Symmetries of operator algebras and subfactors

• Project/Area Number: 15H03623
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: Izumi Masaki 京都大学, 理学研究科, 教授 (80232362)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥16,640,000 (Direct Cost: ¥12,800,000、Indirect Cost: ¥3,840,000); Fiscal Year 2019: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2018: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2017: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2016: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2015: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
• Keywords: 作用素環 / フュージョン圏 / 群作用 / 部分因子環 / 分類空間 / Cuntz環 / モジュラーテンソル圏 / 核型C*環 / 函数解析 / 作用素環論 / von Neumann 環 / C*環 / 関数解析学 / 解析学 / 数理物理学

Outline of Final Research Achievements

We deduced a system of equations satisfied by structure constants of near-group categories and generalized Haagerup categories by using Cuntz algebras. As applications, we determined the structure of the outer automorphism groups, and the fusion categories obtained by equivariantization and de-euivariantization. We obtained a complete invariant for the factors arising from KMS-states for gauge actions of the Pimsner algebras. We studied continuous fields of the Cuntz algebras, and determined the number of actions of poly-Z groups on the Cuntz algebras.

Academic Significance and Societal Importance of the Research Achievements

作用素環は無限次元かつ非可換という著しい特性をもつ数学的対象であり、このような特性を持つ数学的あるいは物理的な対象のモデルとしても資するものである。作用素環の構造の理解のためには対称性の理解が重要であり、本研究では作用素環の対称性の理解を進展させる複数の結果が得られた点で学術てきな意義を持つ。対称性として数学で古くから研究されている群作用に加え、近年量子対称性の名でも呼ばれているフュージョン圏の研究を進展させた点においても重要である。

Research Products

• [Int'l Joint Research] University of New South Wales(オーストラリア)
• [Int'l Joint Research] University of New South Wales(オーストラリア)
• [Int'l Joint Research] University of New South Wales(オーストラリア)
• [Int'l Joint Research] University of New Sounth Wales(オーストラリア)
• [Int'l Joint Research] University of New South Wales(オーストラリア)
• [Int'l Joint Research] Louisiana State University(米国)
• [Int'l Joint Research] Muenster University(ドイツ)
• [Journal Article] The group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra (2019)
  • Author(s): Izumi, Masaki and Sogabe, Taro
  • Journal Title: International Journal of Mathematics Volume: 30 Issue: 11 Pages: 19500571-19
  • DOI: 10.1142/s0129167x19500575
  • Peer Reviewed
• [Journal Article] The K-theory of the flip automorphisms (2019)
  • Author(s): Izumi, Masaki
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 80 Pages: 123-137
  • DOI: 10.2969/aspm/08010123
  • Peer Reviewed
• [Journal Article] The classification of 3^n subfactors and related fusion categories (2018)
  • Author(s): M. Izumi
  • Journal Title: Quantum topology Volume: 3 Issue: 3 Pages: 473-562
  • DOI: 10.4171/qt/113
  • Peer Reviewed
• [Journal Article] The Asaeda-Haagerup fusion categories (2018)
  • Author(s): P. Grossman, M. Izumi, N. Snyder
  • Journal Title: Journal fur die Reine und Angewandte Mathematik Volume: 743 Issue: 743 Pages: 261-305
  • DOI: 10.1515/crelle-2015-0078
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The flow of weights and the Cuntz-Pimsner algebras (2018)
  • Author(s): Masaki Izumi
  • Journal Title: Communications in Mathematical Physics Volume: 357 Issue: 1 Pages: 203-229
  • DOI: 10.1007/s00220-017-2948-0
  • Peer Reviewed
• [Journal Article] A Cuntz algebra approach to the classification of near-group categories (2016)
  • Author(s): Masaki Izumi
  • Journal Title: CMA Proceedings Volume: 46 Pages: 222-343
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] A topological approach to the classification of discrete amenable group actions on nuclear C*-algebras (2019)
  • Author(s): Izumi, Masaki
  • Organizer: Operator Algebras and Applications
  • Int'l Joint Research / Invited
• [Presentation] The flow of weights and the Cuntz-Pimsner algebras (2018)
  • Author(s): Masaki Izumi
  • Organizer: Approximation Properties in Operator Algebras and Ergodic Theory
  • Invited
• [Presentation] Five infinite families of potential modular data related to quadratic categories (2018)
  • Author(s): Masaki Izumi
  • Organizer: Fusion Categories and Subfactors
  • Invited
• [Presentation] Topics in K-theory of operator algebras: Dadarlat-Pennig's generalization of Dixmier-Douady theory and group actions on Kirchberg algebras I,II (2018)
  • Author(s): 泉 正己
  • Organizer: 空間の代数的・幾何的モデルとその周辺
  • Invited
• [Presentation] The classification of poly-Z group actions on Kirchberg algebras I,II (2018)
  • Author(s): 泉 正己
  • Organizer: 作用素論・作用素環論研究集会2018
  • Invited
• [Presentation] Group actions on C*-algebras and obstruction theory (2017)
  • Author(s): Masaki Izumi
  • Organizer: Seminar, Issac Newton Institute
• [Presentation] An operator algebra approach to the classification of certain fusion categories I,II,III (2016)
  • Author(s): Masaki Izumi
  • Organizer: Operator Algebras and Mathematical Physics
  • Place of Presentation: 東北大学
  • Year and Date: 2016-08-01
  • Invited
• [Presentation] An operator algebra approach to the classification of certain fusion categoties I,II,III (2016)
  • Author(s): Masaki Izumi
  • Organizer: The Fourteenth Annual Spring Institute on Noncommutative Geometry and Operator algebras
  • Place of Presentation: Hausdorff Center for Mathematics
  • Year and Date: 2016-05-21
  • Invited
• [Presentation] Indecomposable characters of infinite dimensional groups associated with operator algebras (2016)
  • Author(s): Masaki Izumi
  • Organizer: International conference on structure of C*-algebras and tracial approximation
  • Place of Presentation: 河北師範大学
  • Invited
• [Presentation] The classification of fusion categories via operator algebras (2015)
  • Author(s): Masaki Izumi
  • Organizer: International Meeting and the Annual Meeting of the Taiwanese Mathematical Society
  • Place of Presentation: 国立高雄大学 高雄 台湾
  • Year and Date: 2015-12-19
  • Int'l Joint Research / Invited
• [Presentation] Quadratic categories (2015)
  • Author(s): Masaki Izumi
  • Organizer: International Conference on Subfactor Theory in Mathematics and Physics
  • Place of Presentation: 宣和酒店 秦皇島 中国
  • Year and Date: 2015-07-13
  • Int'l Joint Research / Invited
• [Presentation] Noncommutative Poisson boudaries and Cuntz algebra states (2015)
  • Author(s): Masaki Izumi
  • Organizer: Special Week on Operator algebras
  • Place of Presentation: 華東師範大学 上海 中国
  • Year and Date: 2015-06-01
  • Int'l Joint Research / Invited
• [Funded Workshop] Operator Algebras and Mathematical Physics (2016)
  • Place of Presentation: 東北大学
  • Year and Date: 2016-08-01