Stability of average roughness, octahedrality, and strong diameter $2$ properties of Banach spaces with respect to absolute sums

Abstract

We prove that, if Banach spaces X and Y are δaverage rough, then their direct sum with respect to an absolute norm N is δ/N (1, 1)-average rough. In particular, for octahedral X and Y and for p in (1, ∞) the space X ⊕ p Y is 2 1−1/p -average rough, which is in general optimal. Another consequence is that for any δ in (1, 2] there is a Banach space which is exactly δ-average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the absolute sums, the diametral strong diameter 2 property is stable only for 1-and ∞-sums.2010 Mathematics Subject Classification. Primary 46B20, 46B22.

Summands in locally almost square and locally octahedral spaces

“…They prove that X ⊕ F Y is octahedral whenever X and Y are octahedral and F is positively octahedral, and, conversely, if X ⊕ F Y is octahedral for some nontrivial Banach spaces X and Y , then F has to be positively octahedral. Analogous results for the SD2P in absolute sums are also obtained in [14].…”

“…However, this result already 10 2. Results and proofs follows from the more general results on octahedrality in absolute sums that were proved in [14] and that we have already mentioned in the introduction.…”

“…In the recent paper [14] the stability of average roughness (which is a generalisation of octahedrality) with respect to absolute sums is investigated. The authors of [14] also introduce the notion of positive octahedrality for an absolute, normalised norm F on R 2 , meaning that there exist c, d ≥ 0 with F (c, d) = 1 and F (c + 1, d) = F (c, d + 1) = 2.…”

“…In the recent paper [14] the stability of average roughness (which is a generalisation of octahedrality) with respect to absolute sums is investigated. The authors of [14] also introduce the notion of positive octahedrality for an absolute, normalised norm F on R 2 , meaning that there exist c, d ≥ 0 with F (c, d) = 1 and F (c + 1, d) = F (c, d + 1) = 2. They prove that X ⊕ F Y is octahedral whenever X and Y are octahedral and F is positively octahedral, and, conversely, if X ⊕ F Y is octahedral for some nontrivial Banach spaces X and Y , then F has to be positively octahedral.…”

Symmetric Strong Diameter Two Property

“…We start our investigation in Section 2 by giving equivalent formulations of the SSD2P, which are often more convenient to use. Recently in [13], it was proven that the SD2P is preserved by a lot of absolute normalized norms. However, in Section 3, we show that the only direct sums of Banach spaces that can have the SSD2P are the ℓ ∞ -sums.…”