On Gel'fand-Kolmogoroff type results

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 \(M.\)