Automorphisms of \(\mathcal C(K)\)-spaces and extension of linear operators
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\(^*\) continuous on bounded sets such that \(\Phi(x^*)\) extends \(x^*\) with a bound \(\|\Phi(x^*)\|\leq \lambda \|x^*\|\), for all \(x^*\in X^*\). Such mappings were introduced by \textit{M.\,Zippin} [``Extension of bounded linear operators'', in: W.\ B.\ Johnson (ed.)~et al., Handbook of the geometry of Banach spaces.\ Amsterdam: North-Holland.\ Volume 2, 1703--1741 (2003; Zbl 1048.46016)] and are called Zippin selectors here. Based on this notion, the following technical key result on \(X=\ell_p\) or, more generally, an \(\ell_p\)-sum of finite-dimensional spaces, \(1<p<\infty\), is proved: For a separable superspace \(Z\supset X\), the pair \((X,Z)\) has the \(C\)-extension property if and only if \(Z\) can be renormed so as to contain \(X\) isometrically and such that \[ \lim_{n\to\infty} \|z+x_n\| \geq \lim_{n\to\infty} (\|z\|^p+ \|x_n\|^p)^{1/p} \] for all \(z\in Z\) and all weakly null sequences \((x_n)\subset X\), provided both limits exist.
scientific article; zbMATH DE number 5567374
scientific article