A characterization of Banach spaces containing ℓ1(κ) via ball-covering properties

A renorming characterization of Banach spaces containing $\ell_1 (\kappa)$

“…In the recent preprint [1], a characterisation in the same spirit was obtained for copies of 1 (κ). In order to present this characterisation we need a bit of language.…”

On the transfinite symmetric strong diameter two property

“…Since ℓ 3 3 is not finitely representable in ℓ 1 , so it is not finitely representable in ℓ 1 (κ) either (notice that each finite dimensional subspace of ℓ 1 (κ) is isometrically isomorphic to some finite dimensional subspace of ℓ 1 , and vice-versa). Thanks to a simple transfinite analogue of [11, Lemma 3.7] (replacing finite dimensional spaces with spaces of density < κ) we conclude that ℓ 1 (κ) ⊗ε ℓ 3 3 * is not κ-octahedral (see [4,Definition 5.3]), moreover, we can infer that X = (ℓ 1 (κ) ⊗ε ℓ 3 3 * ) * [12, Theorem 5.3]. Therefore, by applying [5, Theorem 3.2], we conclude that X fails the SSD2P κ .…”

Transfinite almost square Banach spaces

“…We will be specially interested in the connection between transfinite versions of almost squareness and octahedrality, because, as a consequence of our work, we will solve an open question from [9]. In order to do so, let us start with the following definition from [9]. Definition 5.1 (see [9,Definitions 2.3 and 5.3]).…”

“…In order to do so, let us start with the following definition from [9]. Definition 5.1 (see [9,Definitions 2.3 and 5.3]). Let X be a Banach space and κ an uncountable cardinal.…”

“…At this point, it is natural to look for geometric characterisations of the containment of non-separable versions of ℓ 1 and c 0 . In this spirit, as far as the containment of ℓ 1 (κ) is concerned, transfinite generalisations of octahedral norms were introduced in [9] in various directions and some characterisations of the containment of ℓ 1 (κ) were obtained in [6,9]. To mention the strongest known result, it is proved in [6,Theorem 1.3] that a Banach space X contains an isomorphic copy of ℓ 1 (κ), where κ is an uncountable cardinal, if, and only if, there exists an equivalent norm ||| • ||| such that (X, ||| • |||) fails the (−1)-ball covering property for cardinals < κ ((−1)-BCP <κ , for short), which means that, given any subspace Y ⊂ X such that dens(Y ) < κ, there exists x ∈ S (X,|||•|||) such that |||y + rx||| = |||y||| + |r| for all y ∈ Y and r ∈ R.…”