New examples of noncommutative \(\Lambda(p)\) sets
type: Item
instance of: scholarly article
title: New examples of noncommutative \(\Lambda(p)\) sets
zbMATH Open document ID: 1035.43004
author: William D. Banks
author: Asma Harcharras
published in: Illinois Journal of Mathematics
publication date: 2004-01-19T00:00:00Z
review text: Since 1960 and \textit{W. Rudin}'s paper [J. Math. Mech. 9, 203--227 (1960; Zbl 0091.05802)], the notion of \(\Lambda(p)\) set has generated numerous works; for example: \textit{G. Pisier} [Sémin. Géom. des Espaces de Banach 1977--1978, Exposés No. 12--13 (1978; Zbl 0388.43009)], \textit{J. Bourgain} [Acta Math. 162, 227--245 (1989; Zbl 0674.43004)], see also \textit{M. Talagrand} [Acta Math. 175, 273--300 (1995; Zbl 0917.46006)]. For a survey of interactions of this notion with various parts of mathematics, see \textit{J. Bourgain} [\(\Lambda_p\)-sets in analysis: Results, problems and related aspects, Handbook of the geometry of Banach spaces, Volume 1, Amsterdam: Elsevier, 195--232 (2001; Zbl 1016.43004)]. For \(2<p<+\infty\), a subset \(E\subseteq {\mathbb Z}\) is called a \(\Lambda(p)\) set if the \(L^p\) norm is equivalent to the \(L^2\) norm on \(L^p_E({\mathbb T})\) (where \({\mathbb T}={\mathbb R}/2\pi{\mathbb Z}\) and, as usual \(L^p_E({\mathbb T})= \{f\in L^p({\mathbb T})\); \(\widehat f(n)=0\) for \(n\in {\mathbb Z}\setminus E\}\)). In other words, there exists a constant \(C>0\) such that \(\| f\|_{L^p}\leq C(\sum_{n\in E} | \widehat f(n)|^2)^{1/2}\) for every \(f\in L^p_E({\mathbb T})\). In her thesis [\textit{A. Harcharras}, Stud. Math. 137, 203--260 (1999; Zbl 0948.43002)], the second author, introduced the stronger notion of noncommutative \(\Lambda(p)\) set (or \(\Lambda(p)_{cb}\) set): let \(S_p\) be the Schatten \(p\)-class over the Hilbert space \(\ell_2\); a subset \(E\subseteq {\mathbb Z}\) is a noncommutative \(\Lambda(p)\) set if there exists a constant \(C>0\) such that \[ \| f\|_{L^p(S_p)}\leq C\,\max\Biggl\{\biggl\| \biggl(\sum_{n\in E} \widehat f(n)^\ast\,\widehat f(n)\biggr)^{1/2} \biggr\|_{S_p}, \biggl\| \biggl(\sum_{n\in E} \widehat f(n)\,\widehat f(n)^\ast\biggr)^{1/2} \biggr\|_{S_p}\Biggr\}\,, \] for every \(f\in L^p_E(S_p)\). For the study of \(\Lambda(p)\) sets, Rudin used a combinatorial property, called now \(Z^+(k)\) (where \(k…
reviewed by: Daniel Li
Mathematics Subject Classification ID: 43A46
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 2028859
zbMATH Keywords: noncommutative \(\Lambda(p)\) sets
zbMATH Keywords: prime numbers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2028859