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 $\|A\otimes \operatorname{id}_{F}\|_{E_m\check{\otimes}F\to E_n^*\hat{\otimes}F}\leqslant C \|A\|_{E_m\to E_n^*}$ for all $m,n\in\mathbb{N},$ where $\dim E_n=n$, then both $F$ and $F^*$ 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 $(E_n^*)_{n\geqslant 1}$ 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 $(E,\|\cdot\|)$, we prove the sharp estimate $ \|\varrho_V\|_{\mathrm{cb}}\leq\sqrt{\gamma(E)}, $ $\gamma(E)$ being the positive Grothendieck constant associated with the pair $(E, \ell^n_2)$. %\st{with extremal cases achieving equality.} This yields a new proof of \cite[Theorem 2.1]{Davidchoi} using the lower bound $K_G^+(\ell_\infty^4,\ell_2^2) \geq 1.1658$ obtained in this paper.