The Effros–Ruan conjecture for bilinear forms on C*-algebras

Haagerup, Uffe; Musat, Magdalena

doi:10.1007/s00222-008-0137-7

Cited by 31 publications

(62 citation statements)

References 16 publications

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“…where the second line follows from the normalization assumption Tr(P 2 ) = Tr(Id) = n on elements of S. Combining (34) and (35) gives (33), completing the second step of the proof.…”

Section: Moreover T Can Be Taken With Real Coefficientsmentioning

confidence: 76%

Survey on nonlocal games and operator space theory

Palazuelos

Vidick

2016

Journal of Mathematical Physics

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This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states. C 2016 AIP Publishing LLC. [http://dx

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“…where the second line follows from the normalization assumption Tr(P 2 ) = Tr(Id) = n on elements of S. Combining (34) and (35) gives (33), completing the second step of the proof.…”

Section: Moreover T Can Be Taken With Real Coefficientsmentioning

confidence: 76%

Survey on nonlocal games and operator space theory

Palazuelos

Vidick

2016

Journal of Mathematical Physics

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“…In particular Grothendieck's fundamental work on tensor norms leads to many interesting new problems in the context of operator algebras and operator spaces. Let us mention in particular Shlyakhtenko and Pisier's version of Grothendieck's theorem for operator spaces, Haagerup and Musat's the very recent completion of Grothendieck's theorem for C * -algebras [4] and the results in [7,19,20]. A fundamental object in the theory of operator spaces is Pisier's operator space OH, the only operator space completely isometric to its anti-dual.…”

Section: Introductionmentioning

confidence: 98%

A Maurey type result for operator spaces

Junge

Lee

2008

Journal of Functional Analysis

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The little Grothendieck theorem for Banach spaces says that every bounded linear operator between C(K) and 2 is 2-summing. However, it is shown in [M. Junge, Embedding of the operator space OH and the logarithmic 'little Grothendieck inequality', Invent. Math. 161 (2) (2005) 225-286] that the operator space analogue fails. Not every cb-map v : K → OH is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem: every cb-map v : K → OH is (q, cb)-summing for any q > 2 and hence admits a factorization v(x) c(q) v cb axb q with a, b in the unit ball of the Schatten class S 2q .

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“…We also see in [8,Example 3.6], or in Example 6.5 below, that the bilinear ideals SMB and J CB do not coincide.…”

Section: Example 62 a Multiplicatively Bounded Bilinear Mapping Whimentioning

confidence: 88%

Bilinear ideals in operator spaces

Dimant

Fernández-Unzueta

2015

Journal of Mathematical Analysis and Applications

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Abstract. We introduce a concept of bilinear ideal of jointly completely bounded mappings between operator spaces. In particular, we study the bilinear ideals N of completely nuclear, I of completely integral, E of completely extendible bilinear mappings, MB multiplicatively bounded and its symmetrization SMB. We prove some basic properties of them, one of which is the fact that I is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product. Introduction and PreliminariesLet V, W and X be operator spaces. If we consider the underlying vector space structure, the relationshold through the two natural linear isomorphisms ν, ρ. In order for ν and ρ to induce natural morphisms in the operator space category, it is necessary to have appropriately defined an operator space tensor norm on V ⊗ W and specific classes of linear and bilinear mappings. This is the case, for instance, of the so called projective operator space tensor norm · ∧ , the completely bounded maps and the jointly completely bounded bilinear mappings, where ν and ρ induce the following completely bounded isometric isomorphisms:There are many possible ways to provide V ⊗ W with an operator space tensor norm and, of course, to define classes of mappings. Several authors, inspired by the success that the study of the relations between tensor products and mappings has had in the Banach space setting, have systematically study some analogous relations for operator spaces. This is the case, for instance, of the completely nuclear and completely integral linear mappings (see [7, Section III]).In this paper we follow this approach as well, but with the attention focused on the relations involving ν, the isomorphism in (1) which concerns bilinear mappings. In Section 2 we introduce the notion of an ideal of completely bounded bilinear mappings and study its general properties. In Section 3 we define the ideals of completely nuclear and completely integral bilinear mappings. The main result proved here is that the ideal of completely integral bilinear mappings is naturally identified with the ideal of completely integral linear mappings on the injective operator space tensor product, that is I(V × W, X) ∼ = L I (V ∨ ⊗ W, X) (see Theorem 3.8). This implies that, 2010 Mathematics Subject Classification. 47L25,47L22, 46M05.

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The Effros–Ruan conjecture for bilinear forms on C*-algebras

Cited by 31 publications

References 16 publications

Survey on nonlocal games and operator space theory

Survey on nonlocal games and operator space theory

A Maurey type result for operator spaces

Bilinear ideals in operator spaces

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