P. Jones’ interpolation theorem for noncommutative martingale Hardy spaces

Abstract:

Let $\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal {M}$. For $0<p \leq \infty$, let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$ and the index $p$. We prove that for $0<p<\infty$, the compatible couple $\big (\mathsf {h}_p^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big )$ is $K$-closed in the couple $\big (L_p(\mathcal {N}), L_\infty (\mathcal {N}) \big )$ 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 $(H_1, H_\infty )$.

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 $(0, \infty )$ satisfying some natural conditions, $0<\theta <1$, and $0<r\leq \infty$. If $(E,L_\infty )_{\theta ,r}=F$, then \[ \big (\mathsf {h}_E^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big )_{\theta , r}=\mathsf {h}_{F}^c(\mathcal {M}). \] As an illustration, we obtain that if $\Phi$ is an Orlicz function that is $p$-convex and $q$-concave for some $0<p\leq q<\infty$, then the following interpolation on the noncommutative column Orlicz-Hardy space holds: for $0<\theta <1$, $0<r\leq \infty$, and $\Phi _0^{-1}(t)=[\Phi ^{-1}(t)]^{1-\theta }$ for $t>0$, \[ \big (\mathsf {h}_\Phi ^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big )_{\theta , r}=\mathsf {h}_{\Phi _0, r}^c(\mathcal {M}), \] where $\mathsf {h}_{\Phi _0,r}^c(\mathcal {M})$ is the noncommutative column Hardy space associated with the Orlicz-Lorentz space $L_{\Phi _0,r}$.