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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.

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