  Let $f_1, f_2, ..., f_n$ 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 $1 \le q \le p < \infty$
$$\Big(\int_{\Omega} \Big[ \sum_{k=1}^n |f_k|^q \Big]^{\frac{p}{q}} d \mu
\Big)^{\frac1p} \sim \max_{r \in \{p,q\}} {n^{\frac1r} \Big(\int_\Omega |f|^r
d\mu \Big)^{\frac1r}}.$$ 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 this
inequality as a result for noncommutative L_p in the category of operator
spaces. Moreover, the use of free random variables produces the right
formulation for $p=\infty$, which has not a commutative counterpart.
