Complementation and decompositions in some weakly Lindelöf Banach spaces

Abstract

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 K s such that C (K ) is Lindelöf in the weak topology and we restrict our attention to K s 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 cI + 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 ).

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

“…The first named author and Zieliński have shown that the question "Does this class contain a space on which every bounded linear operator is the sum of a scalar multiple of the identity and an operator with separable range?" is undecidable (see Corollary 3.3 and Theorem 4.1 of [36]). Note that this result does not apply to any space of the form C(αK A ) for an almost disjoint family A ⊆ [N] ω because these spaces are known not to be Lindelöf in their weak topology.…”

On complemented copies of $c_0(\omega _1)$ in $C(K^n)$ spaces

“…In order to establish Theorem 3.1, we need to enrich our terminology. Following the notation of [10], for an ordinal α ≤ ω 1 we put F 0 (α) = α = {β : β < α} and for n > 0 we let F n+1 (α) be the set of all finite sequences of elements of F n (α). Define F (α) = n∈N F n (α).…”

“…The following should be compared with the fact that our space K 1 or the space of [10] are 2-diverse and not weakly pcc by Proposition 3.1 and Theorem 1.7.…”

“…For additional properties of the K n s see Theorem 1.8. In fact our constructions are generalizations of the space from [10] which can serve above as K 1 . As we note in Corollary (4.2) that for K compact scattered if C(K) is weakly pcc, then C(K n ) is weakly pcc for every n, our examples K n for n < ω from Theorem 1.2 give the following:…”

“…Corollary 2.2. Suppose K is the space obtained under the assumption ♣ in Section 4 of [10]. C(K) is half-pcc and contains an isomorphic copy of c 0 (ω 1 ).…”