Investigation of operator algebras associated to number fields

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14551
• 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 Institute of Technology
• Principal Investigator: Takeishi Takuya 京都工芸繊維大学, 基盤科学系, 准教授 (20784490)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: C*-環論 / 代数体 / K-理論 / KMS状態 / 亜群C*-環

Outline of Final Research Achievements

I’m operator algebraist, working on C*-algebras from number fields. Bost--Connes C*-algebras are ones of such C*-algebras, and from my previous research, we know that Bost--Connes C*-algebras completely remembers the underlying number fields. In this research, we proved that the C*-algebras of Arledge--Laca--Raeburn have the same phenomenon. In addition, it is known that Dedekind zeta functions appear as partition functions of Bost--Connes systems. Analogously, ax+b semigroup C*-algebras from number fields have the zeta function as a “summation” of partition functions. There are three other articles related to those two main articles.

Free Research Field

作用素環論

Academic Significance and Societal Importance of the Research Achievements

数学では対象が本質的に同じであるかそうでないかを区別する「同型」という概念がある．簡単な対象であれば同型かどうかは一目でわかることも多いが，私の専門とする作用素環論では無限次元の対象を扱うので，全く違うように見えるものが超越的な理由によって「同じ」になってしまうことが多々ある．そのため，「作用素環の同型・非同型」の決定は伝統的に重要な問題として扱われている．本研究はその一端として，代数体から作られるArledge--Laca--RaeburnのC*-環というクラスのC*-環がすべて「異なる」ということを示した．