P. Jones'Interpolation theorem for noncommutative martingale Hardy spaces

Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $({\mathcal{M}}_n{)}_{n\ge 1}$ of (semifinite) von Neumann subalgebras of $\text{M}$. For $0<p\le \mathrm{\infty }$, let ${\text{h}}_p^c(\mathcal{M})$ denote the noncommutative column conditioned martingale Hardy space associated with the filtration $({\mathcal{M}}_n{)}_{n\ge 1}$ and the index $p$. We prove that for $0<p<\mathrm{\infty }$, the compatible couple $({\text{h}}_p^c(\mathcal{M}),{\text{h}}_{\mathrm{\infty }}^c(\mathcal{M}))$ is $K$-closed in the couple $(L_p(\mathcal{N}),L_{\mathrm{\infty }}(\mathcal{N}))$ for an appropriate amplified semifinite von Neumann algebra $\mathcal{N}\supset \mathcal{M}$. This may be viewed as a noncommutative analogue of P. Jones interpolation of the couple $(H_1,H_{\mathrm{\infty }})$. As an application, we prove a general automatic transfer of real interpolation results from couples of symmetric quasi-Banach function spaces to the corresponding couples of noncommutative conditioned martingale Hardy spaces. More precisely, assume that $E$ is a symmetric quasi-Banach function space on $(0,\mathrm{\infty })$ satisfying some natural conditions, $0<\theta <1$, and $0<r\le \mathrm{\infty }$. If $(E,L_{\mathrm{\infty }}{)}_{\theta ,r}=F$, then $$({\text{h}}_E^c(\mathcal{M}),{\text{h}}_{\mathrm{\infty }}^c(\mathcal{M}){)}_{\theta ,r}={\text{h}}_F^c(\mathcal{M})\mathrm{.}$$ As an illustration, we obtain that if $\mathrm{\Phi }$ is an Orlicz function that is $p$-convex and $q$-concave for some $0<p\le q<\mathrm{\infty }$, then the following interpolation on the noncommutative column Orlicz-Hardy space holds: for $0<\theta <1$, $0<r\le \mathrm{\infty }$, and ${\mathrm{\Phi }}_0^{-1}(t)=[{\mathrm{\Phi }}^{-1}(t){]}^{1-\theta }$ for $t>0$, $$({\text{h}}_{\mathrm{\Phi }}^c(\mathcal{M}),{\text{h}}_{\mathrm{\infty }}^c(\mathcal{M}){)}_{\theta ,r}={\text{h}}_{{\mathrm{\Phi }}_0,r}^c(\mathcal{M})$$ where ${\text{h}}_{{\mathrm{\Phi }}_0,r}^c(\mathcal{M})$ is the noncommutative column Hardy space associated with the Orlicz-Lorentz space $L_{{\mathrm{\Phi }}_0,r}$.