Complementation and decompositions in some weakly Lindelöf Banach spaces (Q624589)

Consider the class \(\mathcal{L}\) of all scattered compact spaces \(K\) of countable height such that \(C(K)\) is Lindelöf. The authors show that the following statement is independent of ZFC: ``For every \(K\in \mathcal{L}\) and for every copy of \(c_0(\omega_1)\) as a subspace of \(C(K)\) there exists an uncountable subset \(E\subset\omega_1\) such that \(c_0(E)\) is a complemented subspace of \(C(K)\)''. This fact was known to be true in ZFC under the additional assumption that \(K\) is an Eberlein compact [\textit{S. A. Argyros, J. F. Castillo, A. S. Granero, M. Jiménez,} and \textit{J. P. Moreno}, Proc. Lond. Math. Soc., III. Ser. 85, No.~3, 742--768 (2002; Zbl 1017.46011)]. Now, the authors prove it to be true under the P-ideal dichotomy, but false under \(\clubsuit\). Indeed, a space \(K\in\mathcal{L}\) is constructed under \(\clubsuit\) such that in any decomposition \(C(K) = A\oplus B\) into infinite-dimensional spaces, one of the factors is isomorphic to \(C(K)\) and the other one is isomorphic to either \(c_0\) or \(C(\omega^\omega)\) (moreover, every operator \(T:C(K)\to C(K)\) is a multiple of the identity plus an operator with separable range). The paper also includes some general facts about compact spaces \(K\in\mathcal{L}\) and their \(C(K)\) spaces. The authors show that, when \(K\in \mathcal{L}\) is uncountable and has finite height, then \(C(K)\) always contains complemented copies of \(c_0(\omega_1)\), but -- unlike the case of Eberlein compacta -- there exists such a space with also an uncomplemented copy of \(c_0(\omega_1)\). They pose the question whether the class \(\mathcal{L}\) coincides with the class of compact scattered spaces of countable height in which the closure of every countable space is countable.