Non-commutative Khintchine type inequalities associated with free groups

Abstract

Let $\Free_n$ denote the free group with $n$ generators $g_1, g_2, ..., g_n$. Let $\lambda$ stand for the left regular representation of $\Free_n$ and let $\tau$ be the standard trace associated to $\lambda$. Given any positive integer $d$, we study the operator space structure of the subspace $\Word_p(n,d)$ of $L_p(\tau)$ generated by the family of operators $\lambda(g_{i_1}g_{i_2} ... g_{i_d})$ 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 $\Word_{\infty}(n,d)$. 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 $1/p = (1-\theta)/2$.