Linear operators on function spaces and geometric topology
| Project/Area Number |
17K05241
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | University of Tsukuba |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| 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 / 関数空間 / Hochschild型コホモロジー / 射影極限 / 野性的空間のホモトピー / Banach環 / Hochschildコホモロジー / 平均次元 / 可換Banach環 / 極大イデアル空間 / 有理ホモトピー / Ｃ１関数空間 / local isometry / 等長写像 / 関数空間の幾何 / 線形作用素 / 幾何学的トポロジー |
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| 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.
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| 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。
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Report
(5 results)
Research Products
(56 results)
