Quasianalytic Banach function algebras

Dales, H. G.; Davie, A. M.

doi:10.1016/0022-1236(73)90065-7

Cited by 60 publications

(116 citation statements)

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“…Dales and Davie in [4] proved that if X is a finite union of uniformly regular sets then D(X, M) is complete. In fact, if X is a finite union of regular sets then D(X, M) is complete.…”

Section: Let X Be a Compact Hausdorff Space And Let A Be A Banach Funmentioning

confidence: 99%

Properties of Certain Subalgebras of Dales-Davie Algebras

Abtahi¹,

Honary²

2007

Glasgow Math. J.

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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].

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“…Dales and Davie in [4] proved that if X is a finite union of uniformly regular sets then D(X, M) is complete. In fact, if X is a finite union of regular sets then D(X, M) is complete.…”

Section: Let X Be a Compact Hausdorff Space And Let A Be A Banach Funmentioning

confidence: 99%

Properties of Certain Subalgebras of Dales-Davie Algebras

Abtahi¹,

Honary²

2007

Glasgow Math. J.

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“…Let D 1 (X) be the algebra of all continuously differentiable complex-valued functions on X, normed by f = f X + f X . The algebra D 1 (X) was considered by Dales and Davie [3]. In general, the normed algebra D 1 (X) is not necessarily complete.…”

Section: Then T Is Compact If and Only If ϕ(X) Is A Norm Compact Subsmentioning

confidence: 99%

Compact homomorphisms of URM algebras

Behrouzi

Mahyar

2004

Proc. Amer. Math. Soc.

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“…Let e\ = fj and P\ = Kj(ε\) for such a j e {1,..., 4}. Then we have e λ e &°°(C) and 0 < e x < l,suppei c P lf \\y\\ < 4||Φ(ei)>;|| 5 and \\D a e { Hoc < Mε~l a{ for all αENj with |α| < k,…”

Section: Yef](t-λ)mentioning

confidence: 99%

“…Actually, the existence of non-trivial divisible subspaces rules out even the weak kind of algebraic representation of the spectral maximal spaces considered in Proposition 1.5 of [13]. This space has been considered in a number of articles; in particular, it follows from the results in [5] and [16] In view of the preceding counterexample, it seems natural to restrict the attention to functional calculi on algebras of functions which contain at least all the ^°°-functions. Hence, let us consider now a natural Banach function algebra A on some compact subset K of C and let us assume that, for some integer k > 0, all the restrictions to K of thê k -functions on C are contained in A.…”

Section: =0mentioning

confidence: 99%