Stability constants of the weak⁎ fixed point property for the space ℓ1

Casini, Emanuele; Miglierina, Enrico; Piasecki, Łukasz; Popescu, Roxana

doi:10.1016/j.jmaa.2017.02.039

Cited by 9 publications

(6 citation statements)

References 18 publications

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“…It is worth emphasizing that this estimate is optimal (see Remark 2.8). This result is a generalization of Theorem 3.7 in [6], where some 1 -preduals X isomorphic to c 0 , for which r * (X) = 1, are considered. Moreover, this result complements Theorem 2.1 in [8] and Theorem 4.1 in [8].…”

mentioning

confidence: 62%

A note on the Banach–Mazur distances between c0 and other ℓ1 -preduals

Gergont

2022

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We prove that if X is an 1-predual isomorphic to the space c0 of sequences converging to zero, then for any isomorphism T : X → c0 we have T T −1 ≥ 1 + 2r * (X), where r * (X) is the smallest radius of the closed ball of the dual space X * containing all the weak * cluster points of the set of all extreme points of the closed unit ball of X * . Introduction.Let X be a real infinite-dimensional Banach space X and let us denote by B X its closed unit ball. If A ⊂ X, then ext A stands for the set of all extreme points of A. The dual of X is denoted by X * . If A ⊂ X * , then A * denotes the weak * closure of A and A stands for the set of all weak * cluster points of A:If f ∈ X * , then ker f denotes the kernel of f , i.e., ker f = {x ∈ X : f (x) = 0}. For any Banach spaces X and Y , X = Y means that X is isometrically isomorphic to Y . A Banach space X is called an L 1 -predual (or a Lindenstrauss space) if X * = L 1 (µ) for some measure µ. In particular, X is named an 1 -predual if X * = 1 . For a given 1 -predual X we put r * (X) = inf{r > 0 : (ext B X * ) ⊂ rB X * } = sup{ e * : e * ∈ (ext B X * ) }.

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confidence: 62%

A note on the Banach–Mazur distances between c0 and other ℓ1 -preduals

Gergont

2022

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“…We conclude our paper by pointing out that some other equivalent conditions for the weak * -FPP are known in the literature. We refer the interested reader to [2,4,5,11,12].…”

Section: Weak * Fixed Point Property In the Dual Of Separable Lindens...mentioning

confidence: 99%

Weak$^*$ fixed point property and the space of affine functions

Casini¹,

Miglierina²,

Piasecki³

2019

Preprint

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First we prove that if a separable Banach space X contains an isometric copy of an infinite-dimensional space A(S) of affine continuous functions on a Choquet simplex S, then its dual X * lacks the weak * fixed point property for nonexpansive mappings. Then, we show that the dual of a separable Lindenstrauss space X fails the weak * fixed point property for nonexpansive mappings if and only if X has a quotient isometric to some space A(S). Moreover, we provide an example showing that "quotient" cannot be replaced by "subspace". Finally, it is worth to be mentioned that in our characterization the space A(S) cannot be substituted by any space C(K) of continuous functions on a compact Hausdorff K.

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“…If in Theorem 4.1 we put e * = e * 1 , then we get the statement of [10, Theorem 2.1].Remark 4.3. From the proof of[7, Proposition 3.8] we have d(W e * , c 0 ) ≤ ln(1 + 2 e * ). It was unknown to the authors of[7] whether this estimate is sharp except for the case when e * = 1.…”

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confidence: 92%

“…From the proof of[7, Proposition 3.8] we have d(W e * , c 0 ) ≤ ln(1 + 2 e * ). It was unknown to the authors of[7] whether this estimate is sharp except for the case when e * = 1. However, by applying Theorem 4.1 we immediately conclude that d(W e * , c 0 ) = ln(1 + 2 e * ) for every e * ∈ B 1 .…”

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confidence: 92%

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

Gergont

Piasecki

2022

Colloq. Math.

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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.

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Stability constants of the weak⁎ fixed point property for the space ℓ1

Cited by 9 publications

References 18 publications

A note on the Banach–Mazur distances between c0 and other ℓ1 -preduals

A note on the Banach–Mazur distances between c0 and other ℓ1 -preduals

Weak$^*$ fixed point property and the space of affine functions

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

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