Grothendieck inequality

In mathematics, the Grothendieck inequality states that there is a universal constant k with the following property. If ai,j is an n by n (real or complex) matrix with

$\left| \sum_{i,j} a_{ij} s_i t_j \right|\le 1$

for all (real or complex) numbers si, tj of absolute value at most 1, then

$\left| \sum_{i,j} a_{ij} \langle S_i , T_j \rangle \right|\le k$,

for all vectors Si, Tj in the unit ball B(H) of a (real or complex) Hilbert space H. The smallest constant k which satisfies this property for all n by n matrices is called a Grothendieck constant and denoted k(n); in fact there are two Grothendieck constants kR(n) and kC(n) for each n depending on whether one works with real or complex numbers, respectively.

The sequences kR(n) and kC(n) are easily seen to be increasing, and Alexander Grothendieck's result states that they are bounded,[1][2] so they have limits.

If we define[3]kR to be supnkR(n) then Grothendieck proved that: $1.57 \approx \frac{\pi}{2} \leq k_{\R} \leq \mathrm{sinh}(\frac{\pi}{2}) \approx 2.3$.

Later Krivine[4] improved the result by proving: 1.67696... ≤ kR ≤ 1.7822139781...=$\frac{\pi}{2 \ln(1+\sqrt{2})}$, conjecturing that the upper bound is tight. However, this conjecture was disproved in a preprint by Braverman, Makarychev, Makarychev and Naor.[5]

References

1. ^ Grothendieck, Alexander (1953), "Résumé de la théorie métrique des produits tensoriels topologiques", Bol. Soc. Mat. Sao Paulo 8: 1–79, MR 0094682
2. ^ Blei, Ron C. (1987), "An elementary proof of the Grothendieck inequality", Proceedings of the American Mathematical Society (American Mathematical Society) 100 (1): 58–60, doi:10.2307/2046119, ISSN 0002-9939, JSTOR 2046119, MR 883401
3. ^ Finch, Steven R. (2003), Mathematical constants, Cambridge University Press, ISBN 978-0-521-81805-6
4. ^ Krivine, J.-L. (1979), "Constantes de Grothendieck et fonctions de type positif sur les sphères", Advances in Mathematics 31 (1): 16–30, doi:10.1016/0001-8708(79)90017-3, ISSN 0001-8708, MR 521464
5. ^ Braverman, Mark; Makarychev, Konstantin; Makarychev, Yury; Naor, Assaf (2011). "The Grothendieck constant is strictly smaller than Krivine's bound". arXiv:1103.6161 [math.FA].
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