Rethinking polyhedrality for lindenstrauss spaces

Casini, Emanuele; Miglierina, Enrico; Piasecki, Łukasz; Veselý, Libor

doi:10.1007/s11856-016-1412-8

Cited by 17 publications

(18 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…they have finitely many extreme points). There are several versions of polyhedality which have been studied in the literature (see [14,16] and references therein) of which we would like to emphasise the following two, named using the notation of [6]. A real Banach space X is said to be:…”

Section: Definition 36 ([40]mentioning

confidence: 99%

Daugavet property in projective symmetric tensor products of Banach spaces

Martín¹,

Zoca²

2020

Preprint

View full text Add to dashboard Cite

We show that all the symmetric projective tensor products of a Banach space X have the Daugavet property provided X has the Daugavet property and either X is an L 1 -predual (i.e. X * is isometric to an L 1 -space) or X is a vector-valued L 1 -space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise "localised" versions of the Daugavet property (i.e. Daugavet points and ∆-points introduced in [1]) for L 1 -preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real L 1preduals in terms of the absence of ∆-points and also to provide new examples of L 1 -preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces (in the sense of [40]) so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for L 1 -preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in [34] about the Daugavet property for projective tensor products is also got.

show abstract

Section: Definition 36 ([40]mentioning

confidence: 99%

Daugavet property in projective symmetric tensor products of Banach spaces

Martín¹,

Zoca²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Let us suppose that X is a separable Lindenstrauss space with nonseparable dual. Then by Corollary 3.4 in [3] X fails the w * -fpp and it also fails property (ii) by Theorem 2.3 in [16] and Theorem 2.1 in [4]. Therefore we limit ourselves to consider the case where X * is isometric to ℓ 1 .…”

Section: A Characterization Of W * -Fixed Point Property In ℓmentioning

confidence: 99%

Weak$^*$ fixed point property in $\ell _1$ and polyhedrality in Lindenstrauss spaces

et al. 2018

Self Cite

View full text Add to dashboard Cite

The aim of this paper is to study the w * -fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of w * -closed subsets of the dual sphere is equivalent to the w * -fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable w * -fixed point property. The last geometrical notion was introduced by Fonf and Veselý as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be related to a polyhedral concept for the predual space. Indeed, we give a hierarchical structure among various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we obtain an improvement of an old result about the norm-preserving compact extension of compact operators.2010 Mathematics Subject Classification. 47H10, 46B45, 46B25. Key words and phrases. w * -fixed point property, stability of the w * -fixed point property, Lindenstrauss spaces, Polyhedral spaces, ℓ 1 space, Extension of compact operators.

show abstract

“…The class H of all 1 -predual hyperplanes in the space c of convergent sequences turned out to be crucial in metric fixed point theory and polyhedral theory in the framework of L 1 -preduals (see [4,5,9,8,7,12,13,10,6,14]).…”

Section: Introductionmentioning

confidence: 99%

Some topological and metric properties of the space of $\ell _1$-predual hyperplanes in $c$

Gergont

Piasecki

2022

Colloq. Math.

Self Cite

View full text Add to dashboard Cite

We study topological and metric properties of the space (H, d), where H is the set of all 1-predual hyperplanes in the space c of convergent sequences, d denotes the Banach-Mazur distance and we identify hyperplanes that are almost isometric. The space c and its subspace c0 of sequences converging to 0 are the simplest examples of elements in H. First we prove that (H, d) is homeomorphic to (K, d 1 ) with K = {x ∈ 1 : x ≤ 1 and x(i) ≥ x(i + 1) ≥ 0 for all i ∈ N}. We provide optimal lower bounds for the distortions T T −1 of isomorphic embeddings T from an arbitrarily chosen element of H into an infinite-dimensional L1-predual X such that (ext BX * ) ⊂ rBX * for some r ∈ [0, 1). Finally, we apply the above results to construct a homotopy contracting H to c0 along the shortest paths in H and we calculate their lengths. For instance, we show that the shortest path in H joining c and c0 has length ln 4.

show abstract

Rethinking polyhedrality for lindenstrauss spaces

Cited by 17 publications

References 20 publications

Daugavet property in projective symmetric tensor products of Banach spaces

Daugavet property in projective symmetric tensor products of Banach spaces

Weak$^*$ fixed point property in $\ell _1$ and polyhedrality in Lindenstrauss spaces

Some topological and metric properties of the space of $\ell _1$-predual hyperplanes in $c$

Contact Info

Product

Resources

About