On a variant of the Grothendieck inequality and estimates on tensor product norms

Abstract

We investigate a Grothendieck-type inequality for pairs of Banach spaces $E,F$ assuming $E$ is finite-dimensional and study the associated Grothendieck-type constant. We prove that if there is a $C>0$ such that $\Vert A\otimes {\mathrm{id}}_F{\Vert }_{E_m\check{\otimes }F\to E_n^{\ast }\hat{\otimes }F}⩽C\Vert A{\Vert }_{E_m\to E_n^{\ast }}$ for all $m,n\in \mathbb{N},$ where $\dim E_n=n$, then both $F$ and $F^{\ast }$ must have finite cotype. Moreover, assuming that $F$ has the bounded approximation property and that the conjecture in \cite{PisierDuality} has an affirmative answer, we show that $(E_n^{\ast }{)}_{n⩾1}$ satisfies G.T. uniformly. We show that the Grothendieck-type constant defined for a pair of Banach spaces $(E,F)$ is closely related to another interesting quantity introduced recently in \cite{XOR games and GT} comparing the projective and injective norms on the tensor product of two finite-dimensional Banach spaces $E$ and $F$. We also study analogously the constants appearing in these extremal problems by restricting only to non-negative tensors. For contractive \emph{little} Parrott homomorphisms ${\varrho }_V:H^{\infty }(\Omega )\to M_n$, where $\Omega$ is the dual unit ball of a finite dimensional Banach space $(E,\Vert \cdot \Vert )$, we prove the sharp estimate $\Vert {\varrho }_V{\Vert }_{\mathrm{cb}}\le \sqrt{\gamma (E)},$ $\gamma (E)$ being the positive Grothendieck constant associated with the pair $(E,{\ell }_2^n)$. %\st{with extremal cases achieving equality.} This yields a new proof of \cite[Theorem 2.1]{Davidchoi} using the lower bound $K_G^{+}({\ell }_{\infty }^4,{\ell }_2^2)\ge 1.1658$ obtained in this paper.

Preliminary version, 27 pages, major improvement in presentation and new results added