Uniform embeddings of Banach spaces

Aharoni, Israel

doi:10.1007/bf02761666

Cited by 8 publications

(4 citation statements)

References 8 publications

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“…Building on work of I. Aharoni, B. Maurey and B. S. Mityagin [3], we show the following equivalence. Restricting to Banach spaces, we have a stronger result, where the equivalence of (1) and (2) is due to N. L. Randrianarivony [51].…”

Section: Question 1 Does the Following Equivalence Hold For All (Sepmentioning

confidence: 99%

“…On the other hand, for all g ∈ G and k, we have φ n − λ(g)φ n ∈ 2W n ⊆ 1 2 k−1 U n,k . Therefore, (2) φ n − λ(g)φ n n,k 1 2 k−1 , for all k and g.…”

Section: Theorem 56 Suppose a Topological Group G Carries A Continuomentioning

confidence: 99%

“…A question first raised by Aharoni [2] (see also the discussion in Chapter 8 [11]) is to determine the class of Banach spaces X uniformly embeddable into their unit ball B X . It follows from [1] that c 0 embeds uniformly into B c 0 and, in fact, every separable Banach space X containing c 0 embeds uniformly into its ball B X .…”

Section: Definition 50 a Topological Group G Is Metrically Stable Ifmentioning

confidence: 99%

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Equivariant Geometry of Banach Spaces and Topological Groups

Rosendal

2017

Forum of Mathematics, Sigma

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Section: Question 1 Does the Following Equivalence Hold For All (Sepmentioning

confidence: 99%

“…On the other hand, for all g ∈ G and k, we have φ n − λ(g)φ n ∈ 2W n ⊆ 1 2 k−1 U n,k . Therefore, (2) φ n − λ(g)φ n n,k 1 2 k−1 , for all k and g.…”

Section: Theorem 56 Suppose a Topological Group G Carries A Continuomentioning

confidence: 99%

Section: Definition 50 a Topological Group G Is Metrically Stable Ifmentioning

confidence: 99%

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Equivariant Geometry of Banach Spaces and Topological Groups

Rosendal

2017

Forum of Mathematics, Sigma

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“…In contrast to Corollary 6.3, Aharoni, Maurey and Mitjagin [2] have shown that Lo(/n) is, however, uniformly homeomorphic to a subset of Hilbert space. See also Aharoni [1] for related results on uniform equivalence to subsets of L p , 1 < p.…”

Section: \E(c)mentioning

confidence: 99%

Linearization of certain uniform homeomorphisms

Weston¹

2007

J. Aust. Math. Soc.

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This article concerns the uniform classification of infinite dimensional real topological vector spaces. We examine a recently isolated linearization procedure for uniform homeomorphisms of the form : X -> Y, where X is a Banach space with non-trivial type and Y is any topological vector space. For such a uniform homeomorphism , we show that Y must be normable and have the same supremal type as X. That Y is normable generalizes theorems of Bessaga and Enflo. This aspect of the theory determines new examples of uniform non-equivalence. That supremal type is a uniform invariant for Banach spaces is essentially due to Ribe. Our linearization approach gives an interesting new proof of Ribe's result.

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Roundness and Metric Type

Lennard

Tonge

Weston

2000

Journal of Mathematical Analysis and Applications

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We prove that if X is a Banach space containing l n uniformly in n, and if Y is a p metric space with metric type q ) p, then the inverse of any uniform homeomorphism T from X onto Y cannot satisfy a Lipschitz condition for large distances of order ␣ -qrp. It follows that if Y is a midpoint-convex subset of a Banach space Z with type q larger than the type supremum of a Banach space X, then X and Y cannot be uniformly homeomorphic. In particular, we prove the non-existence of uniform homeomorphisms between certain non-commutative L -spaces and midp point-convex subsets of another such space. We also prove that if a Banach space X has cotype infimum q larger than two, then it has maximal generalized roundness zero and maximal roundness at most q X . As a consequence, infinite-dimensional C U -algebras are seen to have maximal generalized roundness zero and maximal roundness one. ᮊ

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Uniform embeddings of Banach spaces

Cited by 8 publications

References 8 publications

Equivariant Geometry of Banach Spaces and Topological Groups

Equivariant Geometry of Banach Spaces and Topological Groups

Linearization of certain uniform homeomorphisms

Roundness and Metric Type

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