Operator algebraic approach to infinite-dimensional objects and descriptive set theory

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K17608
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Chiba University
• Principal Investigator: Ando Hiroshi 千葉大学, 大学院理学研究院, 特任助教 (40767266)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 作用素環論 / Polish群

Outline of Final Research Achievements

The following are the main results of our project.(1) In the space of (possibly unbounded) self-adjoint operators, we determined exactly on which closed subsets of the real line the analogue of the Weyl-von Neuman theorme holds.
(2) We constructed a strongly unitarily representable Polish SIN group which is not of finite type (3) We constructed a one-parameter automorphism group of R_+ on the relative bicentralizer algebra defined for an inclusion of type III factors (the smaller one being of type III_1). We used showed that the properties of this flow are closely related to the structure of the underlying type III subfactors.(4) We studied global properties of the unitary groups of C* ana von Neumann algebras. In particular, we showed that there are many unital C* algebras A for which the identity component U_0(A) of its unitary group has property (FH) (any continuous affine isometric action on a real Hilbert space admits a fixed point) but fails to have property (T) of Kazhdan.

Free Research Field

作用素環論

Academic Significance and Societal Importance of the Research Achievements

Bicentralizer flowを巡る今回の研究は今後のIII_1型因子環のbicentralizer問題の研究に対する基礎になると考えられる。(relative) bicentralizer flowとConnes--竹崎のflow of weightsが共役になるのではないかと予想している。この予想はある程度自然であり、今後予想の検証過程で意外な発見もあるかもしれない。自己共役作用素の摂動に関する研究はユニタリ群のPolish群としての構造やその性質に関する興味深い知見をもたらした。今後非局所コンパクトPolish群の大域構造を研究する際に今回の研究結果が応用できるかもしれない。