Let X be a perfect, compact plane set, and let M=(Mn) be a sequence of positive numbers such that M0=1 and Mn/MkMn−k⩾() for all n∈ℕ and k=0, 1, 2, …, n. We consider a remarkable class of Banach function algebras of infinitely differentiable functions f on X such that ∑n=0∞||ffalse(nfalse)||X/Mn<∞ These algebras, called Dales–Davie algebras, are denoted by D(X, M), and they are complete under certain conditions on X. The main aim of this work is to find conditions on the sequence M=(Mn) to guarantee that D(X, M) is natural; that is, its maximal ideal space is identified with X. We present a general result on the naturality of D(X, M) using some formulas from combinatorial analysis. In particular, it is shown that if X is uniformly regular, and if the sequence (Pn)=(Mn/n!) satisfies any one of the following conditions, then D(X, M) is natural: (i) sup {PiPj/Pi+j−1:i, j∈ℕ}<∞; (ii) Pn2⩽Pn−1Pn+1 for all n∈ℕ; and (iii) Bn=max {PkPn−k/Pn: 1⩽k⩽n−1}→0 as n→∞.

Abstract.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].

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ABSTRACT. H. G. Dales has proved in [1] that if A is a Banach function algebra on a compact metrizable space X, then Sc¡(A,X) = T(A,X), where So{A,X) is the set of peak points of A (w.r.t. X) and F(A,X) is the Shilov boundary of A (w.r.t. X). Here, by considering the relation between peak sets and peak points of a Banach function algebra A and its uniform closure A, we present an elementary and constructive proof of the density of peak points in the Shilov boundary.

Abstract.We study an interesting class of Banach function algebras of infinitely differentiable functions on perfect, compact plane sets. These algebras were introduced by H. G. Dales and A. M. Davie in 1973, called Dales-Davie algebras and denoted by D(X, M), where X is a perfect, compact plane set and M = {M n } ∞ n=0 is a sequence of positive numbers such that M 0 = 1 and (be the subalgebra of all f ∈ D(X, M) that can be approximated by the restriction to X of polynomials [rational functions with poles off X]. We show that the maximal ideal space of D P is X d , the polynomial convex hull of X d , and the maximal ideal space of D R is X d . Using some formulae from combinatorial analysis, we find the maximal ideal space of certain subalgebras of Dales-Davie algebras.2000 Mathematics Subject Classification. Primary 46J10, 46J15. Secondary 46J20.1. Introduction. Let X be a compact Hausdorff space. We denote the space of all continuous complex-valued functions on X by C(X). For f ∈ C(X) and a closed subset E of X, we denote the uniform norm of f on E by |f | E . A function algebra on X is a subalgebra A of C(X) that separates the points of X and contains the constant functions. If there is an algebra norm on A such that A is complete under this norm, then A is a Banach function algebra on X, and if the given norm is the uniform norm on X, then A is a uniform algebra on X.We denote by M(A), the maximal ideal space of A. Clearly, for each x ∈ X, the map ε x : A → C, defined by ε x ( f ) = f (x), is a non-zero complex homomorphism on A called the evaluation character at x. The map J : X → M(A), defined by J(x) = ε x , is injective and continuous, and so X is homeomorphic to a compact subset of M(A). If the map J is surjective, then A is a natural Banach function algebra on X.If A is a Banach function algebra on X, thenĀ, the uniform closure of A in C(X), is a uniform algebra on X and, clearly, M(A) ⊆ M(A). The following result in this area is proved in [5].