Non-commutative Khintchine type inequalities associated with free groups Javier ParcetGilles Pisier 46L0746L5246L54non-commutative Khintchine inequalityFell's absorption principlefree group Let $\mathbf{F}_n$ denote the free group with $n$ generators $g_1$, $g_2$, $\dots$, $g_n$. Let $\lambda$ stand for the left regular representation of $\mathbf{F}_n$ and let $\tau$ be the standard trace associated to $\lambda$. Given any positive integer $d$, we study the operator space structure of the subspace $\mathcal{W}_p(n,d)$ of $L_p(\tau)$ generated by the family of operators $\lambda(g_{i_1}g_{i_2}\cdots g_{i_d})$ 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 $1 \le p \le \infty$ a previous result of Buchholz for the space $\mathcal{W}_{\infty}(n,d)$. 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 $\ell_2 otimes \ell_2$ and, moreover, such that $(\sum_{ij}a_{ij}^{*} a_{ij})^{1/2}$ and $(\sum_{ij}a_{ij} a_{ij}^{*})^{1/2}$ both belong to $S_p$. We equip $\mathcal{K}_p$ with the maximum of the three corresponding norms. Then, for $2 \le p \le \infty$, we have $\mathcal{K}_p \simeq (\mathcal{K}_2,\mathcal{K}_{\infty})_{\theta}$ with $1/p = (1 - \theta)/2$. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2612 10.1512/iumj.2005.54.2612 en Indiana Univ. Math. J. 54 (2005) 531 - 556 state-of-the-art mathematics http://iumj.org/access/