A Coding of Bundle Graphs and Their Embeddings Into Banach Spaces

Swift, Andrew

doi:10.1112/s002557931800027x

Cited by 8 publications

(9 citation statements)

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“…It can be easily checked that all the results involving the notion of compatible linear ordering in this paper can be proved by using one of the two conditions above instead and thus that they are compatible with definitions and results either from [20] or [2].…”

Section: Remark 1 For Allmentioning

confidence: 72%

“…One can have a look at [20] for a very detailed survey of properties of bundle graphs (in particular an explicit formula for the distance in such a graph is given and some results of embeddability into Banach spaces are proved). In particular, it was proved in this paper that if one consider the operation ⊘ which consist in replacing every edge of some countably branching bundle graph by another countably branching bundle graph, one gets a new countably branching bundle graph.…”

Section: Bundle Graphs and Embeddability Resultsmentioning

confidence: 99%

“…In particular since every reflexive Banach space is WCG the separability assumption from the original theorem from [20] can be removed. Theorem 5.3.…”

Section: Szlenk Index Of Wcg Spaces Is Determined By Separable Quotientsmentioning

confidence: 99%

“…The proof of the main result consists in pushing forward methods from [2] in order to give informations about the weak * asymptotic structure of dual spaces satisfying the conditions stated in the theorem and then in appealing to known embeddability results. More precisely we show that such a dual space must contain ℓ n ∞ in its n th weak * asymptotic structure for every n and this leads to the construction of a certain family of trees (the so called good-ℓ ∞ trees of arbitrary height) which is known by Swift's work in [20] to allow for the embeddability of the graphs.…”

Section: Introductionmentioning

confidence: 99%

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In this note the result from [20] by A. Swift concerning the embeddability of countably branching bundle graphs into Banach spaces is extended from the context of reflexive spaces with an unconditional asymptotic structure to the context of dual spaces with a weak * unconditional asymptotic structure.

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On the embeddability of the family of countably branching trees into quasi-reflexive Banach spaces

Perreau

2020

Journal of Functional Analysis

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In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk index from the dual are both equal to the first infinite ordinal ω. We also gather results linking these notions with the spreading models of the space. IntroductionLet (T N ) N ≥1 be the family of countably branching trees endowed with the hyperbolic distance. The main result in the present paper is the following theorem.Let us briefly recall the context and the motivation of this theorem. In 1986, J. Bourgain gave in his paper [6] a metric invariant characterizing super-reflexivity: the non equi-Lipschitz embeddability of the family (D N ) N ≥1 of dyadic trees endowed with the hyperbolic distance. His result is the following. Theorem 1.2. Let X be a Banach space. Then X is super-reflexive if and only if the family (D N ) N ≥1 does not equi-Lipschitz embed into X.This was the first step in the so called Ribe program which looks for metric invariants characterizing local properties of Banach spaces. The reader can have a look at [18] for a detailed introduction to the Ribe program and for a survey of results in this direction. A short proof of the non-embeddability of the family of dyadic trees into a super-reflexive space was given more recently by R. Kloeckner in [11] using uniform convexity and a self-improvement argument.In [2], F. Baudier, N. J. Kalton and G. Lancien introduced a new metric invariant in order to give a metric characterization of asymptotic properties of Banach spaces: the non equi-Lipschitz embeddability of the family (T N ) N ≥1 of countably branching trees endowed with the hyperbolic distance. The main tool in their paper is a derivation index called Slzenk index. We will introduce these objects in section 2. They proved the following results.2010 Mathematics Subject Classification. 46B20, 46B80, 46B85, 46T99. The author is supported by the French "Investissements d'Avenir" program, project ISITE-BFC (contract ANR-15-IDEX-03). 1 2 Y. PERREAU Theorem 1.3. Let X be a separable Banach space. If S Z (X) > ω or if S Z (X * ) > ω, then the family (T N ) N ≥1 equi-Lipschitz embeds into X. Theorem 1.4. Let X be a reflexive separable Banach space. If S Z (X) ≤ ω and S Z (X * ) ≤ ω, then the family (T N ) N ≥1 does not equi-Lipschitz embed into X.Note that the assumption of separability can be removed in both theorems by using properties of the Szlenk index. Using an argument à la Kloeckner and the property (β) of Rolewicz, F. Baudier and S. Zhang gave in [4] a shorter proof of the second theorem. However, this argument cannot be extended to a more general setting since (β) implies reflexivity.In [10], N. J. Kalton applied results coming from the study of Orlicz sequence spaces to get estimates on the spreading models of Banach spaces which coarse-Lipschitz embed into asymptotically uniformly convex spaces. Inspired by this method, we will give in section 4 estima...

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Section: Remark 1 For Allmentioning

confidence: 72%

Section: Bundle Graphs and Embeddability Resultsmentioning

confidence: 99%

“…In particular since every reflexive Banach space is WCG the separability assumption from the original theorem from [20] can be removed. Theorem 5.3.…”

Section: Szlenk Index Of Wcg Spaces Is Determined By Separable Quotientsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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