Abstract: Probably the most famous of Grothendieck's contributions to Banach space theory is the result that he himself described as ``the fundamental theorem in the metric theory of tensor products''. That is now commonly referred to as ``Grothendieck's theorem'' (``GT'' for short), or sometimes as ``Grothendieck's inequality''. This had a major impact first in Banach space theory (roughly after 1968), then, later on, in -algebra theory (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of ``operator spaces'' or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell's inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by ``semidefinite programming'' and hence solved in polynomial time. This expository paper (where many proofs are included), presents a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author's 1986 CBMS notes.
15.Ron
Blei, Analysis in integer and fractional dimensions, Cambridge
Studies in Advanced Mathematics, vol. 71, Cambridge University Press,
Cambridge, 2001. MR 1853423
(2003a:46008)
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). MR 753889
(86b:43009)
17.
M. Braverman, K. Makarychev, Y. Makarychev, and A. Naor, The Grothendieck constant is strictly smaller than Krivine's bound. Preprint, March 31, 2011.
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. (eds.), Part I, LNCS 6198, 2010.
19.
J. Briët, F.M. de Oliveira Filho and F. Vallentin, Grothendieck inequalities for semidefinite programs with rank constraint, arXiv:1011.1754v1 [math.OC]
20.
J. Briët, H. Burhman and B. Toner, A generalized Grothendieck inequality and entanglement in XOR games, arXiv:0901.2009v1 [quant-ph].
31.J.
Dixmier, Les anneaux d’opérateurs de classe
finie, Ann. Sci. École Norm. Sup. (3) 66
(1949), 209–261 (French). MR 0032940
(11,370c)
32.Ed
Dubinsky, A.
Pełczyński, and H.
P. Rosenthal, On Banach spaces 𝑋 for which
Π₂(\𝑐𝑎𝑙𝐿_{∞},𝑋)=𝐵(\𝑐𝑎𝑙𝐿_{∞},𝑋),
Studia Math. 44 (1972), 617–648. Collection of
articles honoring the completion by Antoni Zygmund of 50 years of
scientific activity, VI. MR 0365097
(51 #1350)
33.
K. Dykema and K. Juschenko, Matrices of unitary moments, Preprint (arXiv:0901.0288).
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)]. MR
1466414 (98e:46091)
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. MR 1675413
(2000a:46029)
46.
G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light, Cambridge Univ. Press, 2010.
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).
67.
M. Junge and C. Palazuelos, Large violations of Bell's inequalities with low entanglement, (arXiv:1007.3043).
73.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. MR 1468230
(98f:46001b)
74.Jean-Pierre
Kahane, Some random series of functions, 2nd ed., Cambridge
Studies in Advanced Mathematics, vol. 5, Cambridge University Press,
Cambridge, 1985. MR 833073
(87m:60119)
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. MR 2054436
(2004m:46028)
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. MR
843870
80.
S. Khot and A. Naor, Sharp kernel clustering algorithms and their associated Grothendieck inequalities. In Proceedings of SODA'2010, pp. 664-683.
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. MR 1152601
(93b:47062)
86.Hermann
König, On the complex Grothendieck constant in the
𝑛-dimensional case, Geometry of Banach spaces (Strobl, 1989)
London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press,
Cambridge, 1990, pp. 181–198. MR 1110195
(92g:46011)
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. MR 1877506
(2002m:41006)
88.J.
L. Krivine, Théorèmes de factorisation dans les
espaces réticulés, Séminaire Maurey-Schwartz
1973–1974: Espaces 𝐿^{𝑝}, applications radonifiantes
et géométrie des espaces de Banach, Exp. Nos. 22 et 23,
Centre de Math., École Polytech., Paris, 1974, pp. 22 (French).
MR
0440334 (55 #13209)
91.Stanislaw
Kwapień, On operators factorizable through
𝐿_{𝑝} 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. MR 0397464
(53 #1323)
92.Michel
Ledoux, The concentration of measure phenomenon, Mathematical
Surveys and Monographs, vol. 89, American Mathematical Society,
Providence, RI, 2001. MR 1849347
(2003k:28019)
96.
L. Lovász, Semidefinite programs and combinatorial optimization, Lecture Notes, Microsoft Research, Redmont, WA 98052
97.Françoise
Lust-Piquard, Inégalités de Khintchine dans
𝐶_{𝑝}(1<𝑝<∞), C. R. Acad. Sci.
Paris Sér. I Math. 303 (1986), no. 7,
289–292 (French, with English summary). MR 859804
(87j:47032)
101.B.
Maurey, Une nouvelle démonstration d’un
théorème de Grothendieck, Séminaire
Maurey-Schwartz Année 1972–1973: Espaces 𝐿^{𝑝}
et applications radonifiantes, Exp. No. 22, Centre de Math., École
Polytech., Paris, 1973, pp. 7 (French). MR 0399818
(53 #3660)
102.Bernard
Maurey, Théorèmes de factorisation pour les
opérateurs linéaires à valeurs dans les espaces
𝐿^{𝑝}, Société Mathématique de
France, Paris, 1974 (French). With an English summary; Astérisque,
No. 11. MR
0344931 (49 #9670)
103.Alexandre
Megretski, Relaxations of quadratic programs in operator theory and
system analysis, (Bordeaux, 2000) Oper. Theory Adv. Appl.,
vol. 129, Birkhäuser, Basel, 2001, pp. 365–392. MR 1882703
(2003d:90069)
107.A.
M. Olevskiĭ, Fourier series with respect to general
orthogonal systems, Springer-Verlag, New York, 1975. Translated from
the Russian by B. P. Marshall and H. J. Christoffers; Ergebnisse der
Mathematik und ihrer Grenzgebiete, Band 86. MR 0470599
(57 #10347)
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, 1986. MR 829919
(88a:47020)
122.Gilles
Pisier, A simple proof of a theorem of Kirchberg and related
results on 𝐶*-norms, J. Operator Theory 35
(1996), no. 2, 317–335. MR 1401692
(98e:46076)
123.Gilles
Pisier, The operator Hilbert space 𝑂𝐻, complex
interpolation and tensor norms, Mem. Amer. Math. Soc.
122 (1996), no. 585, viii+103. MR 1342022
(97a:46024)
124.Gilles
Pisier, Non-commutative vector valued 𝐿_{𝑝}-spaces
and completely 𝑝-summing maps, Astérisque
247 (1998), vi+131 (English, with English and French
summaries). MR
1648908 (2000a:46108)
126.Gilles
Pisier, Introduction to operator space theory, London
Mathematical Society Lecture Note Series, vol. 294, Cambridge
University Press, Cambridge, 2003. MR 2006539
(2004k:46097)
132.
F. Rădulescu, Combinatorial aspects of Connes's embedding conjecture and asymptotic distribution of traces of products of unitaries, Operator Theory20, 197-205, Theta Ser. Adv. Math.6, Theta, Bucharest, 2006.
142.Jerzy
Sawa, The best constant in the Khintchine inequality for complex
Steinhaus variables, the case 𝑝=1, Studia Math.
81 (1985), no. 1, 107–126. MR 818175
(87d:26024)
143.Robert
Schatten, A Theory of Cross-Spaces, Annals of Mathematics
Studies, no. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
(12,186e)
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. MR
1873025 (2002m:46083) M.
Takesaki, Theory of operator algebras. II, Encyclopaedia of
Mathematical Sciences, vol. 125, Springer-Verlag, Berlin, 2003.
Operator Algebras and Non-commutative Geometry, 6. MR 1943006
(2004g:46079) M.
Takesaki, Theory of operator algebras. III, Encyclopaedia of
Mathematical Sciences, vol. 127, Springer-Verlag, Berlin, 2003.
Operator Algebras and Non-commutative Geometry, 8. MR 1943007
(2004g:46080)
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. MR 788202
(86g:81009)
155.B.
S. Tsirelson, Some results and problems on quantum Bell-type
inequalities, Hadronic J. Suppl. 8 (1993),
no. 4, 329–345. MR
1254597
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. MR 1166197
(93i:46130)
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. MR 1217253
(94c:46133)
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). MR
1060730 (91i:46077)
-algebra theory (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of ``operator spaces'' or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell's inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by ``semidefinite programming'' and hence solved in polynomial time. This expository paper (where many proofs are included), presents a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author's 1986 CBMS notes. 