Complementably universal Banach spaces. II.
type: Item
instance of: scholarly article
title: Complementably universal Banach spaces. II.
zbMATH Open document ID: 1193.46004
published in: Journal of Functional Analysis
publication date: 2009-11-30T00:00:00Z
review text: A Banach space \(U\) is said to be \textit{complementably universal} for a class \({\mathcal B}\) of Banach spaces when every \(B \in {\mathcal B}\) is isomorphic to a complemented subspace of \(U\). The paper is a sequel to the authors' paper [``Complementably universal Banach spaces'', Stud. Math. 58, 91--97 (1976; Zbl 0341.46017)]. The authors prove two main results: (1) If \(U\) is a Banach space complementably universal for the family of all subspaces of \(c_0\) with the bounded approximation property, then \(U^*\) is non-separable. For that, they prove that no separable Banach space is complementably universal for the family \(\{X^*;\;X \subseteq c_0\}\), and use a result which is implicit in \textit{W. B. Johnson} and \textit{T. Oikhberg} [``Separable lifting property and extensions of local reflexivity'', Ill. J. Math. 45, No.~1, 123--137 (2001; Zbl 1004.46008)]: for every subspace \(X\) of \(c_0\), there is a subspace \(Y\) of \(c_0\) with an \(FDD\) such that \(X^*\) is isomorphic to a complemented subspace of \(Y^*\). (2) There is no separable Banach space \(X\) such that every compact operator factors through \(X\). This solves a problem open since the 1970s. The constructions use finite-dimensional blocks and a probabilistic lemma which allows to extract suitable submatrices of a given matrix, in order to get suitable orthogonal systems.
reviewed by: Daniel Li
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 47A68
zbMATH DE Number: 5640317
zbMATH Keywords: universal Banach spaces
zbMATH Keywords: complemented subspaces
zbMATH Keywords: approximation property
zbMATH Keywords: factorization of compact operators
author: William B. Johnson
author: Andrzej Szankowski
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.07.008
OpenAlex ID: W2042029987
cites work: Quotients of c 0 are Almost Isometric to Subspaces of c 0
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DOI: 10.1016/J.JFA.2009.07.008
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description: scientific article; zbMATH DE number 5640317