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Journal of Operator Theory

Volume 71, Issue 2,  Spring  2014  pp. 491-506.

Elementary proofs of Grothendieck theorems for completely bounded norms

Authors:  Oded Regev (1) and Thomas Vidick (2)
Author institution: (1) Ecole normale superieure, Paris, France and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel
(2) Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA U.S.A.

Summary:  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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