Research of symmetries arising from automorphisms of operator algebras

• Project/Area Number: 16K05180
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: Masuda Toshihiko 九州大学, 数理学研究院, 教授 (60314978)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 因子環 / 自己同型群 / 対称性 / 群作用 / 作用素環 / エルゴード理論 / エルゴート変換 / 充足群 / 離散従順亜群 / テンソル圏 / 部分因子環 / 離散亜群 / 流れ / 再中心化環 / C*テンソル圏 / Q-system / 自己同型

Outline of Final Research Achievements

I studied the symmetries of operator algebras by focusing on automorphism groups of operator algebras. I obtained the following results mainly. 1. Classification of Roberts actions of tensor categories on the injective type III_1 factor. 2. I solved the conjecture of Ando-Haagerup-Houdayer-Marakech about the flows of relative bicentralizer associated with inclusion of type III_1 factors when small algebra is injective. 3. I classified outer actions (G-kernels) of discrete amenable groupoids on injective factors. My proof does not require the method of model actions to get rid of obstraction, and it is more natural. 4. I classified actions of discrete amenable groups on orbit equivalence of ergodic theory.

Academic Significance and Societal Importance of the Research Achievements

私の研究によって、作用素環の群作用や自己同型群の対称性の理解がより深まった。特にテンソル圏との関連において深い理解が得られたと考える。またエルゴード理論のように作用素環と関連する分野においても、作用素環論に由来するアイディアが有効に使えることも明確となり、理論の発展に大きく寄与した。

Research Products

• [Journal Article] Classification of outer actions of discrete amenable groupoids on injective factors (2022)
  • Author(s): Toshihiko Masuda
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 74 Issue: 3 Pages: 873-901
  • DOI: 10.2969/jmsj/86328632
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed / Open Access
• [Journal Article] On the relative bicentralizer flows and the relative flow of weights of inclusions of factors of type III_1 (2020)
  • Author(s): Toshihiko Masuda
  • Journal Title: Publications of the Research Institute for Mathematical Sciences Volume: 56 Issue: 2 Pages: 391-400
  • DOI: 10.4171/prims/56-2-4
  • Peer Reviewed / Open Access
• [Journal Article] A simple sufficient condition for triviality of obstructions in the orbifold construction for subfactors (2017)
  • Author(s): Toshihiko Masuda
  • Journal Title: Mathmatica Scandinavia Volume: 121 Issue: 1 Pages: 101-110
  • DOI: 10.7146/math.scand.a-26240
  • Peer Reviewed / Open Access
• [Journal Article] Classification of Roberts actions of strongly amenable C*tensor categories on the injective factor of type III_1 (2017)
  • Author(s): Toshihiko Masuda
  • Journal Title: International Journal of Mathematics Volume: 28 Issue: 07 Pages: 1750002-1750002
  • DOI: 10.1142/s0129167x17500525
  • Peer Reviewed / Open Access
• [Journal Article] A simple sufficient condition for triviality of obstruction in the orbifold construction for subfactors (2017)
  • Author(s): T. Masuda
  • Journal Title: Mathematica Scandinavica Volume: 印刷中
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Rohlin flows on von Neumann algebras (2016)
  • Author(s): T. Masuda, and R. Tomatsu
  • Journal Title: Memoirs of the American Mathematical Society Volume: 244 Issue: 1153 Pages: 0-0
  • DOI: 10.1090/memo/1153
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Classification of actions of discrete Kac algebras on injective factors (2016)
  • Author(s): T. Masuda, R. Tomatsu
  • Journal Title: Memoirs of the American Mathematical Society Volume: 245 Issue: 1160 Pages: 0-0
  • DOI: 10.1090/memo/1160
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] Actions of discrete amenable groups into the normalizers of full groups of ergodic transformations (2023)
  • Author(s): Toshihiko Masuda
  • Organizer: RIMS 共同研究プログラム 「作用素環論における群作用と数理物理の関連」
• [Presentation] Classification of outer actions of discrete amenable groupoids on injective factors (2021)
  • Author(s): 増田俊彦
  • Organizer: 日本数学会函数解析分科会
• [Presentation] Classification of outer actions of discrete amenable groupoids on injective factors (2021)
  • Author(s): 増田俊彦
  • Organizer: 研究集会「作用素環論の最近の進展」
• [Presentation] Outer actions (G-kernels) of discrete amenable groupoids on injective factors (2020)
  • Author(s): 増田俊彦
  • Organizer: 東大京大合同オンライン作用素環セミナー
• [Presentation] On the relative bicentralizer flows and the relative flow of weights of inclusions of factors of type III_1 (2019)
  • Author(s): 増田俊彦
  • Organizer: 日本数学会
• [Presentation] On the relative bicentralizer flows and the relative flow of weights of inclusions of factors of type III_1 (2019)
  • Author(s): 増田俊彦
  • Organizer: 作用素論作用素環研究集会
• [Presentation] Tannaka-Krein-Woronowicz duality from the viewpoint of Q-systems, (2018)
  • Author(s): 増田俊彦
  • Organizer: 日本数学会函数解析分科会
• [Presentation] 強従順なC^*テンソル圏のロバーツ作用の分類について (2017)
  • Author(s): 増田俊彦
  • Organizer: 日本数学会函数解析分科会
  • Place of Presentation: 首都大学東京
  • Year and Date: 2017-03-26
• [Presentation] Classification of Roberts actions of strongly amenable C*-tensor categories on the injective factor of type III_1 (2017)
  • Author(s): 増田俊彦
  • Organizer: 東大作用素環セミナー
• [Presentation] 冨田竹崎理論の紹介とそれに関連する話題 (2017)
  • Author(s): 増田俊彦
  • Organizer: 第56回実函数函数解析合同シンポジウム
• [Presentation] 強従順なC^*テンソル圏のロバーツ作用の分類について (2016)
  • Author(s): 増田俊彦
  • Organizer: 作用素環論エルゴート理論セミナー
  • Place of Presentation: 九州大学
  • Year and Date: 2016-10-17