Ideals in big Lipschitz algebras of analytic functions

Thomas Vils Pedersen. "Ideals in big Lipschitz algebras of analytic functions." Studia Mathematica 161.1 (2004): 33-59. <http://eudml.org/doc/284948>.

@article{ThomasVilsPedersen2004,
abstract = {For 0 < γ ≤ 1, let $Λ⁺_\{γ\}$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_\{γ\}(E,Q)$ be the closed ideal in $Λ⁺_\{γ\}$ consisting of those functions $f ∈ Λ⁺_\{γ\}$ for which (i) f = 0 on E, (ii) $|f(z)-f(w)| = o(|z-w|^\{γ\})$ as d(z,E),d(w,E) → 0, (iii) $f/Q ∈ Λ⁺_\{γ\}$. Also, for a closed ideal I in $Λ⁺_\{γ\}$, let $E_\{I\}$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_\{I\}$ be the greatest common divisor of the inner parts of non-zero functions in I. Our main conjecture about the ideal structure in $Λ⁺_\{γ\}$ is that $J_\{γ\}(E_\{I\},Q_\{I\}) ⊆ I$ for every closed ideal I in $Λ⁺_\{γ\}$. We confirm the conjecture for closed ideals I in $Λ⁺_\{γ\}$ for which $E_\{I\}$ is countable and obtain partial results in the case where $Q_\{I\} = 1$. Moreover, we show that every wk* closed ideal in $Λ⁺_\{γ\}$ is of the form f ∈ $Λ⁺_\{γ\}$: f = 0 on E and f/Q ∈ $Λ⁺_\{γ\}$ for some closed set E ⊆ and some inner function Q.},
author = {Thomas Vils Pedersen},
journal = {Studia Mathematica},
keywords = {big Lipschitz algebra; closed ideal},
language = {eng},
number = {1},
pages = {33-59},
title = {Ideals in big Lipschitz algebras of analytic functions},
url = {http://eudml.org/doc/284948},
volume = {161},
year = {2004},
}

TY - JOUR
AU - Thomas Vils Pedersen
TI - Ideals in big Lipschitz algebras of analytic functions
JO - Studia Mathematica
PY - 2004
VL - 161
IS - 1
SP - 33
EP - 59
AB - For 0 < γ ≤ 1, let $Λ⁺_{γ}$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{γ}(E,Q)$ be the closed ideal in $Λ⁺_{γ}$ consisting of those functions $f ∈ Λ⁺_{γ}$ for which (i) f = 0 on E, (ii) $|f(z)-f(w)| = o(|z-w|^{γ})$ as d(z,E),d(w,E) → 0, (iii) $f/Q ∈ Λ⁺_{γ}$. Also, for a closed ideal I in $Λ⁺_{γ}$, let $E_{I}$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_{I}$ be the greatest common divisor of the inner parts of non-zero functions in I. Our main conjecture about the ideal structure in $Λ⁺_{γ}$ is that $J_{γ}(E_{I},Q_{I}) ⊆ I$ for every closed ideal I in $Λ⁺_{γ}$. We confirm the conjecture for closed ideals I in $Λ⁺_{γ}$ for which $E_{I}$ is countable and obtain partial results in the case where $Q_{I} = 1$. Moreover, we show that every wk* closed ideal in $Λ⁺_{γ}$ is of the form f ∈ $Λ⁺_{γ}$: f = 0 on E and f/Q ∈ $Λ⁺_{γ}$ for some closed set E ⊆ and some inner function Q.
LA - eng
KW - big Lipschitz algebra; closed ideal
UR - http://eudml.org/doc/284948
ER -