Complex convexity and monotonicity in quasi-Banach lattices
type: Item
instance of: scholarly article
title: Complex convexity and monotonicity in quasi-Banach lattices
zbMATH Open document ID: 1129.46001
author: Han Ju Lee
published in: Israel Journal of Mathematics
publication date: 2007-09-03T00:00:00Z
review text: For \(0<p<\infty\), the modulus of \(p\)-monotonicity of a quasi-Banach lattice \(E\) is defined as \(\Pi_p^E(\varepsilon)=\inf\{\| (| x| ^p + | y| ^p)^{1/p}\| - 1: x,y \in E\), \(\| x\| =1\), \(\| y\| \geq \varepsilon\}\). \(E\) is said to be uniformly \(p\)-monotone if \(\Pi_p^E(\varepsilon) > 0\) for \(\varepsilon > 0\) and is said to be uniformly monotone if it is uniformly \(1\)-monotone. It is shown that in an \(\alpha\)-convex quasi-Banach lattice with \(\alpha\)-convexity constant~\(1\), uniform monotonicity is equivalent to complex uniform convexity. A quasi-Banach lattice \(E\) has an equivalent lattice renorming which is complex uniformly convex of power type if and only if for any \(\lambda \geq 1\), \(\ell_\infty\) is not finitely \(\lambda\)-representable in \(E\).
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46A16
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5186272
zbMATH Keywords: quasi-Banach space
zbMATH Keywords: quasi-Banach lattice
zbMATH Keywords: complex uniform convexity
Wikidata QID: Q57618298
MaRDI profile type: Publication
full work available at URL: s11856-007-0038-2
OpenAlex ID: W2047370583
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DOI: 10.1007/S11856-007-0038-2
description: scientific article; zbMATH DE number 5186272