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 $\M$. For $0<p \leq\infty$, let $\h_p^c(\mathcal{M})$ 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 $\big(\h_p^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)$ 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 \[ \big(\h_E^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)_{\theta, r}=\h_{F}^c(\mathcal{M}). \] 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 $t>0$, \[ \big(\h_\Phi^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)_{\theta, r}=\h_{\Phi_0, r}^c(\mathcal{M}) \] where $\h_{\Phi_0,r}^c(\mathcal{M})$ is the noncommutative column Hardy space associated with the Orlicz-Lorentz space $L_{\Phi_0,r}$.