K-closedness results in noncommutative Lebesgue spaces with filtrations

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.