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Theorems of Korovkin type for adapted spaces

Heinz Bauer (1973)

Annales de l'institut Fourier

It is shown that the methods developed in an earlier paper of the author about a Dirichlet problem for the Silov boundary [Annales Inst. Fourier, 11 (1961)] lead in a new and natural way to the most important results about the convergence of positive linear operators on spaces of continuous functions defined on a compact space. Choquet’s notion of an adapted space of continuous functions in connection with results of Mokobodzki-Sibony opens the possibility of extending these results to the case...

Trivial Jensen measures without regularity

J. F. Feinstein (2001)

Studia Mathematica

In this note we construct Swiss cheeses X such that R(X) is non-regular but such that R(X) has no non-trivial Jensen measures. We also construct a non-regular uniform algebra with compact, metrizable character space such that every point of the character space is a peak point.

Un problème d'extension linéaire dans les algèbres uniformes

Nicolas Th. Varopoulos (1971)

Annales de l'institut Fourier

Soit A une algèbre uniforme et soit I un idéal fermé de A tel que A / I soit une algèbre isométriquement isomorphe à C ( X ) , il existe alors une sous-algèbre fermée B ⊂ A telle que A est isométriquement isomorphe à I ⊕ B .

Una nota sobre las álgebras de Banach regulares no-arquimedianas.

Jesús M. Domínguez Gómez (1981)

Revista Matemática Hispanoamericana

Es bien conocido que el conjunto M de los ideales maximales de un álgebra de Banach compleja X es un espacio compacto y Hausdorff para la topología de Gelfand, y que X es isométricamente isomorfa al álgebra C(M,C) de las funciones continuas sobre M si y sólo si X es una B*-álgebra, es decir un álgebra de Banach con involución verificando ||x*x|| = ||x||2 (Gelfand-Naimark). En el caso no-arquimediano, X admite tal representación si y sólo si el subespacio vectorial engendrado por {e ∈ X | e2 = e,...

Uniform algebras and analytic multi­functions

Zbigniew Slodkowski (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Dati due elementi f e g in un'algebra uniforme A , sia G = f ( M A / f ( ∂ A ) . Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione λ → log ⁡ max ⁡ g ⁢ ( f - 1 ⁢ ( λ ) ) è subarmonica su G e che l’applicazione λ → g ⁢ ( f - 1 ⁢ ( λ ) ) è analitica nel senso di Oka.

Uniqueness of complete norms for quotients of Banach function algebras

W. Bade, H. Dales (1993)

Studia Mathematica

We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra L 1 ( G ) of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients A ( Γ ) / J ( E ) ¯ arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed. We construct...

Weak multiplicative operators on function algebras without units

Thomas Tonev (2010)

Banach Center Publications

For a function algebra A let ∂A be the Shilov boundary, δA the Choquet boundary, p(A) the set of p-points, and |A| = |f|: f ∈ A. Let X and Y be locally compact Hausdorff spaces and A ⊂ C(X) and B ⊂ C(Y) be dense subalgebras of function algebras without units, such that X = ∂A, Y = ∂B and p(A) = δA, p(B) = δB. We show that if Φ: |A| → |B| is an increasing bijection which is sup-norm-multiplicative, i.e. ||Φ(|f|)Φ(|g|)|| = ||fg||, f,g ∈ A, then there is a homeomorphism ψ: p(B) → p(A) with respect...

Weak spectral synthesis in Fourier algebras of coset spaces

Eberhard Kaniuth (2010)

Studia Mathematica

Let G be a locally compact group, K a compact subgroup of G and A(G/K) the Fourier algebra of the coset space G/K. Applying results from [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008), 987-1002], we establish injection and localization theorems relating weak spectral sets and weak Ditkin sets for A(G/K) to such sets for A(H/H ∩ K), where H is a closed subgroup of G. We also prove some results towards the analogue of Malliavin's theorem for weak spectral...

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