P. Jones'Interpolation theorem for noncommutative martingale Hardy spaces

Abstract

Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $({\mathcal{M}}_n{)}_{n\ge 1}$ of (semifinite) von Neumann subalgebras of $\M$. For $0<p\le \infty$, let $\h_p^c(\mathcal{M})$ 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 $0<p<\infty$, the compatible couple $\big(\h_p^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)$ is $K$-closed in the couple $(L_p(\mathcal{N}),L_{\infty }(\mathcal{N}))$ 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 $(H_1,H_{\infty })$. 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 $0<r\le \infty$. If $(E,L_{\infty }{)}_{\theta ,r}=F$, then \[ \big(\h_E^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)_{θ, r}=\h_{F}^c(\mathcal{M}). \] As an illustration, we obtain that if $\Phi$ is an Orlicz function that is $p$-convex and $q$-concave for some $0<p\le q<\infty$, then the following interpolation on the noncommutative column Orlicz-Hardy space holds: for $0<\theta <1$, $0<r\le \infty$, and ${\Phi }_0^{-1}(t)=[{\Phi }^{-1}(t){]}^{1-\theta }$ for $t>0$, \[ \big(\h_Φ^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)_{θ, r}=\h_{Φ_0, r}^c(\mathcal{M}) \] where $\h_{Φ_0,r}^c(\mathcal{M})$ is the noncommutative column Hardy space associated with the Orlicz-Lorentz space $L_{{\Phi }_0,r}$.

34 pages