  We show norm estimates for the sum of independent random variables in
noncommutative $L_p$-spaces for $1<p<\infty$ following our previous work. These
estimates generalize the classical Rosenthal inequality in the commutative
case. Among applications, we derive an equivalence for the $p$-norm of the
singular values of a random matrix with independent entries, and characterize
those symmetric subspaces and unitary ideals which can be realized as subspaces
of a noncommutative $L_p$ for $2<p<\infty$.
