Automorphisms of \(\mathcal C(K)\)-spaces and extension of linear operators
type: Item
instance of: scholarly article
title: Automorphisms of \(\mathcal C(K)\)-spaces and extension of linear operators
zbMATH Open document ID: 1184.46010
published in: Illinois Journal of Mathematics
publication date: 2009-06-22T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ijm/1242414132
review text: The author defines a separable Banach space \(X\) to be \(C\)-automorphic if, whenever \(X_1\) and \(X_2\) are subspaces of \(C[0,1]\) isomorphic to \(X\), then there is an automorphism of \(C[0,1]\) mapping \(X_1\) onto \(X_2\). Consequently, there is essentially only one way to embed such a space into \(C[0,1]\). It follows from classical results due to \textit{J.\,Lindenstrauss} and \textit{A.\,Pełczyński} [J.~Funct.\ Anal.\ 8, 225--249 (1971; Zbl 0224.46041)] that subspaces of \(c_0\) are \(C\)-automorphic. In this interesting and deep paper, the author studies \(C\)-automorphic spaces and links his investigations with extension properties; see, for example, his memoir [\textit{N.\,J.\thinspace Kalton}, New York J.~Math.\ 13, 317--381 (2007; Zbl 1134.46004)]. Let us recall from that paper that \(X\) has the universal separable \(C\)-extension property if, whenever \(X\subset Z\), a separable superspace, and \(T: X\to C[0,1]\) is a bounded linear operator, then \(T\) admits an extension to a bounded linear operator \(\tilde T: Z\to C[0,1]\). One of the author's main results is that this property for \(X\) is equivalent to \(X\) being \(C\)-automorphic. This result permits him to prove that \(c_0(X)\) is \(C\)-automorphic if \(X\) is; a case in point is \(X=\ell_1\). He also proves that \(\ell_p\) is not \(C\)-automorphic for \(1<p<\infty\), solving a problem from [\textit{J.\,M.\,F.\thinspace Castillo} and \textit{Y.\,Moreno}, Isr.\ J.\ Math.\ 140, 253--270 (2004; Zbl 1063.46003)]. Indeed, for a certain superreflexive \(Z\supset \ell_p\) with an unconditional basis, the pair \((\ell_p,Z)\) fails the \(C\)-extension property. If, however, \(Z\supset \ell_p\) is a UMD-space with an unconditional basis, then \((\ell_p,Z)\) satisfies the \(C\)-extension property; in the author's own words, ``the appearance of the UMD-condition is quite mysterious.'' A technical device to prove these theorems are homogeneous mappings \(\Phi: X^* \to Z^*\) that are weak\(^*\) continuou…
reviewed by: Dirk Werner
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 5567374
zbMATH Keywords: \(C\)-automorphic Banach spaces
zbMATH Keywords: \(C\)-extension property
zbMATH Keywords: Zippin selector
zbMATH Keywords: UMD-space
zbMATH Keywords: types on Banach spaces
author: N. J. Kalton
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5567374