Idempotents in quotients and restrictions of Banach algebras of functions

Thomas Vils Pedersen

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 4, page 1095-1124
  • ISSN: 0373-0956

Abstract

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Let 𝒜 β be the Beurling algebra with weight ( 1 + | n | ) β on the unit circle 𝕋 and, for a closed set E ⊆ 𝕋 , let J 𝒜 β ( E ) = { f ∈ 𝒜 β : f = 0 on a neighbourhood of E } . We prove that, for β > 1 2 , there exists a closed set E ⊆ 𝕋 of measure zero such that the quotient algebra 𝒜 β / J 𝒜 β ( E ) ‾ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras λ γ and the algebra 𝒜 𝒞 of absolutely continuous functions on 𝕋 , we characterize the closed sets E ⊆ 𝕋 for which the restriction algebras λ γ ( E ) and 𝒜 𝒞 ( E ) are generated by their idempotents.

How to cite

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Pedersen, Thomas Vils. "Idempotents in quotients and restrictions of Banach algebras of functions." Annales de l'institut Fourier 46.4 (1996): 1095-1124. <http://eudml.org/doc/75201>.

@article{Pedersen1996,
abstract = {Let $\{\cal A\}_\beta $ be the Beurling algebra with weight $(1+\vert n\vert )^\beta $ on the unit circle $\{\Bbb T\}$ and, for a closed set $E\subseteq \{\Bbb T\}$, let $J_\{\{\cal A\}_\beta \}(E)=\lbrace f\in \{\cal A\}_\beta :f=0\,\text\{on\} \text\{a\} \text\{neighbourhood\} \text\{of\}\, E\rbrace $. We prove that, for $\beta &gt;\{1\over 2\}$, there exists a closed set $E\subseteq \{\Bbb T\}$ of measure zero such that the quotient algebra $\{\cal A\}_\beta /\overline\{J_\{\{\cal A\}_\beta \}(E)\}$ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras $\lambda _\gamma $ and the algebra $\{\cal A\}\{\cal C\}$ of absolutely continuous functions on $\{\Bbb T\}$, we characterize the closed sets $E\subseteq \{\Bbb T\}$ for which the restriction algebras $\lambda _\gamma (E)$ and $\{\cal A\}\{\cal C\} (E)$ are generated by their idempotents.},
author = {Pedersen, Thomas Vils},
journal = {Annales de l'institut Fourier},
keywords = {Banach algebras of functions; idempotents},
language = {eng},
number = {4},
pages = {1095-1124},
publisher = {Association des Annales de l'Institut Fourier},
title = {Idempotents in quotients and restrictions of Banach algebras of functions},
url = {http://eudml.org/doc/75201},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Pedersen, Thomas Vils
TI - Idempotents in quotients and restrictions of Banach algebras of functions
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 4
SP - 1095
EP - 1124
AB - Let ${\cal A}_\beta $ be the Beurling algebra with weight $(1+\vert n\vert )^\beta $ on the unit circle ${\Bbb T}$ and, for a closed set $E\subseteq {\Bbb T}$, let $J_{{\cal A}_\beta }(E)=\lbrace f\in {\cal A}_\beta :f=0\,\text{on} \text{a} \text{neighbourhood} \text{of}\, E\rbrace $. We prove that, for $\beta &gt;{1\over 2}$, there exists a closed set $E\subseteq {\Bbb T}$ of measure zero such that the quotient algebra ${\cal A}_\beta /\overline{J_{{\cal A}_\beta }(E)}$ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras $\lambda _\gamma $ and the algebra ${\cal A}{\cal C}$ of absolutely continuous functions on ${\Bbb T}$, we characterize the closed sets $E\subseteq {\Bbb T}$ for which the restriction algebras $\lambda _\gamma (E)$ and ${\cal A}{\cal C} (E)$ are generated by their idempotents.
LA - eng
KW - Banach algebras of functions; idempotents
UR - http://eudml.org/doc/75201
ER -

References

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