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

Randrianantoanina, Narcisse

doi:10.1007/s11856-008-0001-x

Cited by 7 publications

(5 citation statements)

References 35 publications

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“…However, this version was only partially satisfactory. A further result is given in [138], but a fully satisfactory non-commutative analogue of Rosenthal's result was finally achieved by Junge and Parcet in [70]. These authors went on to study many related problems involving non-commutative Maurey factorizations, (also in the operator space framework see [71]).…”

Section: Maurey Factorizationmentioning

confidence: 98%

Grothendieck’s Theorem, past and present

Pisier

2012

Bull. Amer. Math. Soc.

213

195

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Abstract. Probably the most famous of Grothendieck's contributions to Banach space theory is the result that he himself described as "the fundamental theorem in the metric theory of tensor products". That is now commonly referred to as "Grothendieck's theorem" ("GT" for short), or sometimes as "Grothendieck's inequality". This had a major impact first in Banach space theory (roughly after 1968), then, later on, in C * -algebra theory (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of "operator spaces" or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell's inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by "semidefinite programming" and hence solved in polynomial time. This expository paper (where many proofs are included), presents a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author's 1986 CBMS notes.

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Section: Maurey Factorizationmentioning

confidence: 98%

Grothendieck’s Theorem, past and present

Pisier

2012

Bull. Amer. Math. Soc.

213

195

View full text Add to dashboard Cite

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“…In [43], Sukochev and Xu studied when L p (N ) embeds into L p (M) for 0 < p < 1, where M and N are semifinite von Neuman algebras. We also refer [37] and [35] for related work. Despite this remarkable development the case when S m q embeds isometrically into S n p is not well studied for 0 < p = q ≤ ∞.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach spaces

Chattopadhyay¹,

Hong²,

Pradhan³

et al. 2022

Preprint

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The existence of isometric embedding of S m q into S n p , where 1 ≤ p = q ≤ ∞ and m, n ≥ 2 has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above mentioned range of p and q. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space ℓ m q (R) into ℓ n p (R), where (q, p) ∈ (0, ∞) × (0, 1) and p = q. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of S m q into S n p , where (q, p) ∈ (0, 2) × (0, 1) ∪ {1} × (0, 1) \ { 1 n : n ∈ N} ∪ {∞} × (0, 1) \ { 1 n : n ∈ N} and p = q. Moreover, in some restrictive cases, we also show that there is no isometric embedding of S m q into S n p , where (q, p) ∈ [2, ∞) × (0, 1). To achieve our goal we significantly use Kato-Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

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“…This follows from the last statement of Corollary 2.22 and Remark 2.23, since by [9, Corollary V.5.2] if X is an infinite dimensional Banach space X which admits a C 2 -smooth bump and is saturated with subspaces of cotype 2, then X is isomorphic to a Hilbert space. generalizes to non-commutative L p -spaces, [20] and [37]. However non-commutative L p -spaces are not stable in general [29].…”

Section: 2mentioning

confidence: 99%

“…We do not know whether Corollary 2.22 generalizes to the setting of non-commutative L p -spaces. Rosenthal's theorem used in the proof of Corollary 2.22 generalizes to non-commutative L p -spaces, [20] and [37]. However non-commutative L p -spaces are not stable in general [29].…”

Section: Convex Transitive Spacesmentioning

confidence: 99%

On an isomorphic Banach–Mazur rotation problem and maximal norms in Banach spaces

Dilworth

Randrianantoanina

2015

Journal of Functional Analysis

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Abstract. We prove that the spaces ℓp, 1 < p < ∞, p = 2, and all infinitedimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the BanachMazur rotation problem, which asks whether a separable Banach space with a transitive norm has to be isometric or isomorphic to a Hilbert space. We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of ℓ 2 2 belongs to the two-dimensional asymptotic structure and we obtain some information about the asymptotic structure in higher dimensions.Further, we prove that the spaces ℓp, 1 < p < ∞, p = 2, have continuum different renormings with 1-unconditional bases each with a different maximal isometry group, and that every symmetric space other than ℓ 2 has at least a countable number of such renormings. On the other hand we show that the spaces ℓp, 1 < p < ∞, p = 2, have continuum different renormings each with an isometry group which is not contained in any maximal bounded subgroup of the group of isomorphisms of ℓp.

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

Cited by 7 publications

References 35 publications

Grothendieck’s Theorem, past and present

Grothendieck’s Theorem, past and present

On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach spaces

On an isomorphic Banach–Mazur rotation problem and maximal norms in Banach spaces

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