2023 Fiscal Year Final Research Report
Theory of operator algebras and functional analytic group theory
| Project/Area Number |
20H01806
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Kyoto University |
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Principal Investigator |
Ozawa Narutaka 京都大学, 数理解析研究所, 教授 (60323466)
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Keywords | 解析的群論 / 離散群 / 作用素環論 |
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| Outline of Final Research Achievements |
Amanability and Kazhdan's peoprty are the two most important concepts in analytic group theory. In the joint work with Yuhei Suzuki, the PI has proved that the several notions of amenability for group actions on operator algebras that have been proposed are all equivalent and given applications of this result. The elementary matrix group EL_d(R) for a finitely generated ring R is the most prominent example of groups with Kazhdan's property. The PI generalizes this fact to a non-unital ring. It is well-known that every operator on (the l_2 space of) a uniformly locally finite metric space that is approximable by finite-propagation operators is quasi-local. Since introduced in 90s, it has been questioned whether the converse also holds true. The PI has answered this in negative by constructing counterexamples.
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| Free Research Field |
作用素環論
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| Academic Significance and Societal Importance of the Research Achievements |
群作用の従順性は群作用の研究において欠かすことのできない道具である。この理論を整備し、新たな例を与えた鈴木氏との共同研究成果には高い学術的価値がある。基本行列群は重要な研究対象であるが、無数の基本行列群を一斉に扱う際に非単位的な環を扱う必要が出てくる。環が非単位的になることにより、大きな技術的問題が生じるが、それを克服する初めての手法を見出したことの意義は大きい。近似的有限伝播性と擬局所性が同値であるか否かは粗距離空間上の作用素論における最重要未解決問題のひとつであった。確率論的手法による反例の構成は未解決問題を解決するのみならず、さらなる研究領域を切り開く重要な学術的進展である。
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