Published online by Cambridge University Press: 25 August 2021
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 <\infty $ , let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and $\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$ .
We prove the following real interpolation identity: if
$0<p <\infty $
and
$0<\theta <1$
, then for
$1/r=(1-\theta )/p$
,
For the case of complex interpolation, we obtain that if
$0<p<q<\infty $
and
$0<\theta <1$
, then for
$1/r =(1-\theta )/p +\theta /q$
,
These extend previously known results from $p\geq 1$ to the full range $0<p<\infty $ . Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned $L_p$ -spaces are also shown to form interpolation scale for the full range $0<p<\infty $ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned $L_p$ -spaces.
We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.