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Real-linear isometries between function algebras

Takeshi Miura (2011)

Open Mathematics

Let A and B be uniformly closed function algebras on locally compact Hausdorff spaces with Choquet boundaries Ch A and ChB, respectively. We prove that if T: A → B is a surjective real-linear isometry, then there exist a continuous function κ: ChB → z ∈ ℂ: |z| = 1, a (possibly empty) closed and open subset K of ChB and a homeomorphism φ: ChB → ChA such that T(f) = κ(f ∘φ) on K and T f = κ f o φ ¯ on ChB K for all f ∈ A. Such a representation holds for surjective real-linear isometries between (not necessarily...

Reflexivity of the isometry group of some classical spaces.

Félix Cabello Sánchez, Lajos Molnár (2002)

Revista Matemática Iberoamericana

We investigate the reflexivity of the isometry group and the automorphism group of some important metric linear spaces and a1gebras. The paper consists of the following sections: 1. Preliminaries. 2. Sequence spaces. 3. Spaces of measurable functions. Hardy spaces. 5. Banach algebras of holomorphic functions. 6. Fréchet algebras of holomorphic functions. 7. Spaces of continuous functions.

Relatively weak* closed ideals of A(G), sets of synthesis and sets of uniqueness

A. Ülger (2014)

Colloquium Mathematicae

Let G be a locally compact amenable group, and A(G) and B(G) the Fourier and Fourier-Stieltjes algebras of G. For a closed subset E of G, let J(E) and k(E) be the smallest and largest closed ideals of A(G) with hull E, respectively. We study sets E for which the ideals J(E) or/and k(E) are σ(A(G),C*(G))-closed in A(G). Moreover, we present, in terms of the uniform topology of C₀(G) and the weak* topology of B(G), a series of characterizations of sets obeying synthesis. Finally, closely related to...

Representations of the direct product of matrix algebras

Daniele Guido, Lars Tuset (2001)

Fundamenta Mathematicae

Suppose B is a unital algebra which is an algebraic product of full matrix algebras over an index set X. A bijection is set up between the equivalence classes of irreducible representations of B as operators on a Banach space and the σ-complete ultrafilters on X (Theorem 2.6). Therefore, if X has less than measurable cardinality (e.g. accessible), the equivalence classes of the irreducible representations of B are labeled by points of X, and all representations of B are described (Theorem 3.3).

Representing measures for the disc algebra and for the ball algebra

Raymond Brummelhuis, Jan Wiegerinck (1991)

Annales Polonici Mathematici

We consider the set of representing measures at 0 for the disc and the ball algebra. The structure of the extreme elements of these sets is investigated. We give particular attention to representing measures for the 2-ball algebra which arise by lifting representing measures for the disc algebra.

Riemann mapping theorem in ℂⁿ

Krzysztof Jarosz (2012)

Annales Polonici Mathematici

The classical Riemann Mapping Theorem states that a nontrivial simply connected domain Ω in ℂ is holomorphically homeomorphic to the open unit disc 𝔻. We also know that "similar" one-dimensional Riemann surfaces are "almost" holomorphically equivalent. We discuss the same problem concerning "similar" domains in ℂⁿ in an attempt to find a multidimensional quantitative version of the Riemann Mapping Theorem

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