Noncommutative analysis based on operator algebras

2023 Fiscal Year Final Research Report

• Project/Area Number: 18H01122
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Nagoya University
• Principal Investigator: Ueda Yoshimichi 名古屋大学, 多元数理科学研究科, 教授 (00314724)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: ランダム行列 / 大偏差原理 / 量子群 / ユニタリ表現論 / 作用素 / 関数計算 / 非可換Lp空間 / 量子ダイバージェンス

Outline of Final Research Achievements

I worked out the random matrix model for liberation process that I introduced and studied before, and the consequence is that many problems concerning the relationship between orbital free entropy and free mutual information are reduced to the full large deviation principle for the random matrix model. I introduced a rather general framework including the spherical representation theory for inductive limits of compact quantum groups and gave its basic theory. I also gave a thorough theory for 2-variable functional calculus for Hilbert space operators. Additionally, I did various studies in non-commutative analysis.

Free Research Field

関数解析，作用素環論，非可換確率論

Academic Significance and Societal Importance of the Research Achievements

非可換解析学はさまざまな意味で受け取り得るが，ヒルベルト空間上の作用素が何らかの形で関わる解析学的問題を作用素環論を踏まえつつ，作用素環論の外部に広がった形で研究を遂行した．自由化過程のランダム行列モデルは新奇なものであり，さまざまな発展が期待できる．また，球表現に関わる研究はここまで徹底した量子群の帰納極限を扱うのに使える枠組みはこれまでに与えておらず，出発点になりうるだろう．さらに２変数の関数計算の研究は基礎中の基礎というべきものなので，これまで散発的な研究で無駄が多かった部分を交通整理しており，近年の量子情報理論での利用を見るとその有用性は明らかと思われる．