The ℒ p spaces

Lindenstrauss, Joram; Rosenthal, Haskell P.

doi:10.1007/bf02788865

Cited by 283 publications

(141 citation statements)

References 22 publications

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“…This is a natural definition when M is a Banach space, since we can view Lip(M) as a nonlinear substitute for the dual space M * (note that in [37] it is shown that there is a norm 1 projection from Lip(M) onto M * ). With this point of view, the above definition of cotype is natural due to the principle of local reflexivity [39], [30]. Unfortunately, Bourgain [8] has shown that under this definition subsets of L 1 need not have finite nonlinear cotype (while L 1 has cotype 2).…”

Section: Introductionmentioning

confidence: 99%

Metric cotype

2008

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We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L p coarsely or uniformly embeds into L q . We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question posed by Arora, Lovász, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matoušek.

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Section: Introductionmentioning

confidence: 99%

Metric cotype

2008

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“…where Y is an L p -space for some 1 ≤ p ≤ ∞. By [74] there is λ > 0, a net {Y α } of finite-dimensional subspaces of Y such that Y = α Y α and, for each α, an isomorphism…”

Section: Localization Techniquesmentioning

confidence: 99%

Banach space techniques underpinning a theory for nearly additive mappings

Sánchez

Castillo

2002

Dissertationes Math.

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Ulam's stability problem for additive maps and the theory of extensions of Banach spaces have remained so far unaware of each other. To support this assertion it is enough to peruse the survey articles [36,52] In fact, both worlds constitute two faces of the same coin. In this spirit, the purpose of this paper is to progress towards a global understanding of the problem, and we shall do so by developing a theory of vector-valued nearly additive mappings on semigroups. While a unified theory seems to be out of reach by now, the part of the theory involving quasi-normed groups and maps with values in Banach spaces seems to be ripe with these pages. We hasten to alert the reader that this paper can by no means be considered a survey, although it contains occasional historical comments; two excellent survey papers are [36,52].

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“…We shall then apply this theorem to Banach algebras whose underlying Banach spaces are L p -spaces in the sense of [13] with p ∈ (1, ∞).…”

Section: Introductionmentioning

confidence: 99%

Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

Runde

2001

Arch. Math.

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It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If A is a reflexive, amenable Banach algebra such that for each maximal left ideal L of A (i) the quotient A/L has the approximation property and (ii) the canonical map from A⊗L ⊥ to (A/L)⊗L ⊥ is open, then A is finite-dimensional. As an application, we show that, if A is an amenable Banach algebra whose underlying Banach space is an L p -space with p ∈ (1, ∞) such that for each maximal left ideal L the quotient A/L has the approximation property, then A is finite-dimensional.

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The ℒ p spaces

Cited by 283 publications

References 22 publications

Metric cotype

Metric cotype

Banach space techniques underpinning a theory for nearly additive mappings

Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

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