# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2012jul02.1947

**Keywords: ** Grothendieck inequality, quantum information theory,
bilinear form, completely bounded norm

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