Grothendieck inequality

From Wikipedia, the free encyclopedia

In mathematics, the Grothendieck inequality relates

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

$\max_{S_i,T_j \in B(H)} \left| \sum_{i,j} a_{ij} \langle S_i , T_j \rangle \right|$ ,

where $B (H)$ is the unit ball of a Hilbert space $H$ . The best constant $k (H)$ in

$\max_{S_i,T_j \in B(H)} \left| \sum_{i,j} a_{ij} \langle S_i , T_j \rangle \right| \leq k(H) \max_{-1 \leq s_i \leq 1, -1 \leq t_j \leq 1 } \left| \sum_{i,j} a_{ij} s_i t_j \right|, \quad a_{i,j} \in \mathbb{R}$

is called the Grothendieck constant of the Hilbert space $H$ .

Alexandre Grothendieck showed that $k (H)$ is bounded by a universal constant, independent of $H$ ; define

$k = \sup_H k(H).$

Grothendieck himself proved that

$1.57 \leq k \leq 2.3.$

Later, Krivine showed that

$1.67696\dots\leq k \leq 1.7822139781\dots;$

in spite of later efforts, the precise value of $k$ is still unknown.

[edit] References

A.Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques (French), Bol. Soc. Mat. São Paulo 8 1953 1--79
J.-L. Krivine, Constantes de Grothendieck et fonctions de type positif sur les spheres., Adv. Math. 31, 16-30, 1979.