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1
Content available remote

Real-linear isometries between certain subspaces of continuous functions

100%
Jamshidi A., Sady F.
Open Mathematics
|
2013
|
tom 11
|
nr 11
2034-2043
EN
In this paper we first consider a real-linear isometry T from a certain subspace A of C(X) (endowed with supremum norm) into C(Y) where X and Y are compact Hausdorff spaces and give a result concerning the description of T whenever A is a uniform algebra on X. The result is improved for the case where T(A) is, in addition, a complex subspace of C(Y). We also give a similar description for the case where A is a function space on X and the range of T is a real subspace of C(Y) satisfying a ceratin separating property. Next similar results are obtained for real-linear isometries between spaces of Lipschitz functions on compact metric spaces endowed with a certain complete norm.
2
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Maps between Banach function algebras satisfying certain norm conditions

100%
Hosseini M., Sady F.
Open Mathematics
|
2013
|
tom 11
|
nr 6
1020-1033
EN
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let \(\bar A\) and \(\bar B\) be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ\{0}, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)−α‖Y=‖ρ(p)τ(p′) − α‖x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M B and a homeomorphism ϕ: M B → M A between the maximal ideal spaces of B and A such that for all f ∈ A, [...] where \(\hat \cdot\) denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from \(\bar A\) onto \(\bar B\) inducing a homeomorphism between \(M_{\bar B}\) and \(M_{\bar A}\) . We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact.
3
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Norm conditions for uniform algebra isomorphisms

100%
Luttman A., Lambert S.
Open Mathematics
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2008
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tom 6
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nr 2
272-280
EN
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 $$ \left\| {T(f)T(g) + \lambda } \right\| = \left\| {fg + \lambda } \right\|\forall f,g \in A, $$ then T is an ℝ-linear isometry and there exist an idempotent e ∈ B, a function κ ∈ B with κ 2 = 1, and an isometric algebra isomorphism $$ \tilde T:{\rm A} \to Be \oplus \bar B(1 - e) $$ such that $$ T(f) = \kappa \left( {\tilde T(f)e + \gamma \overline {\tilde T(f)} (1 - e)} \right) $$ for all f ∈ A, where γ = λ / |λ|. Moreover, if T is unital, i.e. T(1) = 1, then T(i) = i implies that T is an isometric algebra isomorphism whereas T(i) = −i implies that T is a conjugate-isomorphism.
4
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Norm conditions for real-algebra isomorphisms between uniform algebras

100%
Shindo R.
Open Mathematics
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2010
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tom 8
|
nr 1
135-147
EN
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 $$ \tilde S $$: A → B such that $$ \tilde S $$(ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications of this result.
5
Content available remote

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

100%
Jiménez-Vargas A., Lee K., Luttman A., Villegas-Vallecillos M.
Open Mathematics
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2013
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tom 11
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nr 7
1197-1211
EN
Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all $\mathbb{K}$-valued Lipschitz functions on X - where $\mathbb{K}$ is either‒or ℝ - that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = {f(x): |f(x)| = ‖f‖∞} of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y → $\mathbb{K}$ with φ1(y)φ2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T j(f)(y) = φ j(y)S j(f)(ψ(y)) for all f ∈ Lip0(X), y ∈ Y, and j = 1, 2. In particular, if S 1 and S 2 are identity functions, then T 1 and T 2 are weighted composition operators.
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