On Gel'fand-Kolmogoroff type results

Abstract

We prove that a vector bundle $ E \to M$ is characterized by the associative structure of the space of symbols of the Lie algebra generated by all differential operators on $E$ which are eigenvectors of the Lie derivative in the direction of the Euler vector field. We also obtain similar result with the $\mathbb{R}-$ algebra of smooth functions which are polynomial along the fibers of $E.$ This allows us to deduce a Gel'fand-Kolmogoroff type result for the $\mathbb{R}-$algebra ${\rm Pol}(T^*(M))$ of symbols of the differential operators of $M.$