Deformation/rigidity theory and Tomita-Takesaki theory

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14324
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Kyoto University
• Principal Investigator: Isono Yusuke 京都大学, 数理解析研究所, 准教授 (80783076)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: フォンノイマン環 / 冨田・竹崎理論 / 離散群の非特異作用 / エルゴード理論

Outline of Final Research Achievements

I am interested in von Neumann algebras arising from group actions on measure spaces. In the case that the action does not preserve any measure, the algebra becomes of type III, which is the most difficult class in von Neumann algebras. I obtained the following results.
First, I succeeded to generalize Popa's techniques to type III algebras. This is a technical result but is the most important part of my research. As an application, I obtained a W*-superrigidity phenomena.
Second, I proved solidity of nonsingular Bernoulli actions. This example is new in the sense that the action does not preserve any sigma-finite measure.

Free Research Field

作用素環論

Academic Significance and Societal Importance of the Research Achievements

本研究は（III型でない）フォンノイマン環論において知られている理論を，冨田・竹崎理論を用いて，III型のフォンノイマン環に拡張する，というものである．III型フォンノイマン環は非常に扱いが難しく敬遠されがちであるが，一方で数理物理学などにも表れる自然な研究対象である．また幾何学的・確率論的に興味深い具体例も数多く存在している．本研究はそのような自然な対象の理解を深めるものであり，基礎的な研究として意義があると考えている．