K-closedness results in noncommutative Lebesgue spaces with filtrations

TL;DR

This paper proves a new $K$-closedness result for noncommutative Lebesgue spaces with filtrations, advancing the understanding of noncommutative martingale Hardy spaces through interpolation techniques.

Contribution

It introduces a general $K$-closedness theorem in noncommutative Lebesgue spaces and applies it to various noncommutative martingale Hardy spaces, solving a previously open problem.

Findings

Established a new $K$-closedness result for noncommutative Lebesgue spaces.

Derived $K$-closedness results for noncommutative martingale Hardy spaces.

Adapted Bourgain's approach to noncommutative martingales.

Abstract

In this paper, we establish a new general $K$-closedness result in the context of real interpolation of noncommutative Lebesgue spaces involving filtrations. As an application, we derive $K$-closedness results for various classes of noncommutative martingale Hardy spaces, addressing a problem raised by Randrianantoanina. The proof of this general result adapts Bourgain's approach to the real interpolation of classical Hardy spaces on the disk within the framework of noncommutative martingales.