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Systematic study of quantum groups from the viewpoint of operator algebras

Research Project

Project/Area Number 12640199
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 University, Department of Mathematics ; Assistant Professor, 大学院・理学研究科, 助教授 (30241293)

Co-Investigator(Kenkyū-buntansha) UEDA Yoshimichi  Kyushu University, Graduate School of Mathematics ; Assistant Professor, 数理学研究院, 助教授 (00314724)
ARAI Asao  Hokkaido University, Department of Mathematics ; Professor, 大学院・理学研究科, 教授 (80134807)
KISHIMOTO Akitaka  Hokkaido University, Department of Mathematics ; Professor, 大学院・理学研究科, 教授 (00128597)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
KeywordsLocally compact quantum group / von Neumann algebra / Cartan subalgebra / Galois group / Nontype I C^*-algebra / Approximately inner automorphism / Amalgamated free product / Jones index / 融合積 / 作用素環 / 量子群 / 極小作用 / UHF flow / 純粋状態空間 / 基底状態 / 自由積 / カッツ環 / 擬ヴォロノヴィッツ環 / Rohlin性質 / 量子場
Research Abstract

Yamanouchi has made an intensive study of actions of locally compact quantum groups on von Neumann algebras. He introduced a notion of a Galois group for such an action. He has succeeded in topologically identifying the Galois group of an intergrable, minimal action of a general locally compact quantum group as the intrinsic group of the dual quantum group.
Kishimoto has made a very unique research on automorphisms or one-parameter automorphism groups on nontype I separable C^*-algebras. He showed that, for any irreducible representation of such a C^*-algebra without nonzero intersection with compact operators, there exists an approximately inner one-parameter automorphism group so that the representation is covariant with respect to this one-parameter automorphism group.
Arai has made a progress in his research in mathematical physics. He considered a model of quantum particles coupled to a massless quantum scalar field, and showed that the model has a ground state for all values of the coupling constant.
Ueda has made an anlysis on amalgamated free products of von Neumann algebras over Cartan subalgebras. By utilizing an irreducible inclusion of type III factors coming from his free-product type action of the quantum SU_q(2), he has been ble to solve the long-standing problem that the free group factor of Radulescu possesses irreducible subfactors of arbitrary index greater than 4.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (44 results)

All Other

All Publications (44 results)

  • [Publications] Takehiko YAMANOUCHI: "Galois groups of quantum group actions and regularity of fixed-point algebras"Transactions of the American Mathematical Society. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takehiko YAMANOUCHI: "Construction of a canonical subfactor for an inclusion of factors with a common Cartan subalgebra"Hokkaido Mathematical Journal. 32. 41-58 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshimichi UEDA: "Fullness, Connes' χ-groups, and ultra-products of amalgamated free products over Cartan subalgebras"Transactions of the American Mathematical Society. 355. 349-371 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Akitaka KISHIMOTO: "Approximately inner flows on separable C^*-algebras"Reviews in Mathematical Physics. 14. 649-673 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Akitaka KISHIMOTO: "Quasi-product flows on a C^*-algebra"Communications in Mathematical Physics. 229. 397-413 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takehiko YAMANOUCHI: "Description of the automorphism group Aut(A/A^∧{alpha}) for a minimal action of a compact Kac algebra and its application"Journal of Functional Analysis. 194. 1-16 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 新井 朝雄: "多体系と量子場"岩波書店. 90 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Yamanouchi: "Galois groups of quantum group actions and regularity of fixed-point algebras"Trans. Amer. Math. Soc.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Yamanouchi (with H. Aoi): "Construction of a canonical subfactor for an inclusion of factors with a common Cartan subalgebra"Hokkaido Math. J.. 32. 41-58 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Ueda: "Fullness, Connes' x-groups, and ultra-products of amalgamated free products over Cartan subalgebras"Trans. Amer. Math Soc.. 355. 349-371 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A. Kishimoto: "Approximately inner flows on separble C^*-algebras"Reviews in Math. Phys.. 14. 649-673 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A. Kishimoto: "Quasi-product flows on a -algebra"Commun. Math. Phys.. 229. 397-413 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Yamanouchi: "Description of the automorphism group Aut (A/A^α) for a minimal action of a compact Kac algebra and its application"J. Funct Anal.. 194. 1-16 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takehiko YAMANOUCHI: "Galois groups of quantum group actions and regularity of fixed-point algebras"Transactions of the American Mathematical Society. (掲載予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Takehiko YAMANOUCHI: "Construction of a canonical subfactor for an inclusion of factors with a common Cartan subalgebra"Hokkaido Mathematical Journal. 32. 41-58 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Yoshimichi UEDA: "Fullness, Connes' χ-groups, and ultra-products of amalgamated free products over Cartan subalgebras"Transactions of the American Mathematical Society. 355. 349-371 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Akitaka KISHIMOTO: "Approximately inner flows on separable C^*-algebras"Reviews in Mathematical Physics. 14. 649-673 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Akitaka KISHIMOTO: "Quasi-product : flows on a C^*-algebra"Communications in Mathematical Physics. 229. 397-413 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Takehiko YAMANOUCHI: "Description of the automorphism group Aut(A/A^{alpha}) for a minimal action of a compact Kac algebra and its application"Journal of Functional Analysis. 194. 1-16 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 新井朝雄: "多体系と量子場"岩波書店. 90 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Takehiko Yamanouchi: "Uniqueness of Haar measures for a quasi Woronowicz algebra"Hokkaido Mathematical Journal. 30. 105-112 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Takehiko Yamanouchi: "Description of the automorphism group Aut(A/A^{alpha}) for a minimal action of a compact Kac algebra and its application"Journal of Functional Analysis. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Takehiko Yamanouchi: "Takesaki duality for weights on locally compact quantum group covariant systems"Journal of Operator Theory. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Takehiko Yamanouchi: "Construction of a canonical subfactor for an inclusion of factors with a common Cartan subalgebra"Hokkaido Mathematical Journal. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Akitaka Kishimoto: "Examples of one-parameter automorphism groups of UHF algebras"Communications in Mathematical Physics. 216. 395-408 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Akitaka Kishimoto: "UHF flows and the flip automorphism"Reviews in Mathematical Physics. 13. 1163-1181 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Akitaka Kishimoto: "Homogeneity of the pure state space for separable C^*-algebras"International Journal of Mathematics. 12. 813-845 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Asao Arai: "Exact solutions of multi-component nonlinear Schrodinger and Klein-Gordon equations in two-dimensional space-time"J. Phys. A : Math. Gen.. 34. 4281-4288 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Asao Arai: "Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation"Reviews in Mathematical Physics. 13. 1075-1094 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Asao Arai: "Stability of ground states in sectors and its application to the Wigner-Weisskopf model"Reviews in Mathematical Physics. 13. 513-527 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshimichi Ueda: "Remarks on free products with respect to non-tracial states"Mathematica Scandinavica. 88. 111-125 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshimichi Ueda: "Irreducible subfactors of L(F{infty})) of index lambda>4"J. Reine Angew. Math.. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshimichi Ueda: "Automorphisms of free product-type and their crossed-products"Journal of Operator Theory. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Takehiko Yamanouchi: "Double group construction of quantum groups in the von Neumann algebra framework"Journal of Mathematical Society of Japan. 52. 807-834 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Takehiko Yamanouchi: "Uniqueness of Haar measures for a quasi Woronowicz algebra"Hokkaido Mathematical Journal 掲載予定. (印刷中). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Akitaka Kishimoto: "The Rohlin property for shift automorphisms"Reviews on Mathematical Physics. 12. 965-980 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Akitaka Kishimoto: "Trace scaling automorphisms of certain stable AF algebras, II"Quarterly Journal of Mathematics. 51. 131-154 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Akitaka Kishimoto: "Homogeneity of the pure state space of the Cuntz algebra"Journal of Functional Analysis. 171. 331-345 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Asao Arai: "A particle-field Hamiltonian in relativistic quantum electrodynamics"Journal of Mathematical Physics. 41. 4271-4283 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Asao Arai: "Essential spectrum of a self-adjoint operator on an abstract Hilbert space of Fock type and applications to quantum field Hamiltonians"Journal of Mathematical Analysis and Applications. 246. 189-216 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Asao Arai: "Ground states of a general class of quantum field Hamiltonians"Reviews on Mathematical Physics. 12. 1085-1135 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Yoshimichi Ueda: "A relation between certain interpolated Cuntzalgebras and interpolated free group factors"Proceedings of American Mathematical Society. 128. 1397-1404 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Yoshimichi Ueda: "On the fixed-point algebra under a minimal free product-type action of the quantum group SUq(2)"International Mathematics Research Notices. 1. 35-56 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 新井朝雄: "フォック空間と量子場、上、下"日本評論社. 636 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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