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Analysis and a classification problem on group/quantum group actions on operator algebras

Research Project

Project/Area Number 16540180
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionHokkaido University

Principal Investigator

YAMANOUCHI Takehiko  Hokkaido Univ., Fac. of Sci., Asso. Prof., 大学院理学研究院, 助教授 (30241293)

Co-Investigator(Kenkyū-buntansha) KISHIMOTO Akitaka  Hokkaido Univ., Fac. of Sci., Prof., 大学院理学研究院, 教授 (00128597)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywords1-cocycle / coaction / Cartan subalegbra / discrete equivalence relation / factor / flow of weights / ergodic flow / amenable locally compact group / カルタン環 / 測度付き同値関係 / 正規化擬群 / 通約可能擬群 / フォン・ノイマン環 / コサイクル / エルゴード流 / III型因子環 / フォンノイマン環 / 量子群 / 同値関係
Research Abstract

Under this project, I made an intensive study on a certain relationship between 1-cocycles in ergodic theory and actions of cocommutative quantum groups on von Neumann algebras. This research can be divided roughly into two parts as follows.
The first part is concerned with a correspondence between 1-cocyc les on discrete equivalence relations and coactions on von Neumann algebras having Cartan subalgebras. I was able to show that every such cocycle gives rise to a coact ion on the von Neumann algebra A associated with the equivalence relation, whose fixed-point algebra contains the Cartan subalgebra. Conversely, I succeeded to prove that every coaction on A fixing the Cartan subalgebra pointwise arises in the manner described above. As an application, I was able to classify, up to cocycle conjugacy, coactions of amenable locally compact groups on hyperfinite II_{1} factor with full Connes spectra.
The second part is about some relationship between 1-cocycles on ergodic flow spaces and coactions on type III factors. I clarified how every 1-cocycle on the flow of weights on a type III factor N produces a coaction on N whose fixed-point algebra contains the centralizer of a dominant weight on N, and vice versa. I was also able to clarify the structure of the crossed product factors by such coactions by computing explicitly their flows of weights. As an application, I succeeded to give a complete classification on actions of amenable locally compact groups on hyperfinite type III factors with a certain property.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (12 results)

All 2007 2006 2005 2004

All Journal Article (12 results)

  • [Journal Article] One-cocycles on smooth flows of weights and extended modular coactions2007

    • Author(s)
      T.Yamanouchi
    • Journal Title

      Ergodic Theory and Dynamical Systems 27

      Pages: 285-318

    • NAID

      120000962265

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A characterization of coactions whose fixed-point algebras contain special maximal abelian *-subalgebras2006

    • Author(s)
      T.Yamanouchi
    • Journal Title

      Ergodic Theory and Dynamical Systems 26

      Pages: 1673-1706

    • NAID

      120000970184

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] On the normalizing groupoid and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations-subrelations2006

    • Author(s)
      T.Yamanouchi
    • Journal Title

      Journal of Functional Analysis 240

      Pages: 297-333

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Multiplier cocycles of a flow on a C*- algebra2006

    • Author(s)
      A.Kishimoto
    • Journal Title

      Journal of Functional Analysis 235

      Pages: 271-296

    • NAID

      120000954487

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the normalizing groupoid and the commensurability groupoids for inclusions of factors associated to ergodic equivalence rela- tions-subrelations2006

    • Author(s)
      T.Yamanouchi
    • Journal Title

      Journal of Functional Analysis 240

      Pages: 297-333

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Multiplier cocycles of a flow on a C*-algebra2006

    • Author(s)
      A.Kishimoto
    • Journal Title

      Journal of Functional Analysis 235

      Pages: 271-296

    • NAID

      120000954487

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Multiplier cocycles of a flow on a C*- algebra2006

    • Author(s)
      A.Kishimoto
    • Journal Title

      Journal Functional Analysis 235

      Pages: 271-296

    • NAID

      120000954487

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Approximate AF flows2005

    • Author(s)
      A.Kishimoto
    • Journal Title

      Journal of Evolution Equations 5

      Pages: 153-184

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Apporoximate AF flows2005

    • Author(s)
      A.Kisimoto
    • Journal Title

      Journal of Evolution Equations 5

      Pages: 153-184

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The one-cocycle property for shifts2005

    • Author(s)
      A.Kishimoto
    • Journal Title

      Ergodic Theory & Dynamical Systems 25

      Pages: 823-859

    • NAID

      120000964577

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Core Symmetries of a Flow2004

    • Author(s)
      A.Kishimoto
    • Journal Title

      Reviews in Mathematical Physics 16

      Pages: 479-507

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Central sequence algebras of a purely Infinite simple C^*-algebras2004

    • Author(s)
      A.Kishimoto
    • Journal Title

      Canadian Journal of Mathematics 56

      Pages: 1237-1258

    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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