Displaying similar documents to “Some rationally convex sets”

Norm conditions for uniform algebra isomorphisms

Aaron Luttman, Scott Lambert (2008)

Open Mathematics

Similarity:

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...

Real linear isometries between function algebras. II

Osamu Hatori, Takeshi Miura (2013)

Open Mathematics

Similarity:

We describe the general form of isometries between uniformly closed function algebras on locally compact Hausdorff spaces in a continuation of the study by Miura. We can actually obtain the form on the Shilov boundary, rather than just on the Choquet boundary. We also give an example showing that the form cannot be extended to the whole maximal ideal space.

Real-linear isometries between function algebras

Takeshi Miura (2011)

Open Mathematics

Similarity:

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...

Norm conditions for real-algebra isomorphisms between uniform algebras

Rumi Shindo (2010)

Open Mathematics

Similarity:

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...