Some Banach space properties of the duals of the disk algebra and \(H^\infty\)
type: Item
instance of: scholarly article
title: Some Banach space properties of the duals of the disk algebra and \(H^\infty\)
zbMATH Open document ID: 0979.46037
author: Manuel D. Contreras
author: Santiago Díaz
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: Following Pełczyński, a Banach space \(X\) is said to have property (V) if every subset \(W\) of the dual \(X^\ast\) satisfying \[ \lim_{n}\sup_{x^\ast\in W}|\langle x^\ast,x_n\rangle|=0 \] for every weakly unconditionally Cauchy series \(\sum_nx_n\) in \(X\) is weakly relatively compact in \(X^\ast\). A Banach space \(X\) has the local property (V) if each ultrapower \((X)_{\mathcal U}\) has the property (V). A Banach space \(X\) is said to be a Grothendieck space if every sequence in \(X^\ast\) that is weak-\(\ast\) convergent is also weakly convergent. The authors proved that the disk algebra \(A\) has the local property (V). From this fact it follows that \(H^\infty\) also has the local property (V). This result is an extension of a \textit{J. Bourgain}'s theorem [Stud. Math. 75, 193-216 (1983; Zbl 0533.46035)]. It is also proved that the odd duals of \(A\) and \(H^\infty\) are Grothendieck spaces.
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46M07
Mathematics Subject Classification ID: 32A38
zbMATH DE Number: 1548304
zbMATH Keywords: property weak (V)
zbMATH Keywords: property (V)
zbMATH Keywords: Grothendieck spaces
zbMATH Keywords: ultrapowers
zbMATH Keywords: disk algebra
zbMATH Keywords: odd duals
reviewed by: Andrzej Sołtysiak
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132364
OpenAlex ID: W2030365860
DOI: 10.1307/MMJ/1030132364
description: scientific article; zbMATH DE number 1548304