We consider the following noncommutative version of Lewis’s classical result. Every ndimensional subspace E of Lp(M) (1 < p < ∞) for a von Neumann algebra M satisfies

dcb(E, RCn p ′) ≤ cp · n absfrac12− 1 p

for some constant cp depending only on p, where 1 p + 1 p ′ = 1 and RCn p ′ = [Rn ∩ Cn, Rn + Cn] 1 p′ . Moreover, there is a projection P : Lp(M) → Lp(M) onto E with normPcb ≤ cp · n absfrac12−frac1p . We follow the classical change of density argument with appropriate noncommutative variations in addition to the opposite trick.