Summands in locally almost square and locally octahedral spaces

Hardtke, Jan-David

doi:10.12697/acutm.2018.22.13

Cited by 5 publications

(10 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this subsection we will provide examples of ultrapowers of Banach spaces which are transfinite ASQ. Our motivation comes from [17], where it is proved that, in our language, the ultrapower of a Banach space X is SQ <ℵ 0 if, and only if, X is ASQ.…”

Section: Ultrapowersmentioning

confidence: 99%

Transfinite almost square Banach spaces

Avilés¹,

Ciaci²,

Langemets³

et al. 2023

Studia Math.

View full text Add to dashboard Cite

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)].

show abstract

Section: Ultrapowersmentioning

confidence: 99%

Transfinite almost square Banach spaces

Avilés¹,

Ciaci²,

Langemets³

et al. 2023

Studia Math.

View full text Add to dashboard Cite

show abstract

“…Apart from being interesting by themselves, almost squareness properties have shown to be a powerful tool in order to study D2Ps in certain Banach spaces where there is no good description of the dual space. In this direction, let us mention, for instance, that in [27, Section 4], it is proved that if X is LASQ (respectively, ASQ), then any ultrapower of X is LASQ (respectively, ASQ), and, in particular, has the slice-D2P (respectively, SD2P). Observe that it is unclear whether has the slice-D2P (respectively, SD2P) if X has the slice-D2P (respectively, SD2P).…”

Section: Introductionmentioning

confidence: 99%

“…In particular, is WASQ. On the other hand, Hardtke proved in [28, Theorem 3.1] that the Köthe–Bochner space E ( X ) is LASQ whenever the Banach space X is LASQ, for any Banach function space E .…”

Section: Introductionmentioning

confidence: 99%

On weakly almost square Banach spaces

Rodríguez,

Rueda Zoca

2023

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(\Omega,\Sigma)$ be a measurable space, let E be a Banach lattice and let $\nu:\Sigma \to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(\nu)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue–Bochner space $L_1(\mu,Y)$ is weakly almost square for any Banach space Y and for any non-atomic finite measure µ. On the other hand, we make some progress on the open question of whether there exists a locally almost square Banach space, which fails the diameter two property. In this line, we prove that if X is any Banach space containing a complemented isomorphic copy of c0, then for every $0 \lt \varepsilon \lt 1$ , there exists an equivalent norm $|\cdot|$ on X satisfying the following: (i) every slice of the unit ball $B_{(X,|\cdot|)}$ has diameter 2; (ii) $B_{(X,|\cdot|)}$ contains non-empty relatively weakly open subsets of arbitrarily small diameter and (iii) $(X,|\cdot|)$ is (r, s)-SQ for all $0 \lt r,s \lt \frac{1-\varepsilon}{1+\varepsilon}$ .

show abstract

“…More recent studies about the geometry of ultraproduct Banach spaces can be found in [12] for octahedral and almost square Banach spaces or in [4,17] for the Daugavet property.…”

Section: Introductionmentioning

confidence: 99%

Extremal structure in ultrapowers of Banach spaces

García-Lirola

Guillaume

Zoca

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

View full text Add to dashboard Cite

Given a bounded convex subset C of a Banach space X and a free ultrafilter $${\mathcal {U}}$$ U , we study which points $$(x_i)_{\mathcal {U}}$$ ( x i ) U are extreme points of the ultrapower $$C_{\mathcal {U}}$$ C U in $$X_{\mathcal {U}}$$ X U . In general, we obtain that when $$\{x_i\}$$ { x i } is made of extreme points (respectively denting points, strongly exposed points) and they satisfy some kind of uniformity, then $$(x_i)_{\mathcal {U}}$$ ( x i ) U is an extreme point (respectively denting point, strongly exposed point) of $$C_\mathcal U$$ C U . We also show that every extreme point of $$C_{{\mathcal {U}}}$$ C U is strongly extreme, and that every point exposed by a functional in $$(X^*)_{{\mathcal {U}}}$$ ( X ∗ ) U is strongly exposed, provided that $$\mathcal U$$ U is a countably incomplete ultrafilter. Finally, we analyse the extremal structure of $$C_{\mathcal {U}}$$ C U in the case that C is a super weakly compact or uniformly convex set.

show abstract

Summands in locally almost square and locally octahedral spaces

Abstract: Abstract. We study the question whether properties like local/weak almost squareness and local octahedrality pass down from an absolute sum X ⊕F Y to the summands X and Y .

Cited by 5 publications

References 8 publications

Transfinite almost square Banach spaces

Transfinite almost square Banach spaces

On weakly almost square Banach spaces

Extremal structure in ultrapowers of Banach spaces

Contact Info

Product

Resources

About