Finite dimensional subspaces of noncommutative $L_p$ spaces

TL;DR

This paper extends classical results to noncommutative Lp spaces, showing finite-dimensional subspaces are close to certain operator space structures with bounds depending on dimension and p.

Contribution

It provides a noncommutative analogue of Lewis's theorem, establishing bounds on cb-distances and projections for subspaces of noncommutative Lp spaces.

Findings

Bound on cb-distance between subspace and operator space structure

Existence of a projection with controlled cb-norm onto the subspace

Extension of classical change of density argument to noncommutative setting

Abstract

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.