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Group actions and von Neumann algebras

Research Project

Project/Area Number 17K14201
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

Isono Yusuke  京都大学, 数理解析研究所, 特定助教 (80783076)

Project Period (FY) 2017-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywordsフォンノイマン環 / 冨田・竹崎理論 / 離散群の非特異作用 / エルゴード理論 / アフィン変換 / III型フォンノイマン環 / Intertwining theorem / 融合積フォンノイマン環 / 無限テンソル積フォンノイマン環 / von Neumann algebra / type III factor / group action
Outline of Final Research Achievements

I am interested in von Neumann algebras arising from group actions on measure spaces. In the case that the action does not preserve any measure, the algebra becomes type III, which is the most difficult class in von Neumann algebras. I obtained the following two results.
First, I succeeded to generalize Popa's techniques to type III algebras. This is a technical result but is the most important part of my research.
Second, I established a new way of constructing group actions. This uses affine actions of groups, hence the structure of the action is closely related to geometry of the group.

Academic Significance and Societal Importance of the Research Achievements

離散群の測度空間への作用は,測度を保つ作用が広く研究されている.これは測度を保たない群作用の研究が極めて難解である事が理由であり,よって多くの事が分かっていない.しかし一方で,幾何学的,または確率論的な着眼点から自然にそのような作用が現れる事もあるため,その研究には重要な意義がある.私の研究は,そのような作用に対する技術的な進展と具体例の構成であり,これはこの方面における基礎的な研究の一つとみなせるだろう.

Report

(4 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (19 results)

All 2020 2019 2017 Other

All Int'l Joint Research (5 results) Journal Article (5 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 5 results) Presentation (6 results) (of which Int'l Joint Research: 2 results,  Invited: 5 results) Remarks (3 results)

  • [Int'l Joint Research] パリ11大学/ENS Lyon(フランス)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] TU Delft(オランダ)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] KU Leuven(ベルギー)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] パリ11大学(フランス)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] パリ11大学(フランス)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras2020

    • Author(s)
      Cyril Houdayer, Yusuke Isono
    • Journal Title

      Proc. Roy. Soc. Edinburgh Sect. A.

      Volume: 150 Issue: 3 Pages: 1495-1532

    • DOI

      10.1017/prm.2018.152

    • Related Report
      2019 Annual Research Report 2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] L2-Cohomology, Derivations, and Quantum Markov Semi-Groups on q-Gaussian Algebras2020

    • Author(s)
      Martijn Caspers, Yusuke Isono, Mateusz Wasilewski
    • Journal Title

      Int. Math. Res. Not.

      Volume: 印刷中 Issue: 9 Pages: 6405-6441

    • DOI

      10.1093/imrn/rnaa044

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors2019

    • Author(s)
      Yusuke Isono
    • Journal Title

      Anal. PDE

      Volume: 12 Issue: 5 Pages: 1295-1324

    • DOI

      10.2140/apde.2019.12.1295

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On fundamental groups of tensor product II_1 factors2019

    • Author(s)
      Yusuke Isono
    • Journal Title

      J. Inst. Math. Jussieu.

      Volume: 印刷中 Issue: 4 Pages: 1121-1139

    • DOI

      10.1017/s1474748018000336

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Unique prime factorization for infinite tensor product factors2019

    • Author(s)
      Yusuke Isono
    • Journal Title

      J. Funct. Anal.

      Volume: 276 Issue: 7 Pages: 2245-2278

    • DOI

      10.1016/j.jfa.2018.07.015

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] Ergodic theory of affine isometric actions on Hilbert spaces2020

    • Author(s)
      磯野優介
    • Organizer
      RIMS研究集会「作用素環の分類理論における新展開」
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Tensor product decompositions and rigidity of full factors2019

    • Author(s)
      磯野優介
    • Organizer
      RIMS研究集会「作用素環論の最近の進展」
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Unitary conjugacy for type III subfactors and W*-superrigidity2019

    • Author(s)
      磯野優介
    • Organizer
      日本数学会
    • Related Report
      2019 Annual Research Report
  • [Presentation] Popa's intertwining theory for type III factors2019

    • Author(s)
      Yusuke Isono
    • Organizer
      Classification Problems in von Neumann Algebras
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Prime factorization for infinite tensor product factors2017

    • Author(s)
      Yusuke Isono
    • Organizer
      Von Neumann algebras and measured group theory
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Deformation/rigidity理論とIII型フォンノイマン環2017

    • Author(s)
      磯野優介
    • Organizer
      日本数学会特別講演
    • Related Report
      2017 Research-status Report
    • Invited
  • [Remarks]

    • URL

      http://www.kurims.kyoto-u.ac.jp/~isono/

    • Related Report
      2019 Annual Research Report
  • [Remarks]

    • URL

      http://www.kurims.kyoto-u.ac.jp/~isono/

    • Related Report
      2018 Research-status Report
  • [Remarks]

    • URL

      http://www.kurims.kyoto-u.ac.jp/~isono/

    • Related Report
      2017 Research-status Report

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Published: 2017-04-28   Modified: 2021-02-19  

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