Noncommutative Boyd interpolation theorems

TL;DR

This paper introduces a new elementary proof of Boyd's interpolation theorem, extending it to noncommutative symmetric spaces and deriving several key inequalities for noncommutative martingales.

Contribution

It provides the first elementary proof of noncommutative Boyd interpolation theorems and extends classical inequalities to noncommutative symmetric spaces.

Findings

Established a noncommutative version of Boyd's interpolation theorem.

Derived noncommutative maximal inequalities including Doob's and dual Doob's.

Proved Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative martingales.

Abstract

We present a new, elementary proof of Boyd's interpolation theorem. Our approach naturally yields a noncommutative version of this result and even allows for the interpolation of certain operators on l^1-valued noncommutative symmetric spaces. By duality we may interpolate several well-known noncommutative maximal inequalities. In particular we obtain a version of Doob's maximal inequality and the dual Doob inequality for noncommutative symmetric spaces. We apply our results to prove the Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative martingales in these spaces.