The non-commutative Khintchine inequalities for p<1

TL;DR

This paper establishes non-commutative Khintchine inequalities for all p between 0 and 1, extending classical results to new settings including free probability and non-commutative Lp spaces.

Contribution

It provides the first proof of Khintchine inequalities for 0<p<1 in non-commutative Lp spaces, introduces a new factorization for operators, and shows H"older continuity of Mazur maps in this context.

Findings

Khintchine inequalities hold for 0<p<1 in non-commutative Lp spaces.

New factorization theorem for operators from Hilbert spaces to non-commutative Lp spaces.

Mazur maps are H"older continuous on semifinite von Neumann algebras.

Abstract

We give a proof of the Khintchine inequalities in non-commutative $L_p$-spaces for all $0< p<1$. These new inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, e.g. for the analogues of such random variables in free probability. We also prove a factorization for operators from a Hilbert space to a non commutative $L_p$-space, which is new for $0<p<1$. We end by showing that Mazur maps are H\"older on semifinite von Neumann algebras. The main tool is a new form of H\"older inequality for non commutative Lp spaces with weights.