We prove the following noncommutative version of Lewis's classical result. Every n-dimensional subspace E of Lp(M) (1<p<\infty) for a von Neumann algebra M satisfies d_{ cb}(E, RC^n_{ p'}) \leq c_p n^{ \abs{ 1/2-1/p}} for some constant c_p depending only on $p$, where $1/p +1/p' =1$ and $RC^n_{ p'} = [R_n\cap C_n, R_n+C_n]_{ 1/p'}$. Moreover, there is a projection $P:Lp(M) --> Lp(M)$ onto E with $\norm{ P}_{ cb} \leq c_p n^{ \abs{ 1/2-1/p}}.$ We follow the classical change of density argument with appropriate noncommutative variations in addition to the opposite trick.
Categories: math.FA math.OA