Almost Isometric Ideals in Banach Spaces

Abrahamsen, Trond A.; Lima, Vegard; Nygaard, Olav

doi:10.1017/s0017089513000335

Cited by 20 publications

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“…Remark 3.6. The argument shows that the conclusion in Lemma 3.5 also holds in the more general setting of X being an almost isometric ideal (see [4] for a definition) in Z, replacing X * * with Z. Theorem 3.7. Let X be an (infinite dimensional) L 1 (µ)-predual and x ∈ S X .…”

Section: ∆And Daugavet-points For Different Classes Of Spacesmentioning

confidence: 86%

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Delta- and Daugavet points in Banach spaces

Abrahamsen¹,

Haller²,

Lima³

et al. 2020

Proceedings of the Edinburgh Mathematical Society

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A ∆-point x of a Banach space is a norm one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet-point. A Banach space X has the Daugavet property if and only if every norm one element is a Daugavet-point.We show that ∆-and Daugavet-points are the same in L 1spaces, L 1 -preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are ∆-points, but where none of them are Daugavetpoints.We also study the property that the unit ball is the closed convex hull of its ∆-points. This gives rise to a new diameter two property that we call the convex diametral diameter two property. We show that all C(K) spaces, K infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums. 2010 Mathematics Subject Classification. Primary 46B20, 46B22, 46B04. Key words and phrases. Diametral diameter two property, Daugavet property, L 1 -space, L 1 -predual space, Müntz space. R. Haller and K. Pirk were partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. 1 2 T. A. ABRAHAMSEN, R. HALLER, V. LIMA, AND K. PIRKWe will sometimes need to clarify which Banach space we are working with and write ∆ X ε (x) and ∆ X instead of ∆ ε (x) and ∆ respectively. The starting point of this research was the discovery that if a Banach space X satisfies B X = conv ∆, then X has the LD2P.We study spaces that satisfy the property B X = conv ∆ in Section 5. The case S X = ∆, i.e. x ∈ conv ∆ ε (x) for all x ∈ S X and ε > 0, has already appeared in the literature, but under different names: The diametral local diameter two property (DLD2P) ([5]), the LD2P+ ([1] and [2]), and space with bad projections ([13]). We will use the term DLD2P in this paper. From [18, Corollary 2.3 and (7) p. 95] and [13, Theorem 1.4] the following characterization is known. Proposition 1.1. Let X be a Banach space. The following assertions are equivalent:(1) X has the DLD2P;(2) for all x ∈ S X we have x ∈ conv ∆ ε (x) for all ε > 0;(3) for all projections P : X → X of rank-1 we have Id − P ≥ 2.Related to the DLD2P is the Daugavet property. We have (cf. [18, Corollary 2.3]): Proposition 1.2. Let X be a Banach space. The following assertions are equivalent:(1) X has the Daugavet property, i.e. for all bounded linear rank-1 operators T : X → X we have Id − T = 1 + T ; (2) for all x ∈ S X we have B X = conv ∆ ε (x) for all ε > 0.

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Section: ∆And Daugavet-points For Different Classes Of Spacesmentioning

confidence: 86%

“…(3) for every nonzero x * ∈ X * , the rank-1 operator T = x * ⊗ x satisfies Id − T = 1 + T ; (4) for every x * ∈ S X * the rank-1 norm-…”

Section: Preliminariesmentioning

confidence: 99%

Delta- and Daugavet points in Banach spaces

Abrahamsen¹,

Haller²,

Lima³

et al. 2020

Proceedings of the Edinburgh Mathematical Society

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“…is nonempty and contains a sequence (U k ) ∞ k=1 of pairwise disjoint subsets. 4 Let g k ≥ 0 be such that supp g k ⊆ U k and g k = 1.…”

Section: Characterization Of the Ssd2pmentioning

confidence: 99%

“…Let X be a Banach space and Y a subspace of X. Following [4] we say that Y is an almost isometric ideal (ai-ideal) in X if for every finitedimensional subspace E of X and every ε > 0 there exists a bounded linear operator T : E → Y such that (1 − ε) e ≤ T e ≤ (1 + ε) e and T e = e for all e ∈ E ∩ Y .…”

Section: Subspaces With the Ssd2pmentioning

confidence: 99%

Symmetric Strong Diameter Two Property

et al. 2019

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We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and Põldvere.The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2). We show that the symmetric strong diameter two property is only preserved by ℓ ∞ -sums, and working with weak star slices we show that Lip 0 (M ) have the weak star version of the property for several classes of metric spaces M .2010 Mathematics Subject Classification. Primary 46B20, 46B22. Key words and phrases. strong diameter 2 property, almost square spaces, Lipschitz spaces.R. Haller, J. Langemets, and R. Nadel were partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research.

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“…Note that the Principle of Local Reflexivity means that X is an aiideal in X * * for every Banach space X. Moreover, the Daugavet property, octahedrality and all of the diameter two properties are inherited by ai-ideals (see [1] and [2]).…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Octahedral norms in tensor products of Banach spaces

Langemets

Lima

Zoca

2017

The Quarterly Journal of Mathematics

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We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space Y such that the injective tensor products l 1 ⊗ ε Y and L 1 ⊗ ε Y both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive L-embedded Banach space taking projective tensor products with an arbitrary Banach space.2010 Mathematics Subject Classification. 46B20, 46B04, 46B25, 46B28.

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Almost Isometric Ideals in Banach Spaces

Cited by 20 publications

References 20 publications

Delta- and Daugavet points in Banach spaces

Delta- and Daugavet points in Banach spaces

Symmetric Strong Diameter Two Property

Octahedral norms in tensor products of Banach spaces

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