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Theory of Operator Algebras and Applications to Noncommutative Topological Spaces

Research Project

Project/Area Number 06640281
Research Category

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

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

Principal Investigator

KUSUDA Masaharu  Kansai University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (80195437)

Co-Investigator(Kenkyū-buntansha) MAEDA Toru  Kansai Univ., Fac.of Eng., Associate Professor, 工学部, 助教授 (20199623)
HIRASHIMA Yasumasa  Kansai Univ., Fac.of Eng., Associate Professor, 工学部, 助教授 (80047399)
YAMAMOTO Moboru  Kansai Univ., Fac.of Eng., Professor, 工学部, 教授 (80029628)
KURISU Tadashi  Kansai Univ., Fac.of Eng., Professor, 工学部, 教授 (00029159)
ISII Keiichi  Kansai Univ., Fac.of Eng., Professor, 工学部, 教授 (80029420)
Project Period (FY) 1994 – 1995
Project Status Completed (Fiscal Year 1995)
Budget Amount *help
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1995: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1994: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsC^<**>-algebra / Von Neumann algebra / State / Factor state / Spectrum / Dual C^<**>-algebra
Research Abstract

In general, a positive linear functional of norm 1, on a C^<**>-subalgebra, called a state can extend to a state on the whole C^<**>-algebra. The most important theme on extensions of states is extensions of factor ststes. In this research, we have obtained the theorem that every factor state on a separable abelian C^<**>-subalgebra B of a von Neumann algebra M extends uniquely to a pure state of M if and only if B is generated by minimal projections in M.Thus we can clarify the structure of separable abelian C^<**>-subalgebras, of a von Neumann algebra, on which every factor state uniquely extends to a pure state on the von Neumann algebra.
Now let A be a C^<**>-algebra and let A^^<^> be the spectrum of A,i.e., equvalence classes of nonzero irreducible representations of A.Then A^^<^> is a topological space equipped with the Jacobson topology. The reason of importance of A^^<^> in the theory of C^<**>-algebras is why A^^<^>is a rather large space and contains a lot of imformation on the structure of A and why we can see the structure of A through the topology of A^^<^>. Therefore the topology on A^^<^> has been investigated by many researchers. In this research, we have researched conditions for A^^<^> to be discrete, and we have obtained the following result :
Theorem. Let A be a C^<**>-algebra. Then the following conditions are equivalent :
(1) A^^<^> is discrete in the Jacobson topology.
(2) There exists an ideal I of A such that I^^<^> and <(A/I)>^^^ are discrete in the relative topology of A^^<^> and the open projection p in A^<****> satisfying I=A^<****>p A is a multiplier for A.
(3) A^^<^> is a T_1-space in the Jacobson topology and every open central projection in A^<****> is a multiplier for A.

Report

(3 results)
  • 1995 Annual Research Report   Final Research Report Summary
  • 1994 Annual Research Report
  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] 楠田,雅治: "Unique Prolonegements des Valeurs Absolues des Formes Lineaires Normales sur les Algebres de von Neumann" Mathematica Japonica. 40. 123-126 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 楠田,雅治: "Characterizations of Hereditory C^*-subalgebras" Pure and Applied Mathematika Sciences. 41. 85-94 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 楠田,雅治: "Norm Additivity Conditions for Normal Linear Functionals on von Neumann algebras" Publications of Research Institute for Mathematical Sciences. 31. 721-723 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] C.-H.Chu: "On Factor States of C^*-algebras and their Extensions" The Proceedings of the American Mathematical Society. 124. 207-215 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] KUSUDA,Masaharu: "Unique Prolongements des Valeurs Absolues des Formes Lineaires Normales sur les Algebres de von Neumann." Mathematica Japonica. 40. 123-126 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] KUSUDA,Masaharu: "Characterizations of Hereditary C^<**> -Subalgebras." Pure and Applied Mathematika Sciences. 41. 85-94 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] KUSUDA,Masaharu: "Norm Additivity Ocnditions for Normal Linear Functionals on von Neumann Algebras" Publications of Research Institute for Mathematical Sciences. 31. 721-723 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] CHU,C.-H.: "On Factor States of C^<**> -Algebras and Their Extensions." Proceedings of the American Mathematical Society. 124. 207-215 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 楠田,雅治: "Characterizations of hereditary C^*-subalgebras" Pure and Applied Mathematika Sciences. 41. 85-94 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 楠田,雅治: "Norm additivity conditions for normal hnear functionals on von Neumann algebras" Publications of the Reseach Institute for Mathematical Sciences. 31. 719-721 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] C. H. Chu.楠田,雅治: "On faclor states of C^*-algebras and their extensions" The Proceedings of the American Mathematical Society. 124. 207-215 (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] 栗栖,忠: "Noisy-vs-silent duel and silent-vs-Noisy duel under arbitrary moving" Mathematica Japonica. 44(掲載予定). (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] 楠田 雅治: "Unique prolongements des valeurs absolues des formes lineares normales sur les algebres de von Neumann" Mathematica Japonica. 40. 123-126 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 楠田 雅治: "Characterizations of hereditary C^*-subalgebras" Pure and Applied Mathematika Sciences. 41. 掲載予定 (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 栗栖 忠: "Noisy-vs-silent duel and silent-vs-Noisy duel under arbitrary moving" Mathematica Japonica. 43. 掲載予定 (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 石井 恵一: "線形代数講義" 日本評論社, 195 (1995)

    • Related Report
      1994 Annual Research Report

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Published: 1994-04-01   Modified: 2016-04-21  

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