Rosenthal inequalities in noncommutative symmetric spaces

We give a direct proof of the ‘upper’ Khintchine inequality for a noncommutative symmetric (quasi-)Banach function space with nontrivial upper Boyd index. This settles an open question of C. Le Merdy and the fourth named author (Le Merdy and Sukochev, 2008 [24]). We apply this result to derive a version of Rosenthalʼs theorem for sums of independent random variables in a noncommutative symmetric space. As a result we obtain a new proof of Rosenthalʼs theorem for (Haagerup) Lp-spaces.