The Daugavet property for Lindenstrauss spaces

Becerra, Julio; Martı́n, Miguel

doi:10.1017/cbo9780511721366.004

Cited by 7 publications

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References 11 publications

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“…Classical examples of Banach spaces fulfilling the Daugavet property are C(K), for every perfect compact Hausdorff topological space K, and L 1 (μ), for every non-atomic measure μ. Since Daugavet's pioneering paper [11], the study of Banach spaces having the Daugavet property has attracted the attention of many authors, and today such a study has attained a flourishing development (see [1][2][3][4]15,16,[19][20][21][22][23]). …”

Section: Introductionmentioning

confidence: 99%

Banach spaces with the Daugavet property, and the centralizer

Guerrero

Rodríguez-Palacios

2008

Journal of Functional Analysis

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We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties:(1) Every member of R has the Daugavet property.(2) It Y is a member of R, then, for every Banach space X, both the space L(X, Y ) (of all bounded linear operators from X to Y ) and the complete injective tensor product X ⊗ Y lie in R. (3) If K is a perfect compact Hausdorff topological space, then, for every Banach space Y , and for most vector space topologies τ on Y , the space C(K, (Y, τ )) (of all Y -valued τ -continuous functions on K) is a member of R. (4) If K is a perfect compact Hausdorff topological space, then, for every Banach space Y , most C(K, Y )-superspaces (in the sense of [V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003) 195-206]) are members of R. (5) All dual Banach spaces without minimal M-summands are members of R.

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Section: Introductionmentioning

confidence: 99%

Banach spaces with the Daugavet property, and the centralizer

Guerrero

Rodríguez-Palacios

2008

Journal of Functional Analysis

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Banach spaces which always produce octahedral spaces of operators

Zoca

2023

Collect. Math.

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We characterise those Banach spaces X which satisfy that L(Y, X) is octahedral for every non-zero Banach space Y. They are those satisfying that, for every finite dimensional subspace Z, $$\ell _\infty $$ ℓ ∞ can be finitely-representable in a part of X kind of $$\ell _1$$ ℓ 1 -orthogonal to Z. We also prove that L(Y, X) is octahedral for every Y if, and only if, $$L(\ell _p^n,X)$$ L ( ℓ p n , X ) is octahedral for every $$n\in {\mathbb {N}}$$ n ∈ N and $$1<p<\infty $$ 1 < p < ∞ . Finally, we find examples of Banach spaces satisfying the above conditions like $${\textrm{Lip}}_0(M)$$ Lip 0 ( M ) spaces with octahedral norms or $$L_1$$ L 1 -preduals with the Daugavet property.

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Daugavet property in projective symmetric tensor products of Banach spaces

Martı́n

Zoca

2022

Banach J. Math. Anal.

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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 $$L_1$$ L 1 -predual (i.e., $$X^{*}$$ X ∗ is isometric to an $$L_1$$ L 1 -space) or X is a vector-valued $$L_1$$ 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 $$\Delta$$ Δ -points introduced in Abrahamsen et al. (Proc Edinb Math Soc 63:475–496 2020)] for $$L_1$$ L 1 -preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real $$L_1$$ L 1 -preduals in terms of the absence of $$\Delta$$ Δ -points and also to provide new examples of $$L_1$$ 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 Werner (J Funct Anal 143:117–128, 1997)] so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for $$L_1$$ L 1 -preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in Rueda Zoca (J Inst Math Jussieu 20(4):1409–1428, 2021) about the Daugavet property for projective tensor products is also obtained.

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The Daugavet property for Lindenstrauss spaces

Cited by 7 publications

References 11 publications

Banach spaces with the Daugavet property, and the centralizer

Banach spaces with the Daugavet property, and the centralizer

Banach spaces which always produce octahedral spaces of operators

Daugavet property in projective symmetric tensor products of Banach spaces

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