Study on the preserver problem and the problem on gyro structures on Banach algebras

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03536
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Niigata University
• Principal Investigator: HATORI Osamu 新潟大学, 自然科学系, フェロー (70156363)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 保存問題 / 等距離写像 / GGV / C*環の正凸錐 / Tingley問題

Outline of Final Research Achievements

The Tingley problem is one of the important challenges in the study of the preserver problem, and through this research, several new results have been obtained. We introduced the concept of the complex Mazur-Ulam property and proved that function algebras including the disk algebra have this property, which was subsequently announced. Furthermore, we demonstrated that Banach spaces satisfying certain separation conditions also possess this property. It is known that the convex cone of a unital C*-algebra is a generalized gyrovector space (GGV), a concept introduced by the representative of this study and the other. New insights into mappings on convex cones were also obtained and published. We completed the necessary modified proofs concerning the Mazur-Ulam theorem on GGVs.

Free Research Field

関数解析学

Academic Significance and Societal Importance of the Research Achievements

保存問題において等距離写像については多くの数学者により研究が進められている。その中でTingley問題に関して新たな知見を得られたことは大きな成果である。その中である種の分離条件に着目する方法はこの分野に新しい方法を提示しその意味で学術的な意義がある。GGVの重要な例である単位的C*環の正凸錐上の乗法的にスペクトルあるいはノルム等を保存する写像についてのMolnarの問いに肯定的に答えることができたことは、この方面に新たな研究の方向性を見出した点においても学術的意義が認められる。また、GGV上のgyrometric保存写像に関する定理の証明を改善することができた点も評価できる。