Hereditary approximation property
type: Item
instance of: scholarly article
title: Hereditary approximation property
zbMATH Open document ID: 1272.46012
author: Andrzej Szankowski
author: William B. Johnson
published in: Annals of Mathematics. Second Series
publication date: 2013-01-03T00:00:00Z
review text: Let \(X\) be a Banach space. The authors say that \(X\) has the hereditary approximation property (HAP) or is an HAPpy space if all closed subspaces of \(X\) have the approximation property. Hilbert spaces are clearly HAPpy. The first HAPpy spaces which are not isomorphic to Hilbert spaces were constructed by \textit{W. B. Johnson} [Functional analysis, numerical analysis and optimization, Spec. Top. appl. Math., Proc. Semin. GMD, Bonn 1979, 15--26 (1980; Zbl 0442.46011)]. Denote by \[ d_n(X)=\sup \{ d (E, \ell^n_2): E\subset X, ~\dim E=n\} \] the isomorphism constants of \(X\) to \(\ell^n_2\) from \(n\)-dimensional subspaces of \(X\). (Here, \(d(E,F)\) is the Banach-Mazur distance, i.e., the infimum of \(\| T\|\;\| T^{-1}\|\) as \(T\) ranges over all isomorphisms from \(E\) onto \(F\).) The basic theorem of the article under review is the technical Theorem 2.1. It shows that if \((d_n(X))_n\) goes to infinity sufficiently slowly, then \(X\) is an HAPpy space. The HAPpy spaces constructed in [loc.\,cit.]\ are asymptotically Hilbertian. In particular, as was noted in [loc.\,cit.], they cannot have a symmetric basis unless they are isomorphic to \(\ell_2\). A problem raised in [loc.\,cit.]\ was whether there exist HAPpy spaces with a symmetric basis but not isomorphic to \(\ell_2\). Relying on their basic theorem, the authors give an affirmative answer to this old problem by constructing an HAPpy Orlicz sequence space that is not isomorphic to \(\ell_2\). As another application of the basic theorem, the authors show that there exists a separable infinite-dimensional Banach space \(X\) not isomorphic to a Hilbert space which is complementably universal for all closed subspaces of all of its quotients. In particular, every closed subspace of \(X\) is isomorphic to a complemented subspace of \(X\). Recall that, in contrast, by the classical Lindenstrauss-Tzafriri theorem [\textit{J. Lindenstrauss} and \textit{L. Tzafriri}, Isr. J. Math. 9, 263--269 (1971; Zbl 0211.16301)…
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B07
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 6121654
zbMATH Keywords: Banach spaces
zbMATH Keywords: hereditary uniform approximation property
zbMATH Keywords: asymptotically Hilbertian spaces
zbMATH Keywords: trace formula
zbMATH Keywords: hereditary approximation property
reviewed by: Eve Oja
MaRDI profile type: Publication
full work available at URL: annals.2012.176.3.10
OpenAlex ID: W2146771470
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DOI: 10.4007/ANNALS.2012.176.3.10
description: scientific article; zbMATH DE number 6121654