AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
The Grothendieck inequality revisited
About this Title
Ron Blei, Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 232, Number 1093
ISBNs: 978-0-8218-9855-0 (print); 978-1-4704-1896-0 (online)
DOI: https://doi.org/10.1090/memo/1093
Published electronically: March 11, 2014
Keywords: The Grothendieck inequality,
Parseval-like formulas,
integral representations,
fractional Cartesian products,
projective continuity,
projective boundedness
MSC: Primary 46C05, 46E30; Secondary 47A30, 42C10
Table of Contents
Chapters
- 1. Introduction
- 2. Integral representations: the case of discrete domains
- 3. Integral representations: the case of topological domains
- 4. Tools
- 5. Proof of Theorem 3.5
- 6. Variations on a theme
- 7. More about $\Phi$
- 8. Integrability
- 9. A Parseval-like formula for $\langle {\bf {x}}, {\bf {y}}\rangle$, ${\bf {x}} \in l^p$, ${\bf {y}} \in l^q$
- 10. Grothendieck-like theorems in dimensions $>2$?
- 11. Fractional Cartesian products and multilinear functionals on a Hilbert space
- 12. Proof of Theorem 11.11
- 13. Some loose ends
Abstract
The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved, concerning continuous functions of two variables over general topological domains. The main result is the construction of a continuous map $\Phi$ from $l^2(A)$ into $L^2(\Omega _A, \mathbb {P}_A)$, where $A$ is a set, $\Omega _A = \{-1,1\}^A$, and $\mathbb {P}_A$ is the uniform probability measure on $\Omega _A$, such that \begin{equation}\tag {1} \sum _{\alpha \in A} \textbf {{x}}(\alpha ) \bar {\textbf {{y}}(\alpha )} = \int _{\Omega _A} \Phi (\textbf {{x}})\Phi (\bar {\textbf {{y}}})d \mathbb {P}_A, \ \ \ \textbf {{x}} \in l^2(A), \ \textbf {{y}} \in l^2(A), \end{equation} and \begin{equation}\tag {2} \|\Phi (\textbf {{x}})\|_{L^{\infty }} \leq K \|\textbf {{x}}\|_2, \ \ \ \textbf {{x}} \in l^2(A), \end{equation} for an absolute constant $K > 1$. ($\Phi$ is non-linear, and does not commute with complex conjugation.) The Parseval-like formula in (1) is obtained by iterating the usual Parseval formula in a framework of harmonic analysis on dyadic groups. A modified construction implies a similar integral representation of the dual action between $l^p$ and $l^q$, $\frac {1}{p} + \frac {1}{q} = 1$.
Variants of the Grothendieck inequality in higher dimensions are derived. These variants involve representations of functions of $n$ variables in terms of functions of $k$ variables, $0 < k < n.$ Multilinear Parseval-like formulas are obtained, extending the bilinear formula in (1). The resulting formulas imply multilinear extensions of the Grothendieck inequality, and are used to characterize the feasibility of integral representations of multilinear functionals on a Hilbert space.
- Noga Alon, Konstantin Makarychev, Yury Makarychev, and Assaf Naor, Quadratic forms on graphs, Invent. Math. 163 (2006), no. 3, 499–522. MR 2207233, DOI 10.1007/s00222-005-0465-9
- Noga Alon and Assaf Naor, Approximating the cut-norm via Grothendieck’s inequality, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 72–80. MR 2121587, DOI 10.1145/1007352.1007371
- Ron C. Blei, A tensor approach to interpolation phenomena in discrete abelian groups, Proc. Amer. Math. Soc. 49 (1975), 175–178. MR 361626, DOI 10.1090/S0002-9939-1975-0361626-2
- Ron C. Blei, A uniformity property for $\Lambda (2)$ sets and Grothendieck’s inequality, Symposia Mathematica, Vol. XXII (Convegno sull’Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976) Academic Press, London, 1977, pp. 321–336. MR 0487238
- R. C. Blei, Multidimensional extensions of the Grothendieck inequality and applications, Ark. Mat. 17 (1979), no. 1, 51–68. MR 543503, DOI 10.1007/BF02385457
- Ron Blei, Interpolation sets and extensions of the Grothendieck inequality, Illinois J. Math. 24 (1980), no. 2, 180–187. MR 575057
- Ron Blei, Analysis in integer and fractional dimensions, Cambridge Studies in Advanced Mathematics, vol. 71, Cambridge University Press, Cambridge, 2001. MR 1853423
- Ron Blei, Measurements of interdependence, Lith. Math. J. 51 (2011), no. 2, 141–154. MR 2805733, DOI 10.1007/s10986-011-9114-8
- Ron C. Blei and James H. Schmerl, Combinatorial dimension of fractional Cartesian products, Proc. Amer. Math. Soc. 120 (1994), no. 1, 73–77. MR 1160291, DOI 10.1090/S0002-9939-1994-1160291-6
- Aline Bonami, Étude des coefficients de Fourier des fonctions de $L^{p}(G)$, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 2, 335–402 (1971) (French, with English summary). MR 283496
- J. Bourgain, Subspaces of $l^\infty _N$, arithmetical diameter and Sidon sets, Probability in Banach spaces, V (Medford, Mass., 1984) Lecture Notes in Math., vol. 1153, Springer, Berlin, 1985, pp. 96–127. MR 821978, DOI 10.1007/BFb0074947
- Adam Bowers, A measure-theoretic Grothendieck inequality, J. Math. Anal. Appl. 369 (2010), no. 2, 671–677. MR 2651712, DOI 10.1016/j.jmaa.2010.04.020
- M. Braverman, K. Makarychev, Y. Makarychev, and A. Naor. The Grothendieck constant is strictly smaller than Krivine’s bound. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 453–462. IEEE, 2011.
- T. K. Carne, Banach lattices and extensions of Grothendieck’s inequality, J. London Math. Soc. (2) 21 (1980), no. 3, 496–516. MR 577725, DOI 10.1112/jlms/s2-21.3.496
- M. Charikar and A. Wirth. Maximizing quadratic programs: extending Grothendieck’s inequality. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004.
- A. M. Davie, Quotient algebras of uniform algebras, J. London Math. Soc. (2) 7 (1973), 31–40. MR 324427, DOI 10.1112/jlms/s2-7.1.31
- Maurice Fréchet, Sur les fonctionnelles bilinéaires, Trans. Amer. Math. Soc. 16 (1915), no. 3, 215–234 (French). MR 1501010, DOI 10.1090/S0002-9947-1915-1501010-5
- Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 238, Springer-Verlag, New York-Berlin, 1979. MR 550606
- A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79 (French). MR 94682
- Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
- Uffe Haagerup, A new upper bound for the complex Grothendieck constant, Israel J. Math. 60 (1987), no. 2, 199–224. MR 931877, DOI 10.1007/BF02790792
- Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
- S. Kaijser. Private communication. October 1977.
- 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). MR 1994597, DOI 10.1016/S1631-073X(03)00226-7
- Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition, Dover Publications, Inc., New York, 1976. MR 0422992
- Subhash Khot and Assaf Naor, Grothendieck-type inequalities in combinatorial optimization, Comm. Pure Appl. Math. 65 (2012), no. 7, 992–1035. MR 2922372, DOI 10.1002/cpa.21398
- Jean-Louis Krivine, Sur la constante de Grothendieck, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 8, A445–A446. MR 428414
- J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326
- J. E. Littlewood. On bounded bilinear forms in an infinite number of variables. Quart. J. Math. Oxford, 1:164–174, 1930.
- Gilles Pisier, Grothendieck’s theorem for noncommutative $C^{\ast }$-algebras, with an appendix on Grothendieck’s constants, J. Functional Analysis 29 (1978), no. 3, 397–415. MR 512252, DOI 10.1016/0022-1236(78)90038-1
- Gilles Pisier, Grothendieck’s theorem, past and present, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 2, 237–323. MR 2888168, DOI 10.1090/S0273-0979-2011-01348-9
- Friedrich Riesz, Über die Fourierkoeffizienten einer stetigen Funktion von beschränkter Schwankung, Math. Z. 2 (1918), no. 3-4, 312–315 (German). MR 1544321, DOI 10.1007/BF01199414
- Ronald E. Rietz, A proof of the Grothendieck inequality, Israel J. Math. 19 (1974), 271–276. MR 367628, DOI 10.1007/BF02757725
- R. Salem and A. Zygmund, On a theorem of Banach, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 293–295. MR 21611, DOI 10.1073/pnas.33.10.293
- Robert Schatten, On the direct product of Banach spaces, Trans. Amer. Math. Soc. 53 (1943), 195–217. MR 7568, DOI 10.1090/S0002-9947-1943-0007568-7
- Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
- Anthony Man-Cho So, Jiawei Zhang, and Yinyu Ye, On approximating complex quadratic optimization problems via semidefinite programming relaxations, Math. Program. 110 (2007), no. 1, Ser. B, 93–110. MR 2306132, DOI 10.1007/s10107-006-0064-6
- Andrew Tonge, The von Neumann inequality for polynomials in several Hilbert-Schmidt operators, J. London Math. Soc. (2) 18 (1978), no. 3, 519–526. MR 518237, DOI 10.1112/jlms/s2-18.3.519
- N. Th. Varopoulos, Tensor algebras and harmonic analysis, Acta Math. 119 (1967), 51–112. MR 240564, DOI 10.1007/BF02392079
- N. Th. Varopoulos, On a problem of A. Beurling, J. Functional Analysis 2 (1968), 24–30. MR 0241982, DOI 10.1016/0022-1236(68)90023-2
- N. Th. Varopoulos, Tensor algebras over discrete spaces, J. Functional Analysis 3 (1969), 321–335. MR 0250087, DOI 10.1016/0022-1236(69)90046-9
- N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Functional Analysis 16 (1974), 83–100. MR 0355642, DOI 10.1016/0022-1236(74)90071-8
- David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. MR 1155402