( β )-distortion of some infinite graphs

Journal of Functional Analysis

Dilworth

et al. 2017

Self Cite

In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs $(D_k^\omega)_{k\in\mathbb{N}}$ is investigated. In particular it is shown that for every $\varepsilon>0$ and $k\in\mathbb{N}$, $D_k^\omega$ embeds bi-Lipschiztly with distortion at most $6(1+\varepsilon)$ into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence $(D_k^\omega)_{k\in\mathbb{N}}$ does not admit an equi-bi-Lipschitz embedding into any Banach space that has an equivalent asymptotically midpoint uniformly convex norm. Combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces, within the class of separable reflexive Banach spaces with an unconditional asymptotic structure. Applications to bi-Lipschitz embeddability into $L_p$-spaces and to some problems in renorming theory are also discussed.Comment: 44 pages, 3 tables, 2 figure

Section: Non-embeddability Of the Countably Branching Diamond Graphsmentioning

confidence: 90%

On the geometry of the countably branching diamond graphs

Baudier

Journal of Functional Analysis

Dilworth

et al. 2017

Self Cite

“…The following argument is a modification of the elegant argument of Baudier and Zheng [1], which is itself a modification of Kloeckner's argument [13] for the non-embeddability of binary trees into uniformly convex spaces. By Lemma 3.3 and the choice of y * * , there exists y * * 0 ∈ s 2 n −l 4ε (A * * B X * * ) ∩ V and y * ∈ B Y * ∩ U such that Re y * * 0 (y * ) > ε − δ.…”

Section: Property (β) and Non-linear Characterizationsmentioning

confidence: 99%

ξ-asymptotically uniformly smooth, ξ-asymptotically uniformly convex, and (β) operators

Journal of Functional Analysis

Dilworth

2018

For each ordinal ξ, we define the notion of ξ-asymptotically uniformly smooth and w * -ξ-asymptotically uniformly convex operators. When ξ = 0, these extend the notions of asymptotically uniformly smooth and w * -asymptotically uniformly convex Banach spaces. We give a complete description of renorming results for these properties in terms of the Szlenk index of the operator, as well as a complete description of the duality between these two properties. We also define the notion of an operator with property (β) of Rolewicz which extends the notion of property (β) for a Banach space. We characterize those operators the domain and range of which can be renormed so that the operator has property (β) in terms of the Szlenk index of the operator and its adjoint.

“…If p = ∞, we replace the ℓ p direct sum with the c 0 direct sum. Then S 1 ∞ n=1 ε n , S 3 2 ∞ n=1 ε n , and…”

mentioning

confidence: 99%

Power type asymptotically uniformly smooth and asymptotically uniformly flat norms

2018

Positivity

We provide a short characterization of p-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for 1 < p < ∞, the class of p-asymptotically uniformly smoothable operators can be endowed with an ideal norm making this class a Banach ideal. We also prove that the class of asymptotically uniformly flattenable operators can be endowed with an ideal norm making this class a Banach ideal.