Compact lines and the Sobczyk property

Abstract

Abstract. We show that Sobczyk's Theorem holds for a new class of Banach spaces, namely spaces of continuous functions on linearly ordered compacta.

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

“…Using the classical Sobczyk Theorem [13, p.72] (which states that if H is a linear subspace of a separable Banach space E and T : H → c 0 is a bounded linear operator, then there is a bounded linear operator S : E → c 0 extending T to all of E), one can prove the following characterization showing that for Banach spaces our definition of separable c 0 -extension property is equivalent to that introduced by Correa and Tausk [10,11]. By [10,11] the class of Banach spaces with the separable c 0 -extension property includes all weakly compactly generated Banach spaces, all Banach spaces with the separable complementation property (=every separable subspace is contained in a separable complemented subspace), and all Banach spaces C(K) over ℵ 0 -monolithic compact lines K. Let us recall that a topological space X is ℵ 0monolithic if each separable subspace of X has a countable network. Theorem 6.10.…”

“…The authors express their sincere thanks to Tomek Kania for the suggestion 1 to apply the Szlenk index in the proof of Theorem 5.8 and pointing out 2 the papers [10,11] devoted to the separable c 0 -extension property, to Mikhail Ostrovskii for suggesting the idea 3 of constructing the space E in Example 5.5, and to Grzegorz Plebanek for his suggestion to consider Efimov spaces and some ideas from his joint article with Mirna Džamonja [17] to attack Problem 7.14.…”

Compactifications of $\omega $ and the Banach space $c_0$

“…A Banach space X is said to have the Sobczyk property if every subspace of X isomorphic to c 0 is complemented in X. Moltó [22] singled out a certain topological property of the weak * topology in X * ensuring that X has the Sobczyk property. Correa and Tausk [9] proved that the space C(K) has the Sobczyk property whenever K is a compact line (generalizing an earlier result from [23], where the same was proved for K being the double arrow space); see also [2], [6], [14], [16] for related results.…”

On the c0-extension property for compact lines

“…This definition was introduced by the authors in [3] with the purpose of studying extensions of the celebrated Theorem of Sobczyk [14], which states that every closed subspace of a separable Banach space X has the c 0 EP in X. The quest for generalizations of Sobczyk's Theorem has been engaged upon by many authors [1,2,3,10,11,12,13]. For instance, we have used in [2] the notion of c 0 EP to prove that every isomorphic copy of c 0 in C(K) is complemented, when K is a compact line.…”

“…The quest for generalizations of Sobczyk's Theorem has been engaged upon by many authors [1,2,3,10,11,12,13]. For instance, we have used in [2] the notion of c 0 EP to prove that every isomorphic copy of c 0 in C(K) is complemented, when K is a compact line. By a compact line we mean a linearly ordered set which is compact in the order topology.…”