On démontre un résultat de dichotomie pour les fonctions qui opèrent sur les restrictions d’algèbres de fonctions holomorphes de plusieurs variables. On obtient ce résultat après étude de la séparation par des fonctions holomorphes de compacts sur certaines hypersurfaces de .
Given a subring of the ring of formal power series defined by the growth of the coefficients, we prove a necessary and sufficient condition for it to be a noetherian ring. As a particular case, we show that the ring of Gevrey power series is a noetherian ring. Then, we get a spectral synthesis theorem for some classes of ultradifferentiable functions.
Among all -algebras we characterize those which are algebras of -functions on second countable Hausdorff -manifolds.