Property (quasi‐<i>α</i>) and the denseness of norm attaining mappings

Choi, Yun Sung; Song, Hyun Gwi

doi:10.1002/mana.200510676

Cited by 12 publications

(25 citation statements)

References 14 publications

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“…(10). It follows from Proposition 3.19, whose proof is based on that of [11,Proposition 2.10], where it is proved that property quasi-α implies property A. (11).…”

Section: 3mentioning

confidence: 95%

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On strongly norm attaining Lipschitz maps

Cascales

Chiclana

García-Lirola

et al. 2019

Journal of Functional Analysis

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We study the set SNA(M, Y ) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which (strongly) attain their Lipschitz norm (i.e. the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when M is a length space (or local) or when M is a closed subset of R with positive Lebesgue measure, providing new examples which have very different topological properties than the previously known ones. On the other hand, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space F (M ) over M , and show that all of them actually provide the norm density of SNA(M, Y ) in the space of all Lipschitz maps from M to any Banach space Y . Next, we prove that SNA(M, R) is weakly sequentially dense in the space of all Lipschitz functions for all metric spaces M . Finally, we show that the norm of the bidual space of F (M ) is octahedral provided the metric space M is discrete but not uniformly discrete or M is infinite.J. Lindenstrauss extended such study to general linear operators, showed that this is not always possible, and also gave positive results. If we say that a Banach space X has (Lindenstrauss) property A when NA(X, Y ) = L(X, Y ) for every Banach space Y , it is shown in [36] that reflexive spaces have property A. This result was extended by J. Bourgain [9] showing that Banach spaces X with the RNP also have Lindenstrauss property A, and he also provided a somehow reciprocal result. We refer the interested reader to the survey paper [3] for a detailed account on norm attaining linear operators.

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“…(10). It follows from Proposition 3.19, whose proof is based on that of [11,Proposition 2.10], where it is proved that property quasi-α implies property A. (11).…”

Section: 3mentioning

confidence: 95%

“…Property quasi-α. In [11] it is defined a property which, in spite of being weaker than property α, still implies property A. As in the case of property α, we have slightly modified the original definition to an equivalent one which requires the set {x λ } λ∈Λ ⊆ X bellow to be balanced.…”

Section: 3mentioning

confidence: 99%

On strongly norm attaining Lipschitz maps

Cascales

Chiclana

García-Lirola

et al. 2019

Journal of Functional Analysis

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“…Motivated by it, W. Schachermayer [23] considered the so called property α and showed this property implies property A. Besides, more recently it was introduced property quasi-α and property quasi-β and it was observed that those properties were, in fact, new sufficient conditions for properties A and B, respectively (for more details we recommend [2,9]).…”

Section: Consider the Pointmentioning

confidence: 99%

“…Our goal in the present work has been twofold: on the one hand, we consider group invariant separation theorems motivated by the recent paper [13], where a version of the Hanh-Banach extension theorem for group invariant functionals was provided (see also [5]); on the other hand, we consider the validity of relevant results in the theory of normattaining operators with the extra assumption that the involved operators are also group invariant (we refer the reader to [1,2,8,9,17,18,20,23,24,26]).…”

Section: Introductionmentioning

confidence: 99%

“…These properties will be used in Sections 3 and 4, where we present the main results of the manuscript. More specifically, in Section 3, we consider separation results related to the Hanh-Banach separation theorems; in Section 4, we extend some results from the theory of norm-attaining operators to the context of G-invariant mappings: we start by considering some of the classical counterexamples in the theory in order to see whether these counterexamples remain valid or not depending on the specific choice of the group G. Furthermore, we provide the analogous to properties α of Schachermayer [23] and β of Lindenstrauss [20] (as well as their generalizations, properties quasi-α and quasi-β (see [9])). Finally, we define and study the G-Bishop-Phelps property, which is G-invariant version of the Bishop-Phelps property introduced by Bourgain [8], and show that the Radon-Nikodým property is sufficient to ensure the G-Bishop-Phelps property.…”

Section: Introductionmentioning

confidence: 99%

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Group invariant operators and some applications on norm-attaining theory

Dantas¹,

Falcó²,

Jung³

2021

Preprint

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In this paper, we study geometric properties of the set of group invariant continuous linear functionals and operators between Banach spaces. In particular, we present group invariant versions of the Hahn-Banach separation theorem and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators, establishing group invariant versions of the properties α of Schachermayer and β of Lindenstrauss, and presenting relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain's result, establishing that if X has the Radon-Nikodým property, then X has the G-Bishop-Phelps theorem for G-invariant operators whenever G is a compact group.

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Norm-Attaining Tensors and Nuclear Operators

et al. 2022

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Given two Banach spaces X and Y , we introduce and study a concept of norm-attainment in the space of nuclear operators N (X, Y ) and in the projective tensor product space X ⊗ π Y . We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in N (X, Y ) and in X ⊗ π Y is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in X ⊗ π Y which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of

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Property (quasi‐α) and the denseness of norm attaining mappings

Cited by 12 publications

References 14 publications

On strongly norm attaining Lipschitz maps

On strongly norm attaining Lipschitz maps

Group invariant operators and some applications on norm-attaining theory

Norm-Attaining Tensors and Nuclear Operators

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