论文信息 - Elementary Proofs of Grothendieck Theorems for Completely Bounded Norms

Elementary Proofs of Grothendieck Theorems for Completely Bounded Norms

We provide alternative proofs of two recent Grothendieck theorems for jointly completely bounded bilinear forms, originally due to Pisier and Shlyakhtenko (Invent. Math. 2002) and Haagerup and Musat (Invent. Math. 2008). Our proofs are elementary and are inspired by the so-called embezzlement states in quantum information theory. Moreover, our proofs lead to quantitative estimates.

Thomas Vidick | Oded Regev | O. Regev | Thomas Vidick

[1] Uffe Haagerup,et al. The Effros–Ruan conjecture for bilinear forms on C*-algebras , 2007, 0711.1851.

[2] G. Pisier. Grothendieck's Theorem, past and present , 2011, 1101.4195.

[3] David P. Blecher,et al. Tracially Completely Bounded Multilinear Maps on C∗‐Algebras , 1989 .

[4] Steen Thorbjørnsen,et al. Random matrices and $K$-theory for exact $C^*$-algebras , 1999, Documenta Mathematica.

[5] A. Grothendieck. Résumé de la théorie métrique des produits tensoriels topologiques , 1996 .

[6] Bilinear forms on exact operator spaces and B(H)\otimes B(H) , 1993, math/9308208.

[7] E. Effros,et al. A New Approach to Operator Spaces , 1991, Canadian Mathematical Bulletin.

[8] Uffe Haagerup,et al. The Grothendieck inequality for bilinear forms on C∗-algebras , 1985 .

[9] G. Pisier,et al. Bilinear Forms on Exact Operator Spaces , 1993 .

[10] Gilles Pisier,et al. Grothendieck’s theorem for operator spaces , 2001, math/0108205.

[11] Gilles Pisier. Grothendieck's theorem for noncommutative C∗-algebras, with an Appendix on Grothendieck's constants , 1978 .

[12] P. Hayden,et al. Universal entanglement transformations without communication , 2003 .

[13] Thomas Vidick,et al. Quantum XOR Games , 2012, 2013 IEEE Conference on Computational Complexity.

[14] Gilles Pisier,et al. Introduction to Operator Space Theory , 2003 .

[15] G. Pisier. Grothendieck’s Theorem , 1986 .