FINITE DIMENSIONAL SUBSPACES OF NONCOMMUTATIVE Lp Spaces

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 i...

ar X iv : m at h / 03 07 16 9 v 1 [ m at h . FA ] 1 1 Ju l 2 00 3 On Subspaces of Non - commutative L p - Spaces

We study some structural aspects of the subspaces of the non-commutative (Haagerup) Lp-spaces associated with a general (non necessarily semi-finite) von Neumann algebra a. If a subspace X of Lp(a) contains uniformly the spaces lnp , n ≥ 1, it contains an almost isometric, almost 1-complemented copy of lp. If X contains uniformly the finite dimensional Schatten classes S p , it contains their l...

Noncommutative Burkholder/Rosenthal inequalities II: applications

We show norm estimates for the sum of independent random variables in noncommutative Lp-spaces for 1 < p <∞ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications, we derive an equivalence for the p-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces an...

A New Class of Banach Sequence Spaces

Stability of low-rank matrix recovery and its connections to Banach space geometry

Abstract. There are well-known relationships between compressed sensing and the geometry of the finite-dimensional lp spaces. A result of Kashin and Temlyakov [20] can be described as a characterization of the stability of the recovery of sparse vectors via l1minimization in terms of the Gelfand widths of certain identity mappings between finitedimensional l1 and l2 spaces, whereas a more recen...