Let A be a Banach function algebra on a compact Hausdorff space X . In this paper we consider some relations between the maximal ideal space, the Shilov boundary and the Choquet boundary of A and its uniform closure A . As an application we determine the maximal ideal space, the Shilov boundary and the Choquet boundary of algebras of infinitely differentiable functions which were introduced by Dales and Davie in 1973. For some notations, definitions, elementary and known results, one can refer to [2] and [3] .