Mixed-norm inequalities and operator space $L_p$ embedding theory

Let $f_1, f_2, \ldots, f_n$ be a family of independent copies of a given random variable $f$ in a probability space $(\Omega, \mathcal{F}, \mu)$. Then, the following equivalence of norms holds whenever $1 \le q \le p < \infty$

$$\Big( \int_{\Omega} \Big[ \sum_{k=1}^n \vert f_k\vert^q \Big]^{\f... ...( \int_\Omega \vert f\vert^r d\mu \Big)^{\frac1r} \right\}. \tag{$\Sigma_{pq}$}$$

We prove a noncommutative analogue of this inequality for sums of free random variables over a given von Neumann subalgebra. This formulation leads to new classes of noncommutative function spaces which appear in quantum probability as square functions, conditioned square functions and maximal functions. Our main tools are Rosenthal type inequalities for free random variables, noncommutative martingale theory and factorization of operator-valued analytic functions. This allows us to generalize $(\Sigma_{pq})$ as a result for noncommutative $L_p$ in the category of operator spaces. Moreover, the use of free random variables produces the right formulation of $(\Sigma_{\infty q})$, which has not a commutative counterpart. In the last part of the paper, we use our mixed-norm inequalities to construct a completely isomorphic embedding of $L_q$ -equipped with its natural operator space structure- into some sufficiently large $L_p$ space for $1 \le p < q \le 2$. The construction of such embedding has been open for quite some time. We also show that hyperfiniteness and the QWEP are preserved in our construction.