On a three-space property for Lindelöf $\Sigma$-spaces, (WCG)-spaces and the Sobczyk property

Abstract

Corson's example shows that there exists a Banach space $E$ which is not weakly normal but $E$ contains a closed subspace isomorphic to the Banach space $C[0,1]$ and such that the quotient space $E/C[0,1]$ is isomorphic to the weakly compactly generated Banach space $c_{0}[0,1]$. This applies to show the following two results: (i) The Lindel\"of property is not a three-space property. (ii) The Lindel\"of $\Sigma$-property is not a three-space property. In this note using the lifting property developed by Susanne Dierolf we present a very simple argument providing also (ii), see Theorem 1. This argument used in the proof applies also to show that under Continuum Hypothesis every infinite-dimensional topological vector space $E$ which contains a dense hyperplane admits a stronger vector topology $\upsilon$ with the same topological dual and such that $(E,\upsilon)$ contains a dense non-Baire hyperplane. This partially answers a question of Saxon concerning Arias de Reyna-Valdivia-Saxon theorem. A Banach space $E$ has the Sobczyk Property if it contains an isomorphic copy of $c_0$ and every such a copy is complemented in $E$. The classical Sobczyk's theorem says that every separable Banach space has this property. We give an example of a $C(K)$-space $E$ and its subspace $Y$ isometric to $c_0$ such that $E/Y$ is isomorphic to $c_0(\Gamma)$, with $card(\Gamma )=2^{\aleph_0}$, yet $Y$ is uncomplemented in $E$. This complements Corson's example and shows that the Sobczyk Property (as well as the (WCG)-property, and the Separable Complementation Property) is not a~three-space property. In the last part we recall some facts (partially with a simpler presentation) concerning K-analytic, Lindelöf $\Sigma$ and analytic locally convex spaces. Additionally, a few remarks concerning weakly K-analytic spaces are included.