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Classification of E0-semigroups and W*-semigroups

Research Project

Project/Area Number 19K23403
Research Category

Grant-in-Aid for Research Activity Start-up

Allocation TypeMulti-year Fund
Review Section 0201:Algebra, geometry, analysis, applied mathematics,and related fields
Research InstitutionNagoya University

Principal Investigator

Sawada Yusuke  名古屋大学, 多元数理科学研究科, 博士研究員 (20851439)

Project Period (FY) 2019-08-30 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
KeywordsE0-半群 / プロダクトシステム / ランダムウォーク / 超群 / CP0-半群 / 伸張 / W*-双加群
Outline of Research at the Start

Arvesonは,プロダクトシステムの概念を導入することによってB(H)上のE0-半群の指数理論を構築した.申請者は,その指数理論を一般のvon Neumann環上のE0-半群に拡張することを目指す.申請者のこれまでの研究によって,プロダクトシステムの拡張概念であるW*-双加群から成るプロダクトシステムが導入され,E0-半群はそのプロダクトシステムによって分類されることが分かった.本研究では,Arvesonの指数理論の拡張として,W*-双加群のI型プロダクトシステムと指数を導入し,そのI型プロダクトシステムに対応するE0-半群を指数によって完全に分類することを試みる.

Outline of Final Research Achievements

It is known that product systems of W*-bimodules (W*-product systems) are invariant for E0-semigroups on von Neumann algebras. I clarified a relationship between W*-product systems over von Neumann algebra M and ones over the commutant of M.
Also, we can derive a hypergroup from a random walk on a graph equipped with a good symmetry by Wildberger's method. I investigated distance distributions of such a random walk in the view point of the algebraic structure of the hypergroup associated with the random walk. I obtained an analogous result of the above for an open quantum random walk on a distance set.

Academic Significance and Societal Importance of the Research Achievements

W*-プロダクトシステムがArvesonのプロダクトシステムの拡張であることがより明瞭な形で明らかになり, Arvesonのプロダクトシステムの理論がどのように拡張され得るか考察しやすくなった.また,超群の積を計算することによって,ランダムウォークの距離分布が従来の方法よりも簡単に計算できるようになった.さらに,距離集合上の開放系量子ウォークと超群の理論の基礎を築くことが出来た.

Report

(3 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • Research Products

    (6 results)

All 2021 2020 2019

All Journal Article (2 results) (of which Peer Reviewed: 2 results) Presentation (4 results) (of which Invited: 2 results)

  • [Journal Article] A Connes correspondence approach to the dilation theory2020

    • Author(s)
      Yusuke Sawada
    • Journal Title

      International Journal of Mathematics

      Volume: 31 Issue: 05 Pages: 2050040-2050040

    • DOI

      10.1142/s0129167x20500408

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Hypergroups and distance distributions of random walks on graphs2020

    • Author(s)
      Kenta Endo, Ippei Mimura, Yusuke Sawada
    • Journal Title

      Mathematica Scandinavica

      Volume: 未定

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Presentation] Hypergroups and random walks on graphs2021

    • Author(s)
      澤田友佑
    • Organizer
      山上滋先生退官記念研究集会
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Distance distributions of random walks and open quantum random walks2020

    • Author(s)
      澤田友佑
    • Organizer
      Workshop on “Non-commutative Probability and Related Fields”
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] E0-semigroups, dilations and product systems of W*-bimodules2019

    • Author(s)
      澤田友佑
    • Organizer
      RIMS共同研究, 作用素環論の最近の発展
    • Related Report
      2019 Research-status Report
  • [Presentation] E0-semigroups and product systems of W*-bimodules2019

    • Author(s)
      澤田友佑
    • Organizer
      第54回関数解析研究会
    • Related Report
      2019 Research-status Report

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Published: 2019-09-03   Modified: 2022-01-27  

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