2021 Fiscal Year Final Research Report

Geometric constants of Banach Spaces and their applications

Research Project

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Project/Area Number	17K05287
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Research Field	Basic analysis
Research Institution	Hokkaido University of Education
Principal Investigator	Komuro Naoto 北海道教育大学, 教育学部, 教授 (30195862)
Co-Investigator(Kenkyū-buntansha)	三谷健一岡山県立大学, 情報工学部, 准教授 (00468969) 斎藤吉助新潟大学, 自然科学系, フェロー (30018949)
Project Period (FY)	2017-04-01 – 2022-03-31
Keywords	James constant / NJ constant / Birkhoff orthogonality / Radon space / Symmetric point / Rotation invariant norm / Lassak's conjecture / Isosceles orthogonality
Outline of Final Research Achievements	Geometric constants of Banach spaces have been studied by many authors. Among them we mainly investigated the James constant J(X) and von Neumann-Jordan constant C_{NJ}(X) of a Banach space X. It was known that, for a Banach space of three or more dimensions, the James constant J(X) is root 2 if and only if the norm is induced by an inner product. We obtained a new characterization of 2-dimensional Banach spaces with J(X) = root 2 and some examples. It is shown that this result gives a counter example to Lassak's conjecture. We also investigated the Birkhoff orthogonality which is a notion of generalized orthogonality in normed spaces. There was a result by Turnsek which characterized the symmetric points for Birkhoff orthogonality in B(H). We obtained some generalizations of the result to the cases in von Neumann algebras and C* algebras. We also gave a characterization of the 2-dimensional Radon space which has the symmetric property with respect to the Birkhoff orthogonality.
Free Research Field	関数解析学
Academic Significance and Societal Importance of the Research Achievements	バナハ空間の不動点性や正規構造など、性質の異なるバナハ空間を分類する方法として、幾何学的定数が様々考案され、研究されてきた。その代表的なものである James定数や NJ定数について、未知の課題が多数残されており、その多くで前進することができたことがこの研究の成果の一つである。バナハ空間における直行性概念は、幾何学的定数とも密接に関係するが、その代表ともいえるBirkhoff直交は、対称性を持たないという特徴がある。しかし局所的に対称点が存在することがあり、B(H)での対称点を特徴づけた Turnsek の研究を、フォン・ノイマン環などに拡張する結果が得られた。