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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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