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Study on non-commutative dynamical systems

Research Project

Project/Area Number 04452006
Research Category

Grant-in-Aid for General Scientific Research (B)

Allocation TypeSingle-year Grants
Research Field 解析学
Research InstitutionHokkaido University

Principal Investigator

KISHIMOTO Akitaka  Hokkaido University, Faculty of Science Professor, 理学部, 教授 (00128597)

Co-Investigator(Kenkyū-buntansha) ARAI Asao  Hokkaido University, Faculty of Science Associate Professor, 理学部, 助教授 (80134807)
GIYA Yoshikazu  Hokkaido University, Faculty of Science Professor, 理学部, 教授 (70144110)
HAYASHI Mikihiro  Hokkaido University, Faculty of Science Professor, 理学部, 教授 (40007828)
NAKAZI Takahiko  Hokkaido University, Faculty of Science Professor, 理学部, 教授 (30002174)
OKABE Yasunori  Hokkaido University, Faculty of Science Professor, 理学部, 教授 (30028211)
綿谷 安男  北海道大学, 理学部, 助教授 (00175077)
Project Period (FY) 1992 – 1993
Project Status Completed (Fiscal Year 1993)
Budget Amount *help
¥6,800,000 (Direct Cost: ¥6,800,000)
Fiscal Year 1993: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1992: ¥3,700,000 (Direct Cost: ¥3,700,000)
KeywordsOperator algebras / Non-commutativity / Dynamical systems / Groups / Action / Crossed products / Covariant representations / Rohlin property / 非可換力学系 / エルゴード性 / 自己同型写像 / 正準反交換関係 / 無限テンソル積
Research Abstract

A non-commutative dynamical system is a triple of a non-commutative algebra, a group or group-like object, and its action on the algebra. When the algebra is commutative, we call the system classical and our general attitude is to study the classical case and try to generalize the results obrained to the non-commutative systems, and then to see if there is anything proper for the non-commutative case. In this way we obrained some results which clarifies the relations between actions, crossed products and (covariant) representations in the case the group is compact. In particular the action will be regarded as of quasi-product type in many cases where the algebra is a simple C^*-algebra.
When the algebra is a certain W^*-algebra and the group is an amenable discrete group, one has a classification result for outer conjugate classes of actions. When the algebra is a C^*-algebra, there had been no such without any conditions on the actions themselves. But now we succeeded in proving a Rohlin property for certain systems, whose W^* version playd an important role in the classification theory and which had been thought unlikely to hold, and the situation is rapidly changing. We are currently working on proving this property for a wider class of systems. This has also invoked a classification theory for a new class of C^*-algebras which includes C^*-algebras like Cuntz algebras. On the other hand we now know that there could be various types of actions even for a rather (technically) simple C^*-algebras ; we may need new in variants for a complete answer.

Report

(3 results)
  • 1993 Annual Research Report   Final Research Report Summary
  • 1992 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] Akitaka Kishimoto: "The crossed product of a UHF algebra by a shift" Ergod.Th.& Dynam.Sys.13. 615-626 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Yasunori Okabe: "A new algorithm derived from the view-point of the fluctuation-dissipation principle in the theory of KM_2-O Langevin equations" Hokkaido Math.J.22. 199-209 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Takahiko Nakazi: "Hyponormal Toeplitz operators and extremal problems of Hardy spaces" Trans.Amer.Math.Soc.338. 753-767 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Mikohiro Hayashi: "Two sheeted discs and bounded analytic functions" J.d'Analyse Math.(発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Yoshikazu Giga: "Motion of a graph by nonsmooth weighted curvature" Proc.of World congress of nonlinear analysists. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Asao Arai: "Comparison and nuclearity of spaces of differential forms on topological vector spaces" J.Funct.Anal.111. 278-294 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Akitaka Kishimoto: "The crossed product of a UHF algebra by a shift" Ergod.Th.& Dynam.Sys.13. 615-626 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Yasunori Okabe: "A new algorithm derived from the view-point of the fluctuation-dissipation principle in the theory of KM_2O-Langevin equations" Hokkaido Math.J.22. 199-209 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Takahiko Nakazi: "Hyponormal Toeplitz operators and extremal problems of Hardy spaces" Trans.Amer.Math.Soc.338. 753-767 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Mikihiro Hayashi: "Two sheeted discs and bounded analytic functions" J.d'Analyze Math.to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Yoshikazu Giga: "Motion of a graph by nonsmooth weighted curvature" Proc. of World congress of nonlinear analysists. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Asao Arai: "Comparison and nuclearity of spaces of difderential forms on topological vector spaces" J.Funct.Anal.111. 278-294 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Akitaka Kishimoto: "Quasi-product actions of a compact group on a C^*-algebra" J.Functional Analysis. 115. 313-343 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] Akitaka Kishimoto: "The crossed product of a UHF algebra by a shift" Ergod.Th. & Dynam.Sys.13. 615-626 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] Yasunori Okabe: "A new algorithm derived from the view-point of the fluctuation-dissipation principle in the theory of KM_2O-Langevin equations" Hokkaido Math.J.22. 199-209 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] Takahiko Nakazi: "Hyponormal Toeplitz operators and extremal problems of Hardy spaces" Trans.Amer.Math.Soc.338. 753-767 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] Mikihiro Hayashi: "Two sheeted discs and bounded analytic functions" J.d'Analyse Math.(発表予定).

    • Related Report
      1993 Annual Research Report
  • [Publications] Yoshikazu Giga: "Motion of a graph by nonsmooth weighted curvature" Proc.of World congress of nonlinear analysists. (発表予定).

    • Related Report
      1993 Annual Research Report
  • [Publications] O.Bratteli: "Non-commutative spheres III:irrational rotations" Commun.Math.Phys.147. 605-624 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] W.Arveson: "A note on extensions of semigroups of *-endomorphisms" Proc.AMS. 116. 769-774 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Y.Okabe: "A new algorithm driven the view-point of the fluctuation-dissipation theorem in the theory of KM_2O-Langevin equations" Hokkaido Math.J.(1993)

    • Related Report
      1992 Annual Research Report
  • [Publications] T.Nakazi: "Homogeneous polynomials and invariant subspaces in the polydiscs" Arch der Math.58. 56-63 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Y.Giga: "Global existence of weak solutions for interface equations coupled with diffusion equations" SIAM J.Math.Anal.23. 821-835 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Y.Konishi: "Some remarks on actions of compact matrix quontum groups on C^*-algebras" Pacific J.Math.153. 119-127 (1992)

    • Related Report
      1992 Annual Research Report

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Published: 1992-04-01   Modified: 2016-04-21  

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