Linear operators on function spaces and geometric topology

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K05241
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: University of Tsukuba
• Principal Investigator: Kawamura Kazuhiro 筑波大学, 数理物質系, 教授 (40204771)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 関数環・関数空間 / Banach-Stone型定理 / バナッハ環のHochschildコホモロジー / 一般化射影極限

Outline of Final Research Achievements

We characterized some isometries on the continuously differentiable function spaces over compact Riemannian manifolds as generalized weighted composition operators, and illustrated deformations of the isometry groups under norm-perturbations with some concrete manifolds. Some Banach-Stone type theorems were obtained in joint works with S.Oi, H.Koshimizu, O. Hatori and T.Miura. Also we showed that the topological Hochschild cohomology of Lipschitz algebras over compact geodesic spaces is infinite dimensional, which shows a contrast to the fact that global homological dimension of the smooth function algebra over a compact smooth manifold is equal to the dimension of the manifold.
We studied the mean dimension of the shift maps on generalized inverse limits and obtained an estimate in terms of the lengths of periodic blocks. The result was applied to refine the dichotomy on the topological entropy of the shift map discovered by Erceg-Kennedy.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

従来関数解析学の文脈において研究されてきた過重合成作用素およびバナッハ環のHochshilcdコホモロジーを、野性的空間の一般・幾何学的トポロジー的手法を活用して研究したことによって、新しい視点を導入することができた。