Transfinite almost square Banach spaces

Abstract

Is is known that a Banach space contains an isomorphic copy of c0 if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose of relating them to the containment of isomorphic copies of c0(κ), where κ is some uncountable cardinal. We also provide several examples and stability results for the above properties by taking direct sums, tensor products and ultraproducts. By connecting the above properties with transfinite analogues of the strong diameter 2 property and octahedral norms, we obtain a solution to an open question of Ciaci et al. [Israel J. Math. (online, 2022)].

On the transfinite symmetric strong diameter two property

“…We will later show that ℓ ∞ (κ) has the 1-ASSD2P κ (see Example 3.3), nevertheless, we claim that the Banach space X := ℓ ∞ (κ) ⊗π ℓ 3 3 doesn't enjoy the SSD2P κ .…”

“…Given a sequence of Banach spaces (X n ), it is known that the c 0 sum c 0 (N, X n ) is always ASQ [1, Example 3.1] and, therefore, has the SSD2P. Moreover, it was proved in [3,Proposition 4.3] that the c 0 sum c 0 (A , X α ) of a family of Banach spaces {X α : α ∈ A } is SQ |A | and thus has the 1-ASSD2P |A | , whenever |A | > ℵ 0 . For these reasons, in the following we will focus only on countable c 0 sums with respect to the SSD2P κ , for κ > ℵ 0 .…”

“…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 .…”

“…In [3] transfinite analogues of the SSD2P were defined, but, before recalling these definitions, let us introduce some notation. Given r ∈ (0, 1), B ⊂ B X and A ⊂ S X * , we say that B r-norms A if, for every x * ∈ A, there is x ∈ B such that x * (x) ≥ r. In addition, we say that B norms A if it r-norms it for all r ∈ (0, 1).…”

“…It is clear that every ASQ κ (SQ κ , respectively) Banach space enjoys the SSD2P κ (1-ASSD2P κ , respectively). Moreover, it was shown in [3,Proposition 5.4] that the SSD2P κ (1-ASSD2P κ , respectively) implies the SD2P κ (1-ASD2P κ , respectively). To sum up, the following implications hold true.…”