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Elementary proofs of Grothendieck theorems for completely bounded norms

Oded Regev 1 Thomas Vidick 2
1 CASCADE - Construction and Analysis of Systems for Confidentiality and Authenticity of Data and Entities
DI-ENS - Département d'informatique de l'École normale supérieure, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR 8548
2 ENS Paris - École normale supérieure - Paris
Abstract : We provide alternative proofs of two recent Grothendieck theorems for jointly completely bounded bilinear forms, originally due to Pisier and Shlyakhtenko (Grothendieck's theorem for operator spaces, \textit{Invent. Math.} \textbf{150}(2002), 185--217) and Haagerup and Musat (The Effros-Ruan conjecture for bilinear forms on ${C}^*$-algebras, \textit{Invent. Math.} \textbf{174}(2008), 139--163). Our proofs are elementary and are inspired by the so-called embezzlement states in quantum information theory. Moreover, our proofs lead to quantitative estimates.
Keywords : Grothendieck inequality quantum information theory bilinear form completely bounded norm
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https://hal.inria.fr/hal-01111570
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Submitted on : Friday, January 30, 2015 - 3:43:51 PM
Last modification on : Tuesday, September 22, 2020 - 3:50:28 AM

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Oded Regev, Thomas Vidick. Elementary proofs of Grothendieck theorems for completely bounded norms . Journal of Operator Theory, 2012, 71 (2), pp.491-506. ⟨10.7900/jot.2012jul02.1947⟩. ⟨hal-01111570⟩

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