Quantum Analysis Seminar
This seminar treats anything related to operator algebras. Both research and survey talks are welcomed.
A list of talks before 2018 is available HERE(Professor Yamagami's website).
2018
- Jan. 11Takaaki Moriyama (University of Tokyo)
Weak density of orbit equivalence classes and free products of infinite abelian groups
It has been known that for a countable group, the structure of the space
of probability-measure-preserving (p.m.p.) actions reflects many
properties of the group. In this talk, we will show that if a group is
the free product of infinite abelian groups, then for every free p.m.p.
action of the group, its orbit equivalence class is weakly dense in the
space of p.m.p. actions. This extends Lewis Bowen's result for free
groups.
- Dec. 21Yusuke Sawada (Nagoya University)
E_0-semigroups, minimal dilations and product systems of W*-bimodules
Arvesonは, I-型因子環上のE_0-半群をプロダクトシステムを用いて分類した. この度, W*-双加群から成るプロダクトシステムを導入し,
それにより一般のvon Neumann環上のE_0-半群の分類を行なった.
その過程でCP_0-半群の極小伸張が構成される. これらはBhat-SkeideのHilbert双加群のプロダクトシステムによる分類と伸張の構成に影響を受けている。
- Nov. 16 Yoshimichi Ueda (Nagoya University)
Two projections and Loewner-Kufarev equations
Free probability is a noncommutative analysis, and thus its central problem is to investigate noncommutative phenomena.
A pair of projections in free probability can be viewed as the most elementary ‘noncommutative’ case.
A radial Loewner-Kufarev equation naturally appears and plays a role in the analysis of pairs of projections.
This is an observation made in my joint work with Masaki Izumi some years ago. I will explain some results around this observation,
including some recent works due to Tarek Hamdi.
- Nov. 9 Michio Seto (National Defense Academy)
Theory of continuous de Branges-Rovnyak decomposition and its applications
de Branges-Rovnyak 分解はヒルベルト空間内の直交分解を一般化した概念です。
もう30年以上前の出来事ですが、de Branges-Rovnyak 分解の連続版を応用し、
de Branges が複素解析における有名な予想を解決したことは作用素論関係者の間
ではよく知られた話かと思います。しかし、その議論がとても難解だったためか、
それは現在では忘れられた理論になってしまいました。
さて、今回の発表ではその連続 de Branges-Rovnyak 分解の理論を応用を視野に
入れて再検討します。
特に、Vasyunin-Nikolskii の論文と安藤先生の講義録とを融合した形式で解説し、
Parseval の等式から等式が量産されることの不等式版(行列版)が得られるこ
とを紹介したいと思います。
- Oct. 16 (Tue) Thierry Levy (Paris 6)
Recent progress in 2-dimensional quantum Yang-Mills theory
Quantum Yang-Mills theory is an important part of the Standard model
built by physicists to describe elementary particles and their interactions.
One approach to this theory consists in constructing a probability measure
on an infinite-dimensional space of connections on a principal bundle over space-time.
However, in the physically realistic 4-dimensional situation,
the construction of this measure is still an open mathematical problem.
The subject of this talk will be the physically less realistic 2-dimensional situation,
in which the construction of the measure is possible, and fairly well understood.
In probabilistic terms, the 2-dimensional Yang-Mills measure is the distribution of
a stochastic process with values in a compact Lie group
(for example the unitary group U(N)) indexed by the set of continuous closed curves
with finite length on a compact surface
(for example a disk, a sphere or a torus) on which one can measure areas.
It can be seen as a Brownian motion (or a Brownian bridge) on the chosen compact Lie group
indexed by closed curves, the role of time being played in a sense by area.
In this talk, I will describe the physical context in which the Yang-Mills measure is constructed,
and describe it without assuming any prior familiarity with the subject.
I will then present a set of results obtained in the last few years by
Antoine Dahlqvist, Bruce Driver, Franck Gabriel, Brian Hall, Todd Kemp,
James Norris and myself concerning the limit as N
tends to infinity of the Yang-Mills measure constructed with the unitary group U(N).
- Oct 5 R. Srinivasan (Chennnai)
E_0-semigroups on factors
I will review the current progress on endomorphism semigroups on factors, particularly on non-type-I factors.
This is mostly my joint work with Oliver T Margetts.
- Jul 10 (Tue) at A-358 Hiroshi Ando(Chiba)
Structure of bicentralizer algebras and inclusions of type III factors
Connesは, 1970年代に単射的因子環の分類を, III_1 型を除き完成させた。また彼はIII_1型因子環$M$と忠実正則状態$\varphi$に対して,
Bicentralizerと呼ばれる$M$の部分von Neumann環を導入し、Bicentraliezrが自明な単射的III_1型因子環はAraki--Woods因子環に同型である
(従って全て互いに同型である)事を示した. 後にHaagerupは実際に単射的因子環のbicentralizerが自明
である事を示し、単射的因子環の分類定理が完成した。Connesはまた、単射的と限らない一般のIII_1型因子環のbicentralizerが自明であるかを問題にした。
これは現在も未解決であるが、多くの具体例では肯定的であるという事が知られている。
また、ConnesはConnes-Stormerの推移性定理から、$M$の二つの忠実正則状態を漸近的にintertwineするユニタリの列は、
bicentralizer間の自然な同型を与える事を示した。Haagerupは最近、Connesの同型写像のアイデアを拡張し、
bicentralizer上に自然なR-作用(bicentralizer flowと呼ぶ事にする)が与えられることを発見した。
同じflowは別の観点から独立にMarrakchiによって昨年、再発見された。bicentralizerはIII_1型 因子環の包含$N\subset M$
に対しても定義する事ができる(増田による)。我々は相対bicentralizerの上のflowの性質を調べ、
それが因子環の包含の構造と密接に関わっている事を発見した。講演ではflowの構成や、その性質について述べたい。
(Uffe Haagerup, Cyril Houdayer, Amine Marrakchiとの共同研究)
- Jun 25--29 Yasuo Watatani(Kyushu)
Relative position of subspaces in a Hilbert space、or representations of quivers by Hilbert spaces and operators
(Intensive course)
- Jun 14 Yoshikata Kida(Tokyo)
Central sequences in the full group and its lifting problem
Central sequences are a well-developed and important technical tool in von Neumann algebras,
and often play an essential role in understanding their structure. On the other hand,
although central sequences for orbit equivalence relations are also expected to be useful,
many basic questions remain unsolved for a long time. One of them,
due to Schmidt, asks whether any inner amenable group admits a Schmidt action,
i.e., a free ergodic probability-measure-preserving action with a non-trivial central sequence
in its full group. We discuss some background materials, the lifting problem of central sequences
in a central groupoid-extension, and its application to provide a new class of groups which admit a Schmidt action.
- May 31 Michiya Mori(Tokyo)
Tingley's problem and the Mazur-Ulam property for operator algebras
Tingley's problem asks whether every surjective isometry between the
unit spheres of two Banach spaces admits an extension to a real linear
surjective isometry between the whole spaces.
In this talk, I explain recent progresses of Tingley's problem in the
setting of operator algebras.
Partly joint work with Narutaka Ozawa(RIMS).
- May 10 Yusuke Isono(RIMS)
Factoriality, Connes' invariants and fullness of amalgamated free products
We investigate factoriality, Connes' type III invariants and fullness of
arbitrary amalgamated free product von Neumann algebras using Popa's
deformation/rigidity theory. Among other things, we generalize many
previous structural results on amalgamated free product von Neumann
algebras and we obtain new examples of full amalgamated free product
factors for which we can explicitely compute Connes' type III invariants.
This is joint work with C. Houdayer.
- Apr 26 Mikael Pichot(McGill/RIMS)
Puzzles and automorphisms of free groups
I will discuss geometric constructions associated with the automorphism
group of the free group on two generators.
- Apr 19 Ryosuke Sato(Kyushu/Nagoya)
Quantized Vershik--Kerov Theory and q-deformed Gelfand--Tsetlin graph
無限ユニタリ群 U(\infty) の有限因子表現を決定する問題に対して、
Vershik--Kerov は U(\infty) の端点指標を部分群 U(N) らの既約指標
で近似するアプローチを考えた。本講演では、その理論をコンパクト量子群
の帰納系に対して与え、それを量子ユニタリ群の場合に適用し、Gorin が提案し
た q-deformed Gelfand--Tsetlin graph との関係について解説する。