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Researches on the classification of von Neumann algebras by selfdual cones and operator inequalities

Research Project

Project/Area Number 09640144
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionIwate University

Principal Investigator

MIURA Yasuhide  Iwate University, Faculty of Humanities and Social Sciences, Professor, 人文社会科学部, 教授 (20091647)

Co-Investigator(Kenkyū-buntansha) ONISHSI Yoshihiro  Iwate University, Faculty of Humanities and Social Sciences, Associate Professor, 人文社会科学部, 助教授 (60250643)
ODAI Yoshitaka  Iwate University, Faculty of Humanities and Social Sciences, Associate Professor, 人文社会科学部, 助教授 (10204215)
ISHIKAWA Yoichiro  Iwate University, Faculty of Humanities and Social Sciences, Associate Professor, 人文社会科学部, 助教授 (80004000)
ISHIKAWA Akihiko  Iwate University, Faculty of Humanities and Social Sciences, Professor, 人文社会科学部, 教授 (00084377)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordsmatrix ordered Hilbert space / selfdual cone / completely positive map / type of von Neumann algebra / operator inequality / ノイマン環 / 不等式
Research Abstract

We considered the question how the algebraic structure of a operator algebra determines or is determined by the structure of its underlying Hilbert space. In order to give the answer to this problem, we have introduced the notion of the complete order isomorphisms and complete orthogonal decomposition isomorphisms on the matrix ordered Hilbert space with a family of selfdual cones. Matrix ordered Hilbert spaces are the appropriate objects to which completely positive maps apply and enabled us to handle non-commutative order. In this research, we have proven that a completely positive projection on the Hilbert space induces a conditional expectation on a von Neumann algebra, and a complete order isomorphism induces an automorphism of the von Neumann algebra. We have then investigated the relationship between the type of von Neumann algebras and completely positive maps, and characterized the matrix ordered standard forms of finite von Neumann algebras.
We have also introduced the notion … More of the order of operators on a Hilbert space based on the positivity with respect to the selfdual cone. The different point of this order from the usual order of operators is the compatibility with the products. We have investigated the fundamental properties on the operator inequalities with respect to this order, and sharpened the arithmetic and geometric mean inequality in the category of the ordered Hilbert space associated with a finite dimensional commutative von Neumann algebra.
Furthermore, we have obtained some results in the geometry and the number theory. One of them is the investigation of the relationship between the sphere theorem and the Dehn lemma. The rank of the group of relative units of a finite Galois extension of the algebraic number field has then been calculated. It is necessary to determine the finite groups all whose abelian subgroups are cyclic. We have described by cyclotomic polynomials quite simply a mechanical system of one-dimensional hard core chain which consists of infinite many particles with two different alternating mass Finally, we have generalized and given very generally applicable proof of the formula of Grant, which is a generalization of product formulae for division values of periods of the exponential function or each elliptic function with complex multiplication in a cyclotomic field, to every hyperelliptic function of cyclotomic type. Less

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] 三浦康秀: "非可換L^2空間における完全正値写像について" 京都大学数理解析研究所講究録. 979. 57-73 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Miura: "On a completely positive projection on a non-commutative L^2-space" Far East J.Math.Sci.5. 521-530 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] S-E.Takahasi & Y.Miura: "A generalization of the Alzer-Faiziev inequality" Utilitas Math.51. 3-8 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Ishikawa: "An elementary proof of Dehn's Lemma(in Japanese)" Artes Liberales. 61. 177-212 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] S.Ishiwata, S.Matsutani & Y.Onishi: "Localized state of hard core chain and cyclotomic polynomial" Physics Letters. A231. 208-216 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Miura: "Complete order isomorphisms between non-commutative L^2-spaces" to appear in Math.Scand.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Ishikawa: "On the sphere theorem and Dehn lemma (in Japanese)" Artes Liberales. 62. 97-109 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Onishi: "Complex multiplication formulae for curves of genus three" Tokyo J.Math.21. 381-431 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yasuhide Miura: "On the completely positive maps on non-commutative L^2-spaces (in Japanese)" Kokyuroku RIMS Kyoto University. 979. 57-73 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yasuhide Miura: "On a completely positive projection on a non-commutative L^2-spaces" Far East J.Math.Sci.5. 521-530 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Sin-Ei Takahasi and Yasuhide Miura: "A generalization of the Alzer-Faiziev inequality" Utilitas Math.51. 3-8 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yoichiro Ishikawa: "An elementary proof of Dehn's Lemma (in Japanese)" Artes Liberales. 61. 177-212 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shingo Ishiwata, Shigeki Matsutani and Yoshihiro Onishi: "Localized state of hard core chain and cyclotomic polynomial" Physics Letters. A231. 208-216 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yasuhide Miura: "Complete order isomorphisms between non-commutative L^2-spaces" Math.Scand.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yoichiro Ishikawa: "On the sphere theorem and Dehn lemma (in Japanese)" Artes Liberales. 62. 97-109 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yoshihiro Onishi: "Complex multiplication formulae for curves of genus three" Tokyo J.Math.21. 381-431 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Miura: "Complete order isomorphisms between non-commutative L^2-spaces" to appear in Math.Scand.

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Ishikawa: "On the sphere theorem and Dehn lemma (in Japanese)" Artes Liberales. 62. 97-109 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Onishi: "Complex multiplication formulae for curves of genus three" Tokyo J.Math.21. 381-431 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Yasuhide Miura: "On a completely positive projection on a non-commutative L^2-space" Far East Journal of Mathematical Sciences. 5. 521-530 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] S-E.Takahasi & Y.Miura: "A generalization of the Alzer-Faziev' inequality." Utilitas Mathematica. 51. 3-8 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Yoichiro Ishikawa: "An elementary proof of Dehn's Lemma.(in Japanese)" Artes Liberales. 61. 177-212 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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