Relations between Banach function algebras and their uniform closures

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]. Let X be a compact Hausdorff space and let C(X) denote the space of all continuous complex valued functions on X. A function algebra on X is a subalgebra of C(X) which contains the constants and separates the points of X. If there is an algebra norm on A so that Ilf gII < IlfH *HgII for all f, g E A, then A is called a normed function algebra. A complete normed function algebra on X is called a Banach function algebra on X. If the norm of a Banach function algebra is the uniform norm on X; i.e. Ilf IX = supxEX If(x)I, it is called a uniform algebra on X. If A is a function algebra on X, then A, the uniform closure of A, is a uniform algebra on X. If (A, 11 11) is a Banach function algebra on X, for every x E X the map ox: A -* C, defined by Ox(f) = f(x), is a nonzero continuous complex homomorphism on A and so Ox E MA, where MA is the maximal ideal space of A. We call ox the evaluation homomorphism at x. Clearly for every || fKx = sup If(x)I = sup IO'(f)I < sup 10(f)l = 1flM ? llfll xEX XEX OEMA where E is the Gelfand transform of f. The Banach function algebra A on X is called natural, if every 0 E MA is given by an evaluation homomorphism ox at some x E X; or, in other Received by the editors February 9, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46J10; Secondary 46J20.