Classification of operator algebras and applications to dynamical systems

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17553
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: Sato Yasuhiko 京都大学, 理学研究科, 助教 (70581502)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: C*-環 / 分類可能性 / Jiang-Su 環 / 核型次元 / 力学系 / 従順群

Outline of Final Research Achievements

A mathematical object in which xy≠yx can happen is called non-commutative. In this research, we aim to classify operator algebras (C*-algebras) which were introduced in order to represent non-commutative mathematics in infinite dimensional spaces. As a consequence of this study, we generalized my previous answer to Toms-Winter conjecture of the case of a unique tracial state, to the case of compact finite dimensional spaces much larger situation. Furthermore, investigating discrete countable amenable group actions on classifiable C*-algebras, this research showed the classifiability of the crossed product C*-algebras, which are known as C*-algebras containing information on given dynamical systems.

Free Research Field

作用素環論

Academic Significance and Societal Importance of the Research Achievements

作用素環の分類理論における重要未解決問題の1つに　Toms-Winter予想とよばれる問題がある。本研究では、この予想の肯定解を「トレースの端点空間がコンパクト有限次元」という基礎的な場合に与えた。これまで日本の作用素環の研究が力学系に主眼を置いていたのに対し、この結果は分類理論そのものへ重要な応用を与えたといえる。また、この結果を更に逆輸入する形で、接合積とよばれる力学系の情報を含む作用素環の分類可能性を示した。現在 S. White, A. Tikuisis, W. Winter, G. Szabo, をはじめ多くの研究者がこの研究成果を元にした結果を与えている。