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

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 $C_b^1(X)$.