Embeddings of non-commutative L p -spaces into non-commutative L 1 -spaces, 1 &lt; p &lt; 2

Junge, Marius

doi:10.1007/s000390050012

Cited by 12 publications

(8 citation statements)

References 17 publications

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“…It remains a prominent question to give a complete generalization of this result to the setting of non-commutative L p -spaces. See, for example, Junge [15].…”

Section: Introduction and Synopsismentioning

confidence: 99%

Strict p-negative type of a metric space

Weston

2009

Positivity

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Abstract. Doust and Weston [8] have introduced a new method called enhanced negative type for calculating a non-trivial lower bound ℘ T on the supremal strict p-negative type of any given finite metric tree (T, d). In the context of finite metric trees any such lower bound ℘ T > 1 is deemed to be non-trivial. In this paper we refine the technique of enhanced negative type and show how it may be applied more generally to any finite metric space (X, d) that is known to have strict p-negative type for some p ≥ 0. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given in [8, Corollary 5.5] and, moreover, leads in to one of our main results: The supremal p-negative type of a finite metric space cannot be strict. By way of application we are then able to exhibit large classes of finite metric spaces (such as finite isometric subspaces of Hadamard manifolds) that must have strict p-negative type for some p > 1. We also show that if a metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q ∈ [0, p). This generalizes Schoenberg [27, Theorem 2] and leads to a complete classification of the intervals on which a metric space may have strict p-negative type.

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“…It remains a prominent question to give a complete generalization of this result to the setting of non-commutative L p -spaces. See, for example, Junge [15].…”

Section: Introduction and Synopsismentioning

confidence: 99%

Strict p-negative type of a metric space

Weston

2009

Positivity

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“…There are also results along these lines which deal with the less tractable (commutative) case p > 2, and with certain of the non commutative L p -spaces. See, for example, the papers of Koldobsky and König [17], and Junge [16], respectively. General references on the interplay between p-negative type inequalities and isometric embeddings include Deza and Laurent [8], and Wells and Williams [31].…”

Section: Introduction and Synopsismentioning

confidence: 99%

Enhanced negative type for finite metric trees

Doust

Weston

2008

Journal of Functional Analysis

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Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities that quantify the extent of the "strictness" of the 1-negative type inequalities for finite metric trees. These inequalities of "enhanced 1-negative type" are sufficiently strong to imply that any given finite metric tree must have strict p-negative type for all values of p in an open interval that contains the number 1. Moreover, these open intervals can be characterized purely in terms of the unordered distribution of edge weights that determine the path metric on the particular tree, and are therefore largely independent of the tree's internal geometry. From these calculations we are able to extract a new non linear technique for improving lower bounds on the maximal p-negative type of certain finite metric spaces. Some pathological examples are also considered in order to stress certain technical points.Comment: 35 pages, no figures. This is the final version of this paper sans diagrams. Please note the corrected statement of Theorem 4.16 (and hence inequality (1)). A scaling factor was omitted in Version #

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“…Recent work in this line of research can be found for instance in [13,18,32,35,36]. The recent survey article [29] provides an up-to-date information on the latest developments.…”

Section: Introductionmentioning

confidence: 97%

“…181-215] for more information on this line of research. As in the classical case, the method of the proof in [18] is probabilistic. Theorem 1.1 says a lot more than the classical situation.…”

Section: Introductionmentioning

confidence: 98%

Embeddings of non-commutative L p -spaces into preduals of finite von Neumann algebras

Randrianantoanina

2008

Isr. J. Math.

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Let R be a (not necessarily semi-finite) σ-finite von Neumann algebra. We prove that there exists a finite von Neumann algebra N so that for every 1 < p < 2, the Haagerup L p -space associated with R embeds isomorphically into N * . We also provide a proof of the following non-commutative generalization of a classical result of Rosenthal: if M is a semi-finite von Neumann algebra then every reflexive subspace of M * embeds isomorphically into L r (M) for some r > 1.

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Embeddings of non-commutative L p -spaces into non-commutative L 1 -spaces, 1 < p < 2

Cited by 12 publications

References 17 publications

Strict p-negative type of a metric space

Strict p-negative type of a metric space

Enhanced negative type for finite metric trees

Embeddings of non-commutative L p -spaces into preduals of finite von Neumann algebras

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