New applications of extremely regular function spaces

Abrahamsen, Trond A.; Nygaard, Olav; Põldvere, Märt

doi:10.2140/pjm.2019.301.385

Cited by 17 publications

(16 citation statements)

References 15 publications

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“…The previous result motivates the following definition: if M is a metric space which can be written as M = γ∈Γ M γ satisfying (1) and (2) in the statement of the previous proposition for C = 1, we say that M is the 1 -sum of the family {M γ } γ∈Γ .…”

Section: 2])mentioning

confidence: 99%

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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Section: 2])mentioning

confidence: 99%

On strongly norm attaining Lipschitz maps

Cascales

Chiclana

García-Lirola

et al. 2019

Journal of Functional Analysis

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“…(a) Lindenstrauss spaces (this follows by inspecting the proof of Proposition 4.6 in [4]); (b) uniform algebras (see Theorem 4.2 in [3]); (c) ASQ-spaces, in particular, Banach spaces which are M-ideals in their bidual (see [1]); (d) Banach spaces with an infinite-dimensional centralizer (this follows by inspecting the proof of Proposition 3.3 in [6]); (e) somewhat regular linear subspaces of C 0 (L), whenever L is an infinite locally compact Hausdorff topological space [5]; (f) Müntz spaces (this follows by inspecting the proof of Theorem 2.5 in [2]).…”

Section: Introductionmentioning

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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“…In both cases, it is extremely important having a c 0 behaviour in which X has the SD2P because this allows us to construct weakly null sequences in the projective symmetric tensor product (see [11,Lemma 3.1]). Bearing this fact in mind we will introduce the sequential strong diameter two property (sequential SD2P) in Definition 2.6, a stregthening of the SD2P motivated by the symmetric strong diameter two property (see definition below) introduced in [3], proving that if a Banach space X has such property then every projective symmetric tensor product of X has the SD2P in Theorem 2.7. In spite of the fact that the sequential SD2P seems to be quite technical property to check in a Banach space, we will prove that it can be applied to several Banach spaces which are well known to have the SD2P as are infinite-dimensional uniform algebras or Banach spaces with an infinite-dimensional centralizer and whose unit ball contains any extreme point, and also the space Lip 0 (M ) whenever M is a metric space with infinitely-many cluster points.…”

Section: Introductionmentioning

confidence: 99%

“…For else standard notation about Banach spaces we refer to [6]. According to [3] a Banach space X is said to have the symmetric strong diameter two property (SSD2P) if, for every n ∈ N, slices S 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Almost squareness and strong diameter two property in tensor product spaces

Zoca

2020

RACSAM

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We study almost squareness and the strong diameter two property in the setting of projective (symmetric) tensor product. We prove that almost squareness is preserved by taking projective tensor product, providing non-trivial examples of ASQ projective tensor product spaces. Furthermore, we give sufficient conditions for a projective symmetric tensor product to have the strong diameter two property which extend most of the known results and provide new examples of such spaces with the strong diameter two property.

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New applications of extremely regular function spaces

Cited by 17 publications

References 15 publications

On strongly norm attaining Lipschitz maps

On strongly norm attaining Lipschitz maps

Symmetric Strong Diameter Two Property

Almost squareness and strong diameter two property in tensor product spaces

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