Sum of Free Variables in Fully Symmetric Spaces

TL;DR

This paper develops a method to estimate norms of sums of free variables in symmetric spaces, enabling interpolation of inequalities for non-commutative martingales and improving classical results like Johnson-Schechtman and Khintchine inequalities.

Contribution

It introduces a new approach based on Voiculescu's inequality for norm estimates, applicable to various non-commutative settings and inequalities.

Findings

Established norm estimates for free variables in symmetric spaces.

Interpolated Burkholder inequalities for non-commutative martingales.

Improved Johnson-Schechtman and Khintchine inequalities for free groups.

Abstract

We give a method to obtain, from Voiculescu's inequality, norm estimates for sums of free variables with amalgamation in general fully symmetric spaces. We use these estimates to interpolate the Burkholder inequalities for non commutative martingales. The method is also applicable to other similar settings. In that spirit, we improve known results on the non commutative Johnson-Schechtman inequalities and recover Khintchine inequalities associated to free groups.