Complementation and decompositions in some weakly Lindelِf Banach spaces

Let Γ denote an uncountable set. We consider the questions if a Banach space X of the form C ( K ) of a given class (1) has a complemented copy of c 0 ( Γ ) or (2) for every c 0 ( Γ ) ⊆ X has a complemented c 0 ( E ) for an uncountable E ⊆ Γ or (3) has a decomposition X = A ⊕ B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely Ks such that C ( K ) is Lindelöf in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C ( K ) s depend on additional combinatorial axioms which are independent of ZFC ± CH. If we assume the P-ideal dichotomy, for every c 0 ( Γ ) ⊆ C ( K ) there is a complemented c 0 ( E ) for an uncountable E ⊆ Γ , which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C ( K ) for K of height ω + 1 where every operator is of the form c I + S for c ∈ R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C ( K ) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space C p ( K ) .