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Complementation and decompositions in some weakly Lindelöf Banach spaces. (English) Zbl 1225.46012
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 [S. A. Argyros, J. F. Castillo, A. S. Granero, M. Jiménez, and 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.

46B26 Nonseparable Banach spaces
54C35 Function spaces in general topology
03E35 Consistency and independence results
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
Full Text: DOI
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