Rosenthal sets for Banach-valued functions
type: Item
instance of: scholarly article
title: Rosenthal sets for Banach-valued functions
zbMATH Open document ID: 0854.43007
DOI: 10.1007/BF01268867
author: Frédérique Watbled
published in: Archiv der Mathematik
publication date: 1996-10-27T00:00:00Z
review text: Let \(G\) be a compact metric abelian group with doubling normalized Haar measure \(\mu\). Let \(X\) be a Banach space not containing \(c_0\). We show that if \(H\) is a dense subgroup of \(G\), if \(F \in L^\infty (G,X)\), and if all the differences \(\Delta_h F(t) = F(t + h) - (t)\), \(h \in H\), are \(\mu\)-a.e. equal to continuous functions on \(G\) then \(F\) also is \(\mu\)-a.e. equal to a continuous function on \(G\). -- We apply this result in the following problem of harmonic analysis: let \(G\), \(X\), \(H\) be as above, \(\Gamma\) be the dual group of \(G\), and suppose \(H\) is countable. If we denote by \(\widehat H\) the dual group of \(H\), where \(H\) is given the discrete topology, then we can identify \(\Gamma\) with a dense subgroup of \(\widehat H\). Let \(\Lambda\) be a subset of \(\Gamma\) with countable closure in \(\widehat H\). F. Piquard has proved that under such hypothesis we have \(L^\infty_\Lambda (G) = C_\Lambda (G)\). Here we generalize this result to Banach-valued functions by showing that if \(X\) is a Banach space not containing \(c_0\) and if \(\Lambda\) is as above then \(L^\infty_\Lambda (G,X) = C_\Lambda (G,X)\). We show that this happens also in the case where \(\Lambda\) is any Rosenthal set and \(X\) is any Banach space with the Schur property.
Mathematics Subject Classification ID: 43A46
zbMATH DE Number: 938562
zbMATH Keywords: compact metric abelian group
zbMATH Keywords: Haar measure
zbMATH Keywords: Banach space
zbMATH Keywords: harmonic analysis
zbMATH Keywords: dual group
zbMATH Keywords: Banach-valued functions
zbMATH Keywords: Rosenthal set
zbMATH Keywords: Schur property
MaRDI profile type: Publication
cites work: On bases and unconditional convergence of series in Banach spaces
cites work: Generalization of two theorems of M. I. Kadets concerning the indefinite integral of abstract almost periodic functions
cites work: Q4039051
cites work: Q3254057
cites work: Q5556050
cites work: Q3965074
cites work: L'espace des fonctions presque-périodiques dont le spectre est contenu dans un ensemble compact dénombrable a la propriété de Schur
cites work: Q3997989
full work available at URL: https://doi.org/10.1007/bf01268867
OpenAlex ID: W2001603302
description: scientific article; zbMATH DE number 938562