Rosenthal's theorem for subspaces of noncommutative Lp

We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative Lp space for some p>1. This is a noncommutative version of Rosenthal's result for commutative Lp spaces. Similarly for 1 < q < 2, an infinite dimensional subspace X of a noncommutative Lq space either contains lq or embeds in Lp for some q < p < 2. The novelty in the noncommutative setting is a double sided change of density.