Espace des parties réelles des éléments d'une algèbre de Banach de fonctions

In this paper we study the functions which operate on Re A (the space of real parts of elements of a Banach function algebra A). We prove as our main result that if A is uniform or if A is ultraseparating, then only linear functions operate boundedly. We finally obtain a dichotomy for symbolic calculus on C̃(K), where K is a compact subset of the unit circle T andC̃(K) denotes the space of those continuous functions ƒ on K which admit a continuous extension to T whose Fourier conjugate is continuous.