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

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In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete C k C^{k} Finsler manifold M M is determined by the normed algebra C b k ( M ) C_b^k(M) of all real-valued, bounded and C k C^k smooth functions with bounded derivative defined on M M . As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete C k C^{k} Finsler manifold M M is determined by the algebra C b k ( M ) C_b^k(M) ; (ii) the weak Finsler structure of a separable and complete C k C^{k} Finsler manifold M M modeled on a Banach space with a Lipschitz and C k C^k smooth bump function is determined by the algebra C b k ( M ) C^k_b(M) ; (iii) the weak Finsler structure of a C 1 C^1 uniformly bumpable and complete C 1 C^{1} Finsler manifold M M modeled on a Weakly Compactly Generated (WCG) Banach space is determined by the algebra C b 1 ( M ) C^1_b(M) ; and (iv) the isometric structure of a WCG Banach space X X with a C 1 C^1 smooth bump function is determined by the algebra C b 1 ( X ) C_b^1(X) .