Banach space properties of strongly tight uniform algebras

Scott Saccone

Studia Mathematica (1995)

  • Volume: 114, Issue: 2, page 159-180
  • ISSN: 0039-3223

Abstract

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We use the work of J. Bourgain to show that some uniform algebras of analytic functions have certain Banach space properties. If X is a Banach space, we say X is strongif X and X* have the Dunford-Pettis property, X has the Pełczyński property, and X* is weakly sequentially complete. Bourgain has shown that the ball-algebras and the polydisk-algebras are strong Banach spaces. Using Bourgain’s methods, Cima and Timoney have shown that if K is a compact planar set and A is R(K) or A(K), then A and A* have the Dunford-Pettis property. Prior to the work of Bourgain, it was shown independently by Wojtaszczyk and Delbaen that R(K) and A(K) have the Pełczyński property for special classes of sets K. We show that if A is R(K) or A(K), where K is arbitrary, or if A is A(D) where D is a strictly pseudoconvex domain with smooth C 2 boundary in ℂ n , then A is a strong Banach space. More generally, if A is a uniform algebra on a compact space K, we say A is strongly tight if the Hankel-type operator S g : A → C / A defined by f → fg + A is compact for every g ∈ C(K). Cole and Gamelin have shown that R(K) and A(K) are strongly tight when K is arbitrary, and their ideas can be used to show A(D) is strongly tight for the domains D considered above. We show strongly tight uniform algebras are strong Banach spaces.

How to cite

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Saccone, Scott. "Banach space properties of strongly tight uniform algebras." Studia Mathematica 114.2 (1995): 159-180. <http://eudml.org/doc/216186>.

@article{Saccone1995,
abstract = {We use the work of J. Bourgain to show that some uniform algebras of analytic functions have certain Banach space properties. If X is a Banach space, we say X is strongif X and X* have the Dunford-Pettis property, X has the Pełczyński property, and X* is weakly sequentially complete. Bourgain has shown that the ball-algebras and the polydisk-algebras are strong Banach spaces. Using Bourgain’s methods, Cima and Timoney have shown that if K is a compact planar set and A is R(K) or A(K), then A and A* have the Dunford-Pettis property. Prior to the work of Bourgain, it was shown independently by Wojtaszczyk and Delbaen that R(K) and A(K) have the Pełczyński property for special classes of sets K. We show that if A is R(K) or A(K), where K is arbitrary, or if A is A(D) where D is a strictly pseudoconvex domain with smooth $C^2$ boundary in $ℂ^n$, then A is a strong Banach space. More generally, if A is a uniform algebra on a compact space K, we say A is strongly tight if the Hankel-type operator $S_g: A → C/A$ defined by f → fg + A is compact for every g ∈ C(K). Cole and Gamelin have shown that R(K) and A(K) are strongly tight when K is arbitrary, and their ideas can be used to show A(D) is strongly tight for the domains D considered above. We show strongly tight uniform algebras are strong Banach spaces. },
author = {Saccone, Scott},
journal = {Studia Mathematica},
keywords = {uniform algebras of analytic functions; Dunford-Pettis property; Pełczyński property; ball-algebras; polydisk-algebras; strong Banach spaces; pseudoconvex domain; Hankel-type operator; strongly tight uniform algebras are strong Banach spaces},
language = {eng},
number = {2},
pages = {159-180},
title = {Banach space properties of strongly tight uniform algebras},
url = {http://eudml.org/doc/216186},
volume = {114},
year = {1995},
}

TY - JOUR
AU - Saccone, Scott
TI - Banach space properties of strongly tight uniform algebras
JO - Studia Mathematica
PY - 1995
VL - 114
IS - 2
SP - 159
EP - 180
AB - We use the work of J. Bourgain to show that some uniform algebras of analytic functions have certain Banach space properties. If X is a Banach space, we say X is strongif X and X* have the Dunford-Pettis property, X has the Pełczyński property, and X* is weakly sequentially complete. Bourgain has shown that the ball-algebras and the polydisk-algebras are strong Banach spaces. Using Bourgain’s methods, Cima and Timoney have shown that if K is a compact planar set and A is R(K) or A(K), then A and A* have the Dunford-Pettis property. Prior to the work of Bourgain, it was shown independently by Wojtaszczyk and Delbaen that R(K) and A(K) have the Pełczyński property for special classes of sets K. We show that if A is R(K) or A(K), where K is arbitrary, or if A is A(D) where D is a strictly pseudoconvex domain with smooth $C^2$ boundary in $ℂ^n$, then A is a strong Banach space. More generally, if A is a uniform algebra on a compact space K, we say A is strongly tight if the Hankel-type operator $S_g: A → C/A$ defined by f → fg + A is compact for every g ∈ C(K). Cole and Gamelin have shown that R(K) and A(K) are strongly tight when K is arbitrary, and their ideas can be used to show A(D) is strongly tight for the domains D considered above. We show strongly tight uniform algebras are strong Banach spaces.
LA - eng
KW - uniform algebras of analytic functions; Dunford-Pettis property; Pełczyński property; ball-algebras; polydisk-algebras; strong Banach spaces; pseudoconvex domain; Hankel-type operator; strongly tight uniform algebras are strong Banach spaces
UR - http://eudml.org/doc/216186
ER -

References

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