Some points of view on Grothendieck's inequalities
Erik Christensen
TL;DR
The paper investigates finite-dimensional Grothendieck-type inequalities through the lens of operator-space theory and cross norms, linking the little and full inequalities via completely bounded maps. It develops Haagerup's constructive factorization in the abelian, finite-dimensional setting, showing $\|X\|_{cbF} \le \sqrt{k_G^{\mathbb C}}\|X\|_F$ and proving $\|X\|_{cbF}=\|X\|_F$ for matrices with nonnegative entries (giving $k_G^{\mathbb C}=1$ in that case). A key analytic relation is established: $\|P\|_{cbB} \le k_G^{\mathbb C}\|P\|_B$ for positive $P$, which together with a block-factorization yields the bound $K_G^{\mathbb C} \le \dfrac{k_G^{\mathbb C}}{2 - k_G^{\mathbb C}}$ on the Grothendieck constant, connecting to a geometric reformulation via ${\mathcal Q}_n$ and ${\mathcal R}_n$. The geometrical characterization identifies $k_G^{\mathbb C}$ as the smallest constant ensuring ${\mathcal Q}_n \subseteq k_G^{\mathbb C}{\mathcal R}_n - M_n({\mathbb C})_+$ for all $n$, and yields a Schur-multiplier version of Grothendieck's inequality. Overall, the work provides constructive, algebraic and geometric insights into Grothendieck-type inequalities in the finite-dimensional, commutative setting and clarifies the role of positivity and convex geometry in determining the constants.
Abstract
Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps implies that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constant in the complex case.