Uniqueness of unconditional bases in nonlocally convex \(\ell _{1}\)-products
type: Item
instance of: scholarly article
title: Uniqueness of unconditional bases in nonlocally convex \(\ell _{1}\)-products
zbMATH Open document ID: 1210.46005
author: F. Albiac
author: C. Leránoz
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-11-22T00:00:00Z
review text: A quasi-Banach space \(X\) with an unconditional basis is said to have a unique unconditional basis up to permutation if for any two normalized unconditional bases of \(X\) there exists an automorphism of \(X\) which takes one basis to a permutation of the other. In [\textit{J.~Bourgain, P. G.~Casazza, J.~Lindenstrauss} and \textit{L.~Tzafriri}, Mem. Am. Math. Soc. 54, 1--111 (1985; Zbl 0575.46011)] it was posed the problem: Does the space \(\ell_1(X)\) have a unique unconditional basis up to permutation whenever the Banach space \(X\) does? In the paper under review, the authors translate this question to the nonlocally convex setting. They prove, in particular, the following corollary: Suppose that \(X\) is a quasi-Banach space with unconditional basis and whose \(q\)-Banach envelope is isomorphic to \(\ell_q\) for some \(0<q<1\). Then the space \(\ell_1(X)\) has a unique unconditional basis up to permutation. The \(q\)-Banach envelope of a quasi-normed space \((X,\|\;\|)\) is the completion of \(X\) in the quasi-norm, generated by the set \(\{\sum_1^na_ix_i: \sum_i|a_i|^q\leq 1\), \(\|x_i\|\leq 1\), \(n\in \mathbb{N}\}\).
Mathematics Subject Classification ID: 46A35
Mathematics Subject Classification ID: 46A16
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 5817864
zbMATH Keywords: quasi-Banach space
zbMATH Keywords: uniqueness of unconditional basis up to permutation
zbMATH Keywords: lattice anti-Euclidean
reviewed by: A. M. Plichko
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.09.048
OpenAlex ID: W1992314204
cites work: Q5317928
cites work: Uniqueness of the unconditional basis of \(\ell_{1}(\ell_p)\) and \(\ell_p(\ell_1)\), \(0<p<1\)
cites work: Uniqueness of unconditional basis in Lorentz sequence spaces
cites work: Uniqueness of unconditional bases in nonlocally convex \(c_0\)-products
cites work: Banach spaces with a unique unconditional basis, up to permutation
cites work: Uniqueness of unconditional bases in Banach spaces
cites work: Q3925395
cites work: Orlicz sequence spaces without local convexity
cites work: Convexity conditions for non-locally convex lattices
cites work: Banach Envelopes of Non-Locally Convex Spaces
cites work: Q3219291
cites work: Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces
cites work: Q4188284
cites work: Banach spaces with a unique unconditional basis
cites work: Q4440823
cites work: $H_{p}$-spaces, p≤1, and spline systems
cites work: Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces. II
DOI: 10.1016/J.JMAA.2010.09.048
Recommended article: Uniqueness of unconditional basis of \(\ell_{p}(c_{0})\) and \(\ell_{p}(\ell_{2})\), \(0<p<1\)
Recommended article: An alternative approach to the uniqueness of unconditional basis of \(\ell _p(c_{0})\) for \(0<p<1\)
Recommended article: Q4431833
Recommended article: Uniqueness of symmetric basis in quasi-Banach spaces
Recommended article: Uniqueness of the unconditional basis of \(\ell_{1}(\ell_p)\) and \(\ell_p(\ell_1)\), \(0<p<1\)
Recommended article: Banach Envelopes of Non-Locally Convex Spaces
Recommended article: Addendum to: ``Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces''
Recommended article: On the permutative equivalence of squares of unconditional bases
Recommended article: Uniqueness of unconditional basis in quasi-Banach spaces which are not sufficiently Euclidean
Recommended article: Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces. II
description: scientific article; zbMATH DE number 5817864