Study on the nuclear dimension of operator algebras

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03516
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Kyushu University (2020-2023); Kyoto University (2019)
• Principal Investigator: Sato Yasuhiko 九州大学, 数理学研究院, 准教授 (70581502)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 分類理論 / 作用素環論 / 核型次元 / Jiang-Su 環

Outline of Final Research Achievements

We have studied the nuclear dimension, which characterizes the classifiability of nuclear C*-algebras, their dynamical systems and conditional expectations of operator algebras with finite nuclear dimension. In particular, we construct KMS states on operator algebras with the bundle structure under the assumption of finite nuclear dimension with a unique tracial state, as a consequence of this construction we have obtained uncountably many counterexamples to the Powers-Sakai conjecture. We have also constructed an endomorphism without conditional expectation on operator algebras with finite nuclear dimension, solving an open problem by E. Kirchberg. In previous classification theories, it has been fundamental to assume a condition of the simplicity, but in order of the technical obstructions, we have presented a classification theory for nonsimple operator algebras, which we call rationally approximately finite dimensional (RAF) algebras.

Free Research Field

作用素環論

Academic Significance and Societal Importance of the Research Achievements

作用素環論における長年の分類理論の発展により, 近年とても抽象度の高い分類理論が定理として完成した. 本研究はそのパワフルな分類理論を未解決問題へ応用し, 得られた解答の一般化や解決を行ったものである. また技術的な側面として数理物理でよく研究されているKMS状態の基本構造を調べる事で, 今まで議論されてこなかった非単純な作用素環の分類が必要である事がわかった. この成果は, これまで脚光を浴びなかった非単純な分類理論の重要性を示し, 新たな領域を開拓する契機となり得る. さらに分類理論の応用という形で, 分類理論に多大な功績を残したE. Kirchberg氏の残した未解決問題を解決した.