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Norm conditions for real-algebra isomorphisms between uniform algebras

Rumi Shindo (2010)

Open Mathematics

Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism S ˜ : A → B such that S ˜ (ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications...

Norm conditions for uniform algebra isomorphisms

Aaron Luttman, Scott Lambert (2008)

Open Mathematics

In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / 0 and T: A → B is a surjective map, not assumed to be linear, satisfying T ( f ) T ( g ) + λ = f g + λ ∀ f , g ∈ A , then T is an ℝ-linear isometry and there exist an...

Norm continuity of weakly quasi-continuous mappings

Alireza Kamel Mirmostafaee (2011)

Colloquium Mathematicae

Let be the class of Banach spaces X for which every weakly quasi-continuous mapping f: A → X defined on an α-favorable space A is norm continuous at the points of a dense G δ subset of A. We will show that this class is stable under c₀-sums and ℓ p -sums of Banach spaces for 1 ≤ p < ∞.

Norm fragmented weak* compact sets.

J. E. Jayne, I. Namioka, C. A. Rogers (1990)

Collectanea Mathematica

A Banach space which is a Cech-analytic space in its weak topology has fourteen measure-theoretic, geometric and topological properties. In a dual Banach space with its weak-star topology essentially the same properties are all equivalent one to another.

Normability of gamma spaces

Filip Soudský (2016)

Commentationes Mathematicae Universitatis Carolinae

We give a full characterization of normability of Lorentz spaces Γ w p . This result is in fact known since it can be derived from Kamińska A., Maligranda L., On Lorentz spaces, Israel J. Funct. Anal. 140 (2004), 285–318. In this paper we present an alternative and more direct proof.

Normability of Lorentz spaces—an alternative approach

Amiran Gogatishvili, Filip Soudský (2014)

Czechoslovak Mathematical Journal

We study normability properties of classical Lorentz spaces. Given a certain general lattice-like structure, we first prove a general sufficient condition for its associate space to be a Banach function space. We use this result to develop an alternative approach to Sawyer’s characterization of normability of a classical Lorentz space of type Λ . Furthermore, we also use this method in the weak case and characterize normability of Λ v ∞ . Finally, we characterize the linearity of the space Λ v ∞ by a simple...

Normal bases for the space of continuous functions defined on a subset of Zp.

Ann Verdoodt (1994)

Publicacions Matemàtiques

Let K be a non-archimedean valued field which contains Qp and suppose that K is complete for the valuation |·|, which extends the p-adic valuation. Vq is the closure of the set {aqn|n = 0,1,2,...} where a and q are two units of Zp, q not a root of unity. C(Vq → K) is the Banach space of continuous functions from Vq to K, equipped with the supremum norm. Our aim is to find normal bases (rn(x)) for C(Vq → K), where rn(x) does not have to be a polynomial.

Normal cones and C * - m -convex structure

El Kinani, A., Mohamed Amine Nejjari, Mohamed Oudadess (2002)

Commentationes Mathematicae Universitatis Carolinae

The notion of normal cones is used to characterize C * - m -convex algebras among unital, symmetric and complete m -convex algebras.

Normal Hilbert modules over the ball algebra A(B)

Kunyu Guo (1999)

Studia Mathematica

The normal cohomology functor E x t ℵ is introduced from the category of all normal Hilbert modules over the ball algebra to the category of A(B)-modules. From the calculation of E x t ℵ -groups, we show that every normal C(∂B)-extension of a normal Hilbert module (viewed as a Hilbert module over A(B) is normal projective and normal injective. It follows that there is a natural isomorphism between Hom of normal Shilov modules and that of their quotient modules, which is a new lifting theorem of normal Shilov...

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