Approximation property for operator algebras and its application

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K05278
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Osaka Kyoiku University
• Principal Investigator: Okayasu Rui 大阪教育大学, 教育学部, 准教授 (70362746)
• Co-Investigator(Kenkyū-buntansha): 縄田 紀夫 大阪大学, 大学院情報科学研究科, 准教授 (90614040)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: von Neumann環 / 因子環

Outline of Final Research Achievements

We studied injectivity and Haagerup approximation, which are approximation　propertoes in operator algebras. The Haagerup approximation property was not defined for type III von Neumann algebras for many years, but through joint work with Tomatsu, it was generalized. Moreover, in joint work with Ozawa and Tomatsu, some problems in the Haagerup approximation property of quantum groups were also resolved.
We also gave an alternative proof that an injective factor on a Hilbert space with trivial bicentralizer is ITPFI.

Free Research Field

作用素環

Academic Significance and Societal Importance of the Research Achievements

von Neumann 環のHaagerup近似性は有限型の場合は以前から導入されたにも関わらず、一般の場合には全くの手付かずであった。しかし近年の研究によりその重要性が改めて認識されるようになり、一般的な定義の導入が期待されていた。そこで戸松氏との共同研究により、当初の予想された困難さを回避して、簡潔な定義の導入に成功した。