Extreme points, positive Grothendieck constants and tensor product norms

Abstract

We study several interrelated problems arising from the interplay between extreme point theory, Grothendieck-type inequalities, and tensor product norms. We develop a general framework for characterizing the extreme points of the set of positive contractions ${\mathcal{A}}_{X\to Y}$ between finite-dimensional Banach spaces, with explicit results for $X={\ell }_1^n$, $Y={\ell }_{\infty }^n$ and vice versa. These characterizations are applied to evaluate several constants exactly. We show that the positive Grothendieck constant $K_G^{+,\mathbb{R}}(3)$ equals $9/8$ and that the smallest constant ${\rho }^{+}(X)$ for which $\Vert A{\Vert }_{\pi }⩽{\rho }^{+}(X)\Vert A{\Vert }_{\epsilon }$ holds for all $A⩾0$ equals $5/4$ when $X={\ell }_{\infty }^3(\mathbb{R})$. We also prove that ${\rho }^{+}(X)=1$ when $X={\ell }_{\infty }^n(\mathbb{C})$ and $n⩽3$. Finally, we prove that ${\rho }^{+}(X)=1$ for every 2-dimensional subspace $X$ of ${\ell }_{\infty }^3(\mathbb{C})$; since this is stronger than the 2-summing property, it recovers Proposition~4.4 of \cite{AFJS95}.

27 pages