Non-commutative Khintchine type inequalities associated with free groups

Let ${\text{Free}}_n$ denote the free group with $n$ generators $g_1,g_2,\mathrm{.}\mathrm{.}\mathrm{.},g_n$. Let $\lambda$ stand for the left regular representation of ${\text{Free}}_n$ and let $\tau$ be the standard trace associated to $\lambda$. Given any positive integer $d$, we study the operator space structure of the subspace ${\text{Word}}_p(n,d)$ of $L_p(\tau )$ generated by the family of operators $\lambda (g_{i_1}{g}_{i_2}\mathrm{.}\mathrm{.}\mathrm{.}{g}_{i_d})$ with $1\le i_k\le n$. Moreover, our description of this operator space holds up to a constant which does not depend on $n$ or $p$, so that our result remains valid for infinitely many generators. We also consider the subspace of $L_p(\tau )$ generated by the image under $\lambda$ of the set of reduced words of length $d$. Our result extends to any exponent $1\le p\le \mathrm{\infty }$ a previous result of Buchholz for the space ${\text{Word}}_{\mathrm{\infty }}(n,d)$. The main application is a certain interpolation theorem, valid for any degree $d$ (extending a result of the second author restricted to $d=1$). In the simplest case $d=2$, our theorem can be stated as follows: consider the space ${\mathcal{K}}_p$ formed of all block matrices $a=(a_{ij})$ with entries in the Schatten class $S_p$, such that $a$ is in $S_p$ relative to ${\mathrm{\ell }}_2\otimes {\mathrm{\ell }}_2$ and moreover such that $(\sum_{ij} a_{ij}^* a_{ij} )^{1/2}$and$(\sum_{ij} a_{ij} a_{ij}^*)^{1/2}$bothbelongto$S_p$\mathrm{.}Weequip$$\mathcal{K}$_p$withthemaximumofthethreecorrespondingnorms\mathrm{.}Then,for$2 \le p \le \infty$wehave$$\mathcal{K}$p \simeq ($\mathcal{K}$2, $\mathcal{K}$\infty)\theta$with$1/p = (1-\theta)/2$.