Properties of Certain Subalgebras of Dales-Davie Algebras

Abstract

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].

φ-contractibility of some classes of Banach function algebras

“…Let X be a uniformly regular perfect compact plane set. Similar to the case of Lip n R (X, α), the natural subalgebra D n R (X) of D n (X) is defined (see, [1] and [6]). The algebras D n R (X) and D n (X) satisfy conditions of Theorem 2.6 and Corollary 2.7.…”

Algebrability of some subsets of the disk algebra

“…. These algebras were introduced and studied by Dales and Davie in [10], and they have been investigated by Abtahi and Honary in [1,2,13].…”

“…We refer the interested reader to [5,9] for a wider range of results in the topic of lineability and algebrability, and to [1,2,10,13] for further informations on the Dales-Davie algebras.…”