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


Authors: J. A. Jaramillo, M. Jiménez-Sevilla and L. Sánchez-González
Journal: Proc. Amer. Math. Soc. 142 (2014), 1075-1087
MSC (2010): Primary 58B10, 58B20, 46T05, 46T20, 46E25, 46B20, 54C35
DOI: https://doi.org/10.1090/S0002-9939-2013-11834-4
Published electronically: December 17, 2013
MathSciNet review: 3148541
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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^1$ uniformly bumpable and complete $ C^{1}$ Finsler manifold $ M$ modeled on a Weakly Compactly Generated (WCG) Banach space is determined by the algebra $ C^1_b(M)$; and (iv) the isometric structure of a WCG Banach space $ X$ with a $ C^1$ smooth bump function is determined by the algebra $ C_b^1(X)$.


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Additional Information

J. A. Jaramillo
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: jaramil@mat.ucm.es

M. Jiménez-Sevilla
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: marjim@mat.ucm.es

L. Sánchez-González
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: lfsanche@mat.ucm.es

DOI: https://doi.org/10.1090/S0002-9939-2013-11834-4
Received by editor(s): September 1, 2011
Received by editor(s) in revised form: April 23, 2012
Published electronically: December 17, 2013
Additional Notes: The third author was supported by grant MEC AP2007-00868
This work was supported in part by DGES (Spain) Project MTM2009-07848
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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