On the Nonseparable Subspaces of J ( η ) and C ( [ 1 , η ] )

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens(Y ) = η, has a copy of `2(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens(Y ) = η, then Y has a copy of c0(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω1) (resp. C([1, ω1])) contains a complemented copy of `2(ω1) (resp. c0(ω1)). As a consequence, we give examples (J(ω1), C([1, ω1]), C(V ), V being the ”long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens(Y ) = w∗-Dens(Y ∗)), even though these spaces are not weakly countably determined (WCD). 1. Notations and preliminaries Throughout, (X, ‖ · ‖) will be a real Banach space, B(X) the closed unit ball of X, S(X) the unit sphere and X∗ its topological dual. Let O denote the ordinal numbers, LO the limit ordinals and C the cardinals. If η ∈ C, the cofinality cf(η) of η is the smallest cardinal τ for which there exists a sequence of ordinals {βi}1≤i<τ , βi ∈ O, βi < η and η = sup{βi : 1 ≤ i < τ}. A cardinal η is said to be regular if cf(η) = η. Denote by RC the family of regular cardinals. If A,B are subsets of the ordinal η, we write A < B iff, ∀α ∈ A, ∀β ∈ B, we have α < β. A transfinite sequence {Aα}1≤α<θ, θ ∈ O, of subsets of η is said to be a skipped (transfinite) sequence iff Aα < Aβ , whenever α < β < θ, and for each α < θ, ∃nα < η such that Aα < nα < Aα+1. A subset A ⊆ η is said to be nice if min(A) / ∈ LO. We say that S is a segment of η if S ⊆ η and [κ, λ] ⊆ S whenever κ, λ ∈ S, κ ≤ λ. Note that, if S is a nice segment of η, then S = [α, β), where α = min(S) / ∈ LO and 1991 Mathematics Subject Classification. Primary 46B25; Secondary 46B26