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Grothendieck’s theorem, past and present. (English) Zbl 1244.46006

Grothendieck’s theorem in the title refers to the result from A. Grothendieck’s “Résumé de la théorie métrique des produits tensoriels topologiques” [Bol. Soc. Mat. São Paulo 8, 1–79 (1956; Zbl 0074.32303)] that he himself called “théorème fondamental de la théorie métrique des produits tensoriels”; technically it asserts the equivalence of the “Hilbertian” and the projective tensor norm on \(C(K)\otimes C(K)\). Since Grothendieck’s paper was published in an arcane journal and covered a topic considered likewise arcane at the time, it was forgotten for more than a decade until J. Lindenstrauss and A. Pełczyński [“Absolutely summing operators in \({\mathcal L}_p\)-spaces and their applications”, Stud. Math. 29, 275–326 (1968; Zbl 0183.40501)] recast it in terms of an inequality which is nowadays known as the Grothendieck inequality.
It reads as follows: There is a universal constant \(K_G\) such that whenever \((a_{ij})\) is a square matrix satisfying \[ \sup \biggl\{\Bigl| \sum_{i,j} a_{ij} s_i t_j \Bigr| : |s_i|, |t_j|\leq1 \biggr\} \leq 1 \eqno(1) \] for scalars \(s_i\) and \(t_j\), then \[ \sup \biggl\{ \Bigl|\sum_{i,j} a_{ij} \langle x_i, y_j \rangle \Bigr| : \|x_i\|,\|y_j\|\leq1 \biggr\} \leq K_G \eqno(2) \] for vectors \(x_i\), \(y_j\) in a Hilbert space. The best possible choice of this constant is called the Grothendieck constant.
The Grothendieck inequality had a profound influence on Banach space theory in the 1970s and 1980s, and the author has contributed significantly to this circle of ideas, see for instance [G. Pisier, “Counterexamples to a conjecture of Grothendieck”, Acta Math. 151, 181–208 (1983; Zbl 0542.46038)] and [G. Pisier, Factorization of linear operators and geometry of Banach spaces. Reg. Conf. Ser. Math. 60 (1986; Zbl 0588.46010)]. In this survey paper he first reviews various proofs of the Grothendieck inequality and reformulations in Banach space theory; e.g., every bounded linear operator from \(L_1\) to \(L_2\) is absolutely summing. Incidentally, as A. Pietsch has put it, nobody needs to know the exact value of the Grothendieck constant, but everyone likes to know it. In this regard work by M. Braverman, K. Makarychev, Yu. Makarychev and A. Naor [“The Grothendieck constant is strictly smaller than Krivine’s bound”, arXiv:1103.61061] is mentioned that shows that in the case of real scalars \(K_G < \frac{\pi}{2\log(1+\sqrt2)}\), refuting a long-standing conjecture.
But the main emphasis of the survey is on noncommutative versions of Grothendieck’s inequality, first in the setting of \(C^*\)-algebras where it was originally proved by the author [G. Pisier, “Grothendieck’s theorem for noncommutative \(C^*\)-algebras, with an appendix on Grothendieck’s constants”, J. Funct. Anal. 29, 397–415 (1978; Zbl 0388.46043)] and by U. Haagerup [“The Grothendieck inequality for bilinear forms on \(C^*\)-algebras”, Adv. Math. 56, 93–116 (1985; Zbl 0593.46052)], and then for operator spaces, where the fundamental reference is [G. Pisier and D. Shlyakhtenko, “Grothendieck’s theorem for operator spaces”, Invent. Math. 150, No. 1, 185–217 (2002; Zbl 1033.46044)].
Additionally, the relevance of the Grothendieck inequality for quantum mechanics and discrete mathematics is discussed, where every graph supports a Grothendieck inequality of its own. Interestingly, the point of view of discrete mathematics is opposite to the one of analysis: Whereas in analysis the quantity in (2) is the “hard” one that is dominated by \(K_G\) times the “easy” quantity in (1), in discrete mathematics the roles have changed in that now (2) is considered “easy” (by what is called semi-definite programming) and gives, up to \(1/K_G\), a lower bound for the “hard” quantity (1). For more on this the paper [S. Khot and A. Naor, “Grothendieck-type inequalities in combinatorial optimization”, Commun. Pure Appl. Math. 65, No. 7, 992–1035 (2012; Zbl 1248.46047)] should be consulted.
G. Pisier’s survey is written in the lucid and elegant style that is the hallmark of this author. It is a highlight of expository writing and a must-read for everyone interested in contemporary functional analysis. The author intends to maintain an updated and expanded version of his article on his home page; see http://www.math.tamu.edu/~pisier/grothendieck.UNCUT.pdf.

MSC:

46B28 Spaces of operators; tensor products; approximation properties
46L05 General theory of \(C^*\)-algebras
46L07 Operator spaces and completely bounded maps
46B07 Local theory of Banach spaces
05C12 Distance in graphs
46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
68W25 Approximation algorithms
81P40 Quantum coherence, entanglement, quantum correlations
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References:

[1] Antonio Acín, Nicolas Gisin, and Benjamin Toner, Grothendieck’s constant and local models for noisy entangled quantum states, Phys. Rev. A (3) 73 (2006), no. 6, 062105, 5. · doi:10.1103/PhysRevA.73.062105
[2] A. B. Aleksandrov and V. V. Peller, Hankel and Toeplitz-Schur multipliers, Math. Ann. 324 (2002), no. 2, 277 – 327. · Zbl 1030.47019 · doi:10.1007/s00208-002-0339-z
[3] Noga Alon and Eli Berger, The Grothendieck constant of random and pseudo-random graphs, Discrete Optim. 5 (2008), no. 2, 323 – 327. · Zbl 1151.90055 · doi:10.1016/j.disopt.2006.06.004
[4] Noga Alon, Konstantin Makarychev, Yury Makarychev, and Assaf Naor, Quadratic forms on graphs, Invent. Math. 163 (2006), no. 3, 499 – 522. · Zbl 1082.05051 · doi:10.1007/s00222-005-0465-9
[5] Noga Alon and Assaf Naor, Approximating the cut-norm via Grothendieck’s inequality, SIAM J. Comput. 35 (2006), no. 4, 787 – 803. · Zbl 1096.68163 · doi:10.1137/S0097539704441629
[6] S. Arora, E. Berger, E. Hazan, G. Kindler, and M. Safra, On non-approximability for quadratic programs, Preprint, to appear.
[7] Alain Aspect, Bell’s theorem: the naive view of an experimentalist, Quantum [un]speakables (Vienna, 2000) Springer, Berlin, 2002, pp. 119 – 153.
[8] Alain Aspect, Testing Bell’s inequalities, Quantum reflections, Cambridge Univ. Press, Cambridge, 2000, pp. 69 – 78.
[9] J. Audretsch, Entangled Systems, Wiley-Vch, VerlagGmbH & Co., KGaA, Weinheim, 2007. · Zbl 1136.81001
[10] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071
[11] Robert S. Doran , Selfadjoint and nonselfadjoint operator algebras and operator theory, Contemporary Mathematics, vol. 120, American Mathematical Society, Providence, RI, 1991. · Zbl 0741.00053
[12] D. P. Blecher, Tracially completely bounded multilinear maps on \?*-algebras, J. London Math. Soc. (2) 39 (1989), no. 3, 514 – 524. · Zbl 0671.46027 · doi:10.1112/jlms/s2-39.3.514
[13] Krzysztof Jarosz , Function spaces, Lecture Notes in Pure and Applied Mathematics, vol. 136, Marcel Dekker, Inc., New York, 1992. · Zbl 0804.46030
[14] R. C. Blei, Multidimensional extensions of the Grothendieck inequality and applications, Ark. Mat. 17 (1979), no. 1, 51 – 68. · Zbl 0461.43005 · doi:10.1007/BF02385457
[15] Ron Blei, Analysis in integer and fractional dimensions, Cambridge Studies in Advanced Mathematics, vol. 71, Cambridge University Press, Cambridge, 2001. · Zbl 1006.46001
[16] Marek Bożejko and Gero Fendler, Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Un. Mat. Ital. A (6) 3 (1984), no. 2, 297 – 302 (English, with Italian summary). · Zbl 0564.43004
[17] M. Braverman, K. Makarychev, Y. Makarychev, and A. Naor, The Grothendieck constant is strictly smaller than Krivine’s bound. Preprint, March 31, 2011. · Zbl 1292.90243
[18] J. Briët, F.M. de Oliveira Filho and F. Vallentin, The positive semidefinite Grothendieck problem with rank constraint, pp. 31-42 in Proceedings of the 37th International Colloquium on Automata, Languages and Programming, ICALP 2010 S. Abramsky, et al. , Part I, LNCS 6198, 2010. · Zbl 1287.68178
[19] J. Briët, F.M. de Oliveira Filho and F. Vallentin, Grothendieck inequalities for semidefinite programs with rank constraint, arXiv:1011.1754v1 [math.OC] · Zbl 1297.68261
[20] J. Briët, H. Burhman and B. Toner, A generalized Grothendieck inequality and entanglement in XOR games, arXiv:0901.2009v1 [quant-ph].
[21] Nathanial P. Brown and Narutaka Ozawa, \?*-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. · Zbl 1160.46001
[22] Artur Buchholz, Optimal constants in Khintchine type inequalities for fermions, Rademachers and \?-Gaussian operators, Bull. Pol. Acad. Sci. Math. 53 (2005), no. 3, 315 – 321. · Zbl 1117.46043 · doi:10.4064/ba53-3-9
[23] M. Charikar and A. Wirth, Maximizing quadratic programs: extending Grothendieck’s inequality, FOCS (2004), 54-60.
[24] Man Duen Choi and Edward G. Effros, Nuclear \?*-algebras and injectivity: the general case, Indiana Univ. Math. J. 26 (1977), no. 3, 443 – 446. · Zbl 0378.46052 · doi:10.1512/iumj.1977.26.26034
[25] Benoît Collins and Ken Dykema, A linearization of Connes’ embedding problem, New York J. Math. 14 (2008), 617 – 641. · Zbl 1162.46032
[26] A. Connes, Classification of injective factors. Cases \?\?\(_{1}\), \?\?_{\infty }, \?\?\?_{\?}, \?\?=1, Ann. of Math. (2) 104 (1976), no. 1, 73 – 115. · Zbl 0343.46042 · doi:10.2307/1971057
[27] A. M. Davie, Matrix norms related to Grothendieck’s inequality, Banach spaces (Columbia, Mo., 1984) Lecture Notes in Math., vol. 1166, Springer, Berlin, 1985, pp. 22 – 26. · doi:10.1007/BFb0074689
[28] Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. · Zbl 0774.46018
[29] Joe Diestel, Jan H. Fourie, and Johan Swart, The metric theory of tensor products, American Mathematical Society, Providence, RI, 2008. Grothendieck’s résumé revisited. · Zbl 1186.46004
[30] Joe Diestel, Hans Jarchow, and Albrecht Pietsch, Operator ideals, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 437 – 496. · Zbl 1012.47001 · doi:10.1016/S1874-5849(01)80013-9
[31] J. Dixmier, Les anneaux d’opérateurs de classe finie, Ann. Sci. École Norm. Sup. (3) 66 (1949), 209 – 261 (French). · Zbl 0036.35802
[32] Ed Dubinsky, A. Pełczyński, and H. P. Rosenthal, On Banach spaces \? for which \Pi \(_{2}\)(\cal\?_{\infty },\?)=\?(\cal\?_{\infty },\?), Studia Math. 44 (1972), 617 – 648. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI. · Zbl 0262.46018
[33] K. Dykema and K. Juschenko, Matrices of unitary moments, Preprint (arXiv:0901.0288). · Zbl 1241.46034
[34] Edward G. Effros and Zhong-Jin Ruan, A new approach to operator spaces, Canad. Math. Bull. 34 (1991), no. 3, 329 – 337. · Zbl 0769.46037 · doi:10.4153/CMB-1991-053-x
[35] Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. · Zbl 0969.46002
[36] P. C. Fishburn and J. A. Reeds, Bell inequalities, Grothendieck’s constant, and root two, SIAM J. Discrete Math. 7 (1994), no. 1, 48 – 56. · Zbl 0792.05030 · doi:10.1137/S0895480191219350
[37] T. Fritz, Tsirelson’s problem and Kirchberg’s conjecture, 2010 (arXiv:1008.1168). · Zbl 1250.81023
[38] T. W. Gamelin and S. V. Kislyakov, Uniform algebras as Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 671 – 706. · Zbl 1032.46067 · doi:10.1016/S1874-5849(01)80018-8
[39] D. Pérez-García, M. M. Wolf, C. Palazuelos, I. Villanueva, and M. Junge, Unbounded violation of tripartite Bell inequalities, Comm. Math. Phys. 279 (2008), no. 2, 455 – 486. · Zbl 1157.81008 · doi:10.1007/s00220-008-0418-4
[40] Michel X. Goemans and David P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42 (1995), no. 6, 1115 – 1145. · Zbl 0885.68088 · doi:10.1145/227683.227684
[41] A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Resenhas 2 (1996), no. 4, 401 – 480 (French). Reprint of Bol. Soc. Mat. São Paulo 8 (1953), 1 – 79 [ MR0094682 (20 #1194)]. · Zbl 1019.46038
[42] Alexandre Grothendieck, Sur certaines classes de suites dans les espaces de Banach, et le théorème de Dvoretzky-Rogers, Resenhas 3 (1998), no. 4, 447 – 477 (French). With a foreword by Paulo Cordaro; Reprint of the 1953 original. · Zbl 1098.46506
[43] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), 140 (French). · Zbl 0064.35501
[44] M. Grötschel, L. Lovász, and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), no. 2, 169 – 197. · Zbl 0492.90056 · doi:10.1007/BF02579273
[45] Martin Grötschel, László Lovász, and Alexander Schrijver, Geometric algorithms and combinatorial optimization, 2nd ed., Algorithms and Combinatorics, vol. 2, Springer-Verlag, Berlin, 1993. · Zbl 0837.05001
[46] G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light, Cambridge Univ. Press, 2010. · Zbl 1208.81001
[47] Uffe Haagerup, The best constants in the Khintchine inequality, Studia Math. 70 (1981), no. 3, 231 – 283 (1982). · Zbl 0501.46015
[48] Uffe Haagerup, Solution of the similarity problem for cyclic representations of \?*-algebras, Ann. of Math. (2) 118 (1983), no. 2, 215 – 240. · Zbl 0543.46033 · doi:10.2307/2007028
[49] U. Haagerup, All nuclear \?*-algebras are amenable, Invent. Math. 74 (1983), no. 2, 305 – 319. · Zbl 0529.46041 · doi:10.1007/BF01394319
[50] Uffe Haagerup, The Grothendieck inequality for bilinear forms on \?*-algebras, Adv. in Math. 56 (1985), no. 2, 93 – 116. · Zbl 0593.46052 · doi:10.1016/0001-8708(85)90026-X
[51] Uffe Haagerup, A new upper bound for the complex Grothendieck constant, Israel J. Math. 60 (1987), no. 2, 199 – 224. · Zbl 0646.46019 · doi:10.1007/BF02790792
[52] Uffe Haagerup and Takashi Itoh, Grothendieck type norms for bilinear forms on \?*-algebras, J. Operator Theory 34 (1995), no. 2, 263 – 283. · Zbl 0852.46049
[53] Uffe Haagerup and Magdalena Musat, On the best constants in noncommutative Khintchine-type inequalities, J. Funct. Anal. 250 (2007), no. 2, 588 – 624. · Zbl 1129.46046 · doi:10.1016/j.jfa.2007.05.014
[54] Uffe Haagerup and Magdalena Musat, The Effros-Ruan conjecture for bilinear forms on \?*-algebras, Invent. Math. 174 (2008), no. 1, 139 – 163. · Zbl 1188.46034 · doi:10.1007/s00222-008-0137-7
[55] Uffe Haagerup and Gilles Pisier, Bounded linear operators between \?*-algebras, Duke Math. J. 71 (1993), no. 3, 889 – 925. · Zbl 0803.46064 · doi:10.1215/S0012-7094-93-07134-7
[56] U. Haagerup and S. Thorbjørnsen, Random matrices and \?-theory for exact \?*-algebras, Doc. Math. 4 (1999), 341 – 450. · Zbl 0933.46051
[57] Uffe Haagerup and Steen Thorbjørnsen, A new application of random matrices: \?\?\?(\?*_{\?\?\?}(\?\(_{2}\))) is not a group, Ann. of Math. (2) 162 (2005), no. 2, 711 – 775. · Zbl 1103.46032 · doi:10.4007/annals.2005.162.711
[58] Asma Harcharras, Fourier analysis, Schur multipliers on \?^{\?} and non-commutative \Lambda (\?)-sets, Studia Math. 137 (1999), no. 3, 203 – 260. · Zbl 0948.43002
[59] Johan Håstad, Some optimal inapproximability results, J. ACM 48 (2001), no. 4, 798 – 859. · Zbl 1127.68405 · doi:10.1145/502090.502098
[60] Hoshang Heydari, Quantum correlation and Grothendieck’s constant, J. Phys. A 39 (2006), no. 38, 11869 – 11875. · Zbl 1100.81500 · doi:10.1088/0305-4470/39/38/012
[61] Kiyosi Itô and Makiko Nisio, On the convergence of sums of independent Banach space valued random variables, Osaka J. Math. 5 (1968), 35 – 48. · Zbl 0231.60027
[62] Takashi Itoh, On the completely bounded map of a \?*-algebra to its dual space, Bull. London Math. Soc. 19 (1987), no. 6, 546 – 550. · Zbl 0644.46038 · doi:10.1112/blms/19.6.546
[63] Kamil John, On the compact nonnuclear operator problem, Math. Ann. 287 (1990), no. 3, 509 – 514. · Zbl 0687.47012 · doi:10.1007/BF01446908
[64] William B. Johnson and Joram Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 1 – 84. · Zbl 1011.46009 · doi:10.1016/S1874-5849(01)80003-6
[65] Marius Junge, Embedding of the operator space \?\? and the logarithmic ’little Grothendieck inequality’, Invent. Math. 161 (2005), no. 2, 225 – 286. · Zbl 1092.47060 · doi:10.1007/s00222-004-0421-0
[66] M. Junge, M. Navascues, C. Palazuelos, D. Peréz-García, V.B. Scholz, and R.F. Werner, Connes’ embedding problem and Tsirelson’s problem, 2010, Preprint (arXiv:1008.1142). · Zbl 1314.81038
[67] M. Junge and C. Palazuelos, Large violations of Bell’s inequalities with low entanglement, (arXiv:1007.3043). · Zbl 1230.81011
[68] Marius Junge and Javier Parcet, Rosenthal’s theorem for subspaces of noncommutative \?_{\?}, Duke Math. J. 141 (2008), no. 1, 75 – 122. · Zbl 1176.46059 · doi:10.1215/S0012-7094-08-14112-2
[69] Marius Junge and Javier Parcet, Maurey’s factorization theory for operator spaces, Math. Ann. 347 (2010), no. 2, 299 – 338. · Zbl 1213.46049 · doi:10.1007/s00208-009-0440-7
[70] Marius Junge and Javier Parcet, Mixed-norm inequalities and operator space \?_{\?} embedding theory, Mem. Amer. Math. Soc. 203 (2010), no. 953, vi+155. · Zbl 1220.46041 · doi:10.1090/S0065-9266-09-00570-5
[71] M. Junge and G. Pisier, Bilinear forms on exact operator spaces and \?(\?)\otimes \?(\?), Geom. Funct. Anal. 5 (1995), no. 2, 329 – 363. · Zbl 0832.46052 · doi:10.1007/BF01895670
[72] Marius Junge and Quanhua Xu, Representation of certain homogeneous Hilbertian operator spaces and applications, Invent. Math. 179 (2010), no. 1, 75 – 118. · Zbl 1215.46039 · doi:10.1007/s00222-009-0210-x
[73] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. Elementary theory; Reprint of the 1983 original. Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997. Advanced theory; Corrected reprint of the 1986 original. · Zbl 0888.46039
[74] Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. · Zbl 0571.60002
[75] Sten Kaijser and Allan M. Sinclair, Projective tensor products of \?*-algebras, Math. Scand. 55 (1984), no. 2, 161 – 187. · Zbl 0546.46047 · doi:10.7146/math.scand.a-12074
[76] N. J. Kalton, Rademacher series and decoupling, New York J. Math. 11 (2005), 563 – 595. · Zbl 1107.46002
[77] Boris S. Kashin and Stanislaw J. Szarek, The Knaster problem and the geometry of high-dimensional cubes, C. R. Math. Acad. Sci. Paris 336 (2003), no. 11, 931 – 936 (English, with English and French summaries). · Zbl 1031.46015 · doi:10.1016/S1631-073X(03)00226-7
[78] B. S. Kashin and S. Ĭ. Sharek, On the Gram matrices of systems of uniformly bounded functions, Tr. Mat. Inst. Steklova 243 (2003), no. Funkts. Prostran., Priblizh., Differ. Uravn., 237 – 243 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(243) (2003), 227 – 233.
[79] L. A. Khalfin and B. S. Tsirelson, Quantum and quasiclassical analogs of Bell inequalities, Symposium on the foundations of modern physics (Joensuu, 1985) World Sci. Publishing, Singapore, 1985, pp. 441 – 460.
[80] S. Khot and A. Naor, Sharp kernel clustering algorithms and their associated Grothendieck inequalities. In Proceedings of SODA’2010, pp. 664-683. · Zbl 1288.68272
[81] Guy Kindler, Assaf Naor, and Gideon Schechtman, The UGC hardness threshold of the \?_{\?} Grothendieck problem, Math. Oper. Res. 35 (2010), no. 2, 267 – 283. · Zbl 1216.68340 · doi:10.1287/moor.1090.0425
[82] Eberhard Kirchberg, On nonsemisplit extensions, tensor products and exactness of group \?*-algebras, Invent. Math. 112 (1993), no. 3, 449 – 489. · Zbl 0803.46071 · doi:10.1007/BF01232444
[83] V. P. Havin, S. V. Hruščëv, and N. K. Nikol\(^{\prime}\)skiĭ , Linear and complex analysis problem book, Lecture Notes in Mathematics, vol. 1043, Springer-Verlag, Berlin, 1984. 199 research problems.
[84] S. V. Kislyakov, Absolutely summing operators on the disc algebra, Algebra i Analiz 3 (1991), no. 4, 1 – 77 (Russian); English transl., St. Petersburg Math. J. 3 (1992), no. 4, 705 – 774. · Zbl 0761.47006
[85] S. V. Kislyakov, Banach spaces and classical harmonic analysis, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 871 – 898. · Zbl 1024.46002 · doi:10.1016/S1874-5849(01)80022-X
[86] Hermann König, On the complex Grothendieck constant in the \?-dimensional case, Geometry of Banach spaces (Strobl, 1989) London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 181 – 198. · Zbl 0759.46020
[87] Hermann König, On an extremal problem originating in questions of unconditional convergence, Recent progress in multivariate approximation (Witten-Bommerholz, 2000) Internat. Ser. Numer. Math., vol. 137, Birkhäuser, Basel, 2001, pp. 185 – 192. · Zbl 0994.47029
[88] J. L. Krivine, Théorèmes de factorisation dans les espaces réticulés, Séminaire Maurey-Schwartz 1973 – 1974: Espaces \?^{\?}, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 22 et 23, Centre de Math., École Polytech., Paris, 1974, pp. 22 (French).
[89] Jean-Louis Krivine, Sur la constante de Grothendieck, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 8, A445 – A446. · Zbl 0366.60010
[90] J.-L. Krivine, Constantes de Grothendieck et fonctions de type positif sur les sphères, Adv. in Math. 31 (1979), no. 1, 16 – 30 (French). · Zbl 0413.46054 · doi:10.1016/0001-8708(79)90017-3
[91] Stanislaw Kwapień, On operators factorizable through \?_{\?} space, Actes du Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. de Bordeaux, 1971) Soc. Math. France, Paris, 1972, pp. 215 – 225. Bull. Soc. Math. France, Mém. No. 31 – 32.
[92] Michel Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. · Zbl 0995.60002
[93] Denny H. Leung, Factoring operators through Hilbert space, Israel J. Math. 71 (1990), no. 2, 225 – 227. · Zbl 0724.47012 · doi:10.1007/BF02811886
[94] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in \?_{\?}-spaces and their applications, Studia Math. 29 (1968), 275 – 326. · Zbl 0183.40501
[95] Nati Linial and Adi Shraibman, Lower bounds in communication complexity based on factorization norms, STOC’07 — Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM, New York, 2007, pp. 699 – 708. · Zbl 1232.68067 · doi:10.1145/1250790.1250892
[96] L. Lovász, Semidefinite programs and combinatorial optimization, Lecture Notes, Microsoft Research, Redmont, WA 98052
[97] Françoise Lust-Piquard, Inégalités de Khintchine dans \?_{\?}(1&lt;\?&lt;\infty ), C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 7, 289 – 292 (French, with English summary). · Zbl 0592.47038
[98] Françoise Lust-Piquard, A Grothendieck factorization theorem on 2-convex Schatten spaces, Israel J. Math. 79 (1992), no. 2-3, 331 – 365. · Zbl 0789.47019 · doi:10.1007/BF02808225
[99] Françoise Lust-Piquard and Gilles Pisier, Noncommutative Khintchine and Paley inequalities, Ark. Mat. 29 (1991), no. 2, 241 – 260. · Zbl 0755.47029 · doi:10.1007/BF02384340
[100] Françoise Lust-Piquard and Quanhua Xu, The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators, J. Funct. Anal. 244 (2007), no. 2, 488 – 503. · Zbl 1137.46039 · doi:10.1016/j.jfa.2006.09.003
[101] B. Maurey, Une nouvelle démonstration d’un théorème de Grothendieck, Séminaire Maurey-Schwartz Année 1972 – 1973: Espaces \?^{\?} et applications radonifiantes, Exp. No. 22, Centre de Math., École Polytech., Paris, 1973, pp. 7 (French). · Zbl 0262.47014
[102] Bernard Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces \?^{\?}, Société Mathématique de France, Paris, 1974 (French). With an English summary; Astérisque, No. 11. · Zbl 0278.46028
[103] Alexandre Megretski, Relaxations of quadratic programs in operator theory and system analysis, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 365 – 392. · Zbl 0997.90057
[104] A. Nemirovski, C. Roos, and T. Terlaky, On maximization of quadratic form over intersection of ellipsoids with common center, Math. Program. 86 (1999), no. 3, Ser. A, 463 – 473. · Zbl 0944.90056 · doi:10.1007/s101070050100
[105] Timur Oikhberg, Direct sums of operator spaces, J. London Math. Soc. (2) 64 (2001), no. 1, 144 – 160. · Zbl 1020.46014 · doi:10.1017/S0024610701002174
[106] Timur Oikhberg and Gilles Pisier, The ”maximal” tensor product of operator spaces, Proc. Edinburgh Math. Soc. (2) 42 (1999), no. 2, 267 – 284. · Zbl 0940.46042 · doi:10.1017/S0013091500020241
[107] A. M. Olevskiĭ, Fourier series with respect to general orthogonal systems, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by B. P. Marshall and H. J. Christoffers; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 86.
[108] Narutaka Ozawa, About the QWEP conjecture, Internat. J. Math. 15 (2004), no. 5, 501 – 530. · Zbl 1056.46051 · doi:10.1142/S0129167X04002417
[109] Christos H. Papadimitriou and Mihalis Yannakakis, Optimization, approximation, and complexity classes, J. Comput. System Sci. 43 (1991), no. 3, 425 – 440. · Zbl 0765.68036 · doi:10.1016/0022-0000(91)90023-X
[110] Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. · Zbl 1029.47003
[111] Vern I. Paulsen and Mrinal Raghupathi, Representations of logmodular algebras, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2627 – 2640. · Zbl 1220.47144
[112] Antonio M. Peralta, Little Grothendieck’s theorem for real \?\?*-triples, Math. Z. 237 (2001), no. 3, 531 – 545. · Zbl 1049.46049 · doi:10.1007/PL00004878
[113] Antonio M. Peralta, New advances on the Grothendieck’s inequality problem for bilinear forms on JB*-triples, Math. Inequal. Appl. 8 (2005), no. 1, 7 – 21. · Zbl 1160.46332 · doi:10.7153/mia-08-02
[114] Asher Peres, Quantum theory: concepts and methods, Fundamental Theories of Physics, vol. 57, Kluwer Academic Publishers Group, Dordrecht, 1993. · Zbl 0820.00011
[115] Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. · Zbl 0434.47030
[116] Gilles Pisier, Grothendieck’s theorem for noncommutative \?*-algebras, with an appendix on Grothendieck’s constants, J. Funct. Anal. 29 (1978), no. 3, 397 – 415. · Zbl 0388.46043 · doi:10.1016/0022-1236(78)90038-1
[117] Gilles Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and analysis (Varenna, 1985) Lecture Notes in Math., vol. 1206, Springer, Berlin, 1986, pp. 167 – 241. · Zbl 0606.60008 · doi:10.1007/BFb0076302
[118] Gilles Pisier, Factorization of operators through \?_{\?\infty } or \?_{\?1} and noncommutative generalizations, Math. Ann. 276 (1986), no. 1, 105 – 136. · Zbl 0619.47016 · doi:10.1007/BF01450929
[119] Gilles Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics, vol. 60, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. · Zbl 0588.46010
[120] Gilles Pisier, The dual \?* of the James space has cotype 2 and the Gordon-Lewis property, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 323 – 331. · Zbl 0649.46022 · doi:10.1017/S0305004100064902
[121] Gilles Pisier, Multipliers and lacunary sets in non-amenable groups, Amer. J. Math. 117 (1995), no. 2, 337 – 376. · Zbl 0853.43008 · doi:10.2307/2374918
[122] Gilles Pisier, A simple proof of a theorem of Kirchberg and related results on \?*-norms, J. Operator Theory 35 (1996), no. 2, 317 – 335. · Zbl 0858.46045
[123] Gilles Pisier, The operator Hilbert space \?\?, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103. · Zbl 0932.46046 · doi:10.1090/memo/0585
[124] Gilles Pisier, Non-commutative vector valued \?_{\?}-spaces and completely \?-summing maps, Astérisque 247 (1998), vi+131 (English, with English and French summaries). · Zbl 0937.46056
[125] Gilles Pisier, An inequality for \?-orthogonal sums in non-commutative \?_{\?}, Illinois J. Math. 44 (2000), no. 4, 901 – 923. · Zbl 0976.60016
[126] Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. · Zbl 1093.46001
[127] Gilles Pisier, Completely bounded maps into certain Hilbertian operator spaces, Int. Math. Res. Not. 74 (2004), 3983 – 4018. · Zbl 1079.46041 · doi:10.1155/S1073792804140865
[128] Gilles Pisier, Remarks on the non-commutative Khintchine inequalities for 0&lt;\?&lt;2, J. Funct. Anal. 256 (2009), no. 12, 4128 – 4161. · Zbl 1217.46042 · doi:10.1016/j.jfa.2008.11.014
[129] Gilles Pisier and Dimitri Shlyakhtenko, Grothendieck’s theorem for operator spaces, Invent. Math. 150 (2002), no. 1, 185 – 217. · Zbl 1033.46044 · doi:10.1007/s00222-002-0235-x
[130] Gilles Pisier and Quanhua Xu, Non-commutative \?^{\?}-spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459 – 1517. · Zbl 1046.46048 · doi:10.1016/S1874-5849(03)80041-4
[131] Florin Rădulescu, A comparison between the max and min norms on \?*(\?_{\?})\otimes \?*(\?_{\?}), J. Operator Theory 51 (2004), no. 2, 245 – 253. · Zbl 1111.46037
[132] F. Rădulescu, Combinatorial aspects of Connes’s embedding conjecture and asymptotic distribution of traces of products of unitaries, Operator Theory 20, 197-205, Theta Ser. Adv. Math. 6, Theta, Bucharest, 2006. · Zbl 1199.60032
[133] Prasad Raghavendra, Optimal algorithms and inapproximability results for every CSP? [extended abstract], STOC’08, ACM, New York, 2008, pp. 245 – 254. · Zbl 1231.68142 · doi:10.1145/1374376.1374414
[134] P. Raghavendra and D. Steurer, Towards computing the Grothendieck constant. Proceedings of SODA. 2009, 525-534.
[135] Narcisse Randrianantoanina, Embeddings of non-commutative \?^{\?}-spaces into preduals of finite von Neumann algebras, Israel J. Math. 163 (2008), 1 – 27. · Zbl 1155.46032 · doi:10.1007/s11856-008-0001-x
[136] J.A. Reeds, A new lower bound on the real Grothendieck constant, unpublished note, 1991, available at http://www.dtc.umn.edu/reedsj/bound2.dvi.
[137] O. Regev, Bell violations through independent bases games, to appear (arXiv:1101.0576v2). · Zbl 1268.81038
[138] Oded Regev and Ben Toner, Simulating quantum correlations with finite communication, SIAM J. Comput. 39 (2009/10), no. 4, 1562 – 1580. · Zbl 1205.68181 · doi:10.1137/080723909
[139] Ronald E. Rietz, A proof of the Grothendieck inequality, Israel J. Math. 19 (1974), 271 – 276. · Zbl 0321.46018 · doi:10.1007/BF02757725
[140] Haskell P. Rosenthal, On subspaces of \?^{\?}, Ann. of Math. (2) 97 (1973), 344 – 373. · Zbl 0253.46049 · doi:10.2307/1970850
[141] Zhong-Jin Ruan, Subspaces of \?*-algebras, J. Funct. Anal. 76 (1988), no. 1, 217 – 230. · Zbl 0646.46055 · doi:10.1016/0022-1236(88)90057-2
[142] Jerzy Sawa, The best constant in the Khintchine inequality for complex Steinhaus variables, the case \?=1, Studia Math. 81 (1985), no. 1, 107 – 126. · Zbl 0612.60017
[143] Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, no. 26, Princeton University Press, Princeton, N. J., 1950. · Zbl 0039.33503
[144] Rolf Schneider, Zonoids whose polars are zonoids, Proc. Amer. Math. Soc. 50 (1975), 365 – 368. · Zbl 0339.52002
[145] V.B. Scholz and R.F. Werner, Tsirelson’s Problem, arXiv:0812.4305v1 [math-ph]
[146] Allan M. Sinclair and Roger R. Smith, Hochschild cohomology of von Neumann algebras, London Mathematical Society Lecture Note Series, vol. 203, Cambridge University Press, Cambridge, 1995. · Zbl 0826.46050
[147] S. J. Szarek, On the best constants in the Khinchin inequality, Studia Math. 58 (1976), no. 2, 197 – 208. · Zbl 0424.42014
[148] M. Takesaki, Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition; Operator Algebras and Non-commutative Geometry, 5. M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6. M. Takesaki, Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, vol. 127, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 8.
[149] Nicole Tomczak-Jaegermann, The moduli of smoothness and convexity and the Rademacher averages of trace classes \?_{\?}(1\le \?&lt;\infty ), Studia Math. 50 (1974), 163 – 182. · Zbl 0282.46016
[150] Nicole Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. · Zbl 0721.46004
[151] Andrew Tonge, The complex Grothendieck inequality for 2\times 2 matrices, Bull. Soc. Math. Grèce (N.S.) 27 (1986), 133 – 136. · Zbl 0661.47011
[152] J.A. Tropp, Column subset selection, matrix factorization, and eigenvalue optimization, (arXiv:0806.4404v1), 26 June 2008.
[153] B. S. Cirel\(^{\prime}\)son, Quantum generalizations of Bell’s inequality, Lett. Math. Phys. 4 (1980), no. 2, 93 – 100. · doi:10.1007/BF00417500
[154] B. S. Tsirelson, Quantum analogues of Bell’s inequalities. The case of two spatially divided domains, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 142 (1985), 174 – 194, 200 (Russian). Problems of the theory of probability distributions, IX.
[155] B. S. Tsirelson, Some results and problems on quantum Bell-type inequalities, Hadronic J. Suppl. 8 (1993), no. 4, 329 – 345. · Zbl 0788.15008
[156] B.S. Tsirelson, Bell inequalities and operator algebras, Problem 33, 6 July 2006, Open Problems in Quantum Information Theory, Institut für Mathematische Physik, TU Braunschweig, Germany.
[157] A. M. Vershik and B. S. Tsirelson, Formulation of Bell type problems, and ”noncommutative” convex geometry, Representation theory and dynamical systems, Adv. Soviet Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 95 – 114. · Zbl 0765.58004
[158] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. · Zbl 0795.46049
[159] Simon Wassermann, On tensor products of certain group \?*-algebras, J. Functional Analysis 23 (1976), no. 3, 239 – 254. · Zbl 0358.46040
[160] Warren Wogen, On generators for von Neumann algebras, Bull. Amer. Math. Soc. 75 (1969), 95 – 99. · Zbl 0185.21203
[161] Quan Hua Xu, Applications du théorème de factorisation pour des fonctions à valeurs opérateurs, Studia Math. 95 (1990), no. 3, 273 – 292 (French, with English summary). · Zbl 0728.46045
[162] Quanhua Xu, Operator-space Grothendieck inequalities for noncommutative \?_{\?}-spaces, Duke Math. J. 131 (2006), no. 3, 525 – 574. · Zbl 1129.46048 · doi:10.1215/S0012-7094-06-13135-6
[163] Quanhua Xu, Embedding of \?_{\?} and \?_{\?} into noncommutative \?_{\?}-spaces, 1\le \?&lt;\?\le 2, Math. Ann. 335 (2006), no. 1, 109 – 131. · Zbl 1121.46049 · doi:10.1007/s00208-005-0732-5
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