結ケーリーグラフの場合に $h_{n+1}$ と $h_n$ の値の間には非自明な関係があるのではないか”，という問題を提起した．本講演では，この藤原耕二の問題の，講演者による定量的な解決をお話ししたい．講演者の結果から，任意の $n\geq2$ に対し，有限連結ケーリーグラフの列 $\lbrace G_m\rbrace_m$ において「$\inf_m h_n(G_m)$ が正である」ことと「$\inf_m h_2(G_m)$ が正である」ことの同値性が従う．また，有限な頂点推移的なグラフ $G$ と $n\geq2$ に対して，「$h_n(G)$ と $h_{n+1}(G)$ の間に（定量的な）ギャップがあるときには $G$ が（$n$ に依る形で記述できる）ある種の対称性をもつ」ことも明らかになった．
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 $G$.

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) 多分割等周定数，エクスパンダー族と有限ケーリーグラフ Multi-way isoperimetries, expanders, and Cayley graphs 有限正則グラフ $G$ の（通常の）等周定数 $h_2(G)$ とは，グラフの頂点集合 $V$ の（空でない） $2$ 分割 $(A_1,A_2)$ を動かすとき， $\|A_i$の辺境界$\|$ を $\|A_i\|$ で割った量の $i=1,2$ での最大値の分割での最小値をとることで定義される．$2\le n\le \|V\|$ なる整数 $n$ をとるとき，グラフの頂点集合の（空でない）$n$ 分割で同様のことを考えることで，$G$ の $n$ 分割等周定数 $h_n(G)$ が定義される．$h_n(G)$ は $G$ のラプラス作用素の第 $n$ 固有値（$0$ を第 $1$ 固有値とする）$\lambda_n(G)$ と関連することが知られており，「チーガー型の不等式」と呼ばれている（$n=2$ のときは Alon--V. Milman の著名な結果, 一般の場合は Lee--Gharan--Trevisan による最近の結果による）． $h_n(G)$ は $n$ について単調非減少だが，一般に $h_{n+1}(G)$ は $h_n(G)$ に比べいくらでも大きくなりうる．藤原耕二は，“グラフ $G$ が連結ケーリーグラフの場合に $h_{n+1}$ と $h_n$ の値の間には非自明な関係があるのではないか”，という問題を提起した．本講演では，この藤原耕二の問題の，講演者による定量的な解決をお話ししたい．講演者の結果から，任意の $n\geq2$ に対し，有限連結ケーリーグラフの列 $\lbrace G_m\rbrace_m$ において「$\inf_m h_n(G_m)$ が正である」ことと「$\inf_m h_2(G_m)$ が正である」ことの同値性が従う．また，有限な頂点推移的なグラフ $G$ と $n\geq2$ に対して，「$h_n(G)$ と $h_{n+1}(G)$ の間に（定量的な）ギャップがあるときには $G$ が（$n$ に依る形で記述できる）ある種の対称性をもつ」ことも明らかになった． 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

Kyoto Operator Algebra Seminar

2013 Spring/Summer