The \(L_ 1\) structure of weak \(L_ 1\)
type: Item
instance of: scholarly article
title: The \(L_ 1\) structure of weak \(L_ 1\)
zbMATH Open document ID: 0537.46029
DOI: 10.1007/BF01451421
author: Joseph Kupka
author: N. Tenney Peck
published in: Mathematische Annalen
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163933
review text: Let \(\mu\) be a nonatomic probability measure. It is shown that the dual space of weak \(L_ 1\) of \(\mu\) contains a number of order isometric copies of \(L_{\infty}(\mu)\). Moreover, the linear span of these copies is \(weak^*\) dense in the dual space. The normed envelope of weak \(L_ 1\) is shown to be covered, in a certain sense, by order isometric, complemented copies of \(L_ 1[0,1]\) (with Lebesgue measure), of \(\ell_ 1[0,1]\), and also, more generally, of the \(\ell_ 1\) direct sum of continuum many disjoint copies of \(L_ 1[0,1]\). But it is not an abstract \(L_ 1\) space, for it also contains order isometric copies of \(\ell_{\infty}\) and of \(c_ 0[0,1]\). Weak \(L_ 1\) itself contains an order isomorphic copy of \(\ell_{\infty}\), and so it is unlikely that the Nikisin factorization of an \(L_ 0\)-valued operator through weak \(L_ 1\) will yield a direct answer to the \(L_ 0\)-factorization problem.
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 3853777
zbMATH Keywords: Nikisin factorization of an \(L_ 0\)-valued operator through weak \(L_ 1\)
zbMATH Keywords: \(L_ 0\)-factorization problem
MaRDI profile type: Publication
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OpenAlex ID: W3093909398
description: scientific article; zbMATH DE number 3853777