Characterization of a Banach-finsler Manifold in Terms of the Algebras of Smooth Functions

In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete C Finsler manifold M is determined by the normed algebra C b (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 Finsler manifold M is determined by the algebra C b (M); (ii) the weak Finsler structure of a separable and complete C Finsler manifold M modeled on a Banach space with a Lipschitz and C smooth bump function is determined by the algebra C b (M); (iii) the weak Finsler structure of a C uniformly bumpable and complete C Finsler manifold M modeled on a Weakly Compactly Generated (WCG) Banach space with an (equivalent) C smooth norm is determined by the algebra C b (M); and (iv) the isometric structure of a WCG Banach space X with an C smooth bump function is determined by the algebra C b (X).