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Mathematics > Functional Analysis

arXiv:1108.5403 (math)
[Submitted on 26 Aug 2011]

Title:Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions

Authors:J.A. Jaramillo, M. Jimenez-Sevilla, L. Sanchez-Gonzalez
View a PDF of the paper titled Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions, by J.A. Jaramillo and 1 other authors
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Abstract:In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete $C^k$ Finsler manifold $M$ is determined by the normed algebra $C_b^k(M)$ of all real-valued, bounded and $C^k$ smooth functions with bounded derivative defined on $M$. As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete $C^k$ Finsler manifold $M$ is determined by the algebra $C_b^k(M)$; (ii) the weak Finsler structure of a separable and complete $C^k$ Finsler manifold $M$ modeled on a Banach space with a Lipschitz and $C^k$ smooth bump function is determined by the algebra $C^k_b(M)$; (iii) the weak Finsler structure of a $C^k$ uniformly bumpable and complete $C^k$ Finsler manifold $M$ modeled on a Weakly Compactly Generated (WCG) Banach space with an (equivalent) $C^k$ smooth norm is determined by the algebra $C^k_b(M)$; and (iii) the isometric structure of a WCG Banach space $X$ with an $C^1$ smooth bump function is determined by the algebra $C_b^1(X)$.
Comments: 13 pages
Subjects: Functional Analysis (math.FA); Differential Geometry (math.DG); General Topology (math.GN)
MSC classes: 58B10, 58B20, 46T05, 46T20, 46E25, 46B20, 54C35
Cite as: arXiv:1108.5403 [math.FA]
  (or arXiv:1108.5403v1 [math.FA] for this version)
  https://doi.org/10.48550/arXiv.1108.5403
arXiv-issued DOI via DataCite

Submission history

From: Luis Sánchez-González [view email]
[v1] Fri, 26 Aug 2011 22:20:22 UTC (15 KB)
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