Projectional skeletons and Markushevich bases

Abstract

We prove that Banach spaces with a 1‐projectional skeleton form a scriptP‐class and deduce that any such space admits a strong Markushevich basis. We provide several equivalent characterizations of spaces with a projectional skeleton and of spaces having a commutative one. We further analyze known examples of spaces with a noncommutative projectional skeleton and compare their behavior with the commutative case. Finally, we collect several open problems.

Smooth and polyhedral norms via fundamental biorthogonal systems

“…Such a class of Banach spaces contains all WLD Banach spaces, hence all WCG spaces and in particular all reflexive ones; besides, every L 1 (µ) space and every C(K) space, where K is a Valdivia compactum, is a Plichko space (see, e.g., [37, §6.2 and §5.1]). More generally Kalenda [38] recently proved that every Banach space with a projectional skeleton admits a (strong) M-basis. Among the Banach spaces that admit a projectional skeleton we could additionally mention duals of Asplund spaces [39], preduals of Von Neumann algebras [6], or preduals of JBW * -triples [7].…”

“…Among the Banach spaces that admit a projectional skeleton we could additionally mention duals of Asplund spaces [39], preduals of Von Neumann algebras [6], or preduals of JBW * -triples [7]. Additionally, there are several examples of concrete Banach spaces where a fundamental biorthogonal system can be constructed, for example: ℓ ∞ (Γ) for every set Γ [10], ℓ c ∞ (Γ) when |Γ| c [22,49], or C([0, η]) for every ordinal η (this is standard, see for example [38,Proposition 5.11]). More generally, C(T ) has an M-basis, for every tree T , [38, §5.3].…”

Characterization of (semi-)Eberlein compacta using retractional skeletons

“…Given a retractional skeleton (r s ) s∈Γ on a compact space K, it is wellknown that (P s ) s∈Γ defined by P s (f ) := f • r s , s ∈ Γ , is a projectional skeleton on C(K) (see e.g. [21,Proposition 5.3]). Moreover, if ∅ = A ⊂ C(K) is a bounded set and (P s ) s∈Γ is A-shrinking in the sense of [15,Definition 16], it is not very difficult to observe that (r s ) s∈Γ is A-shrinking in the sense of Definition 28.…”

Weakly Corson compact trees

“…For an element t ∈ T , cf(t) denotes the cofinality of ht(t, T ), where cf(t) = 0 if ht(t, T ) is a successor ordinal or ht(t, T ) = 0. Moreover, ims(t) = {s ∈ T : t s, ht(s, T ) = ht(t, T )+1} denotes the set of immediate successors of t. Finally, borrowing the notation from [7] we denote I(T ) = {t ∈ T : cf(t) < ω}.…”

“…The coarse wedge topology on a tree T is the one whose subbase is given by the sets V t and their complements, where t ∈ I(T ). We denote the coarse wedge topology by τ cw ; for further information on it, we refer to [7], [13], or [14]. Let us mention in passing that the coarse wedge topology also coincides with the path topology, [16], [2].…”