P. Jones' interpolation theorem for noncommutative martingale Hardy spaces. II (Q6591575)

Let \(M\) be a von Neumann algebra equipped with a faithful normal semifinite trace \(\tau\) and an increasing family (filtration) of semifinite von Neumann subalgebras \((M_k)_{k=1}^\infty\). Given an eventually constant martingale \((x_k)_{k=1}^\infty\) subordinated to the filtration as above we can first consider a (finite) martingale difference sequence \((dx_k:=x_{k+1}- x_k)_{k=1}^\infty\), and then, for fixed \(p \in [1,\infty]\), the Hardy column norm\N\[\N\|(dx_k)_{k=1}^\infty\|_{H^c_p(M)}:= \tau\left(\left( \sum_{k=1}^\infty |dx_k|^2\right)^{\frac{p}{2}} \right)^{\frac{1}{p}}.\N\]\NCompleting the space of finite martingale difference sequences with respect to the above norm yields the noncommutative martingale Hardy column space \(H^c_p(M)\). The main theorem of the present paper states that the spaces \((H^c_p(M), p\in [1,\infty])\) form a natural interpolation scale with respect to the real interpolation method. The key part of the proof is the estimate of the \(K\)-functional of the pair \((H_2^c(M), H_\infty^c(M))\), involving the dual Cesàro operator. The author provides also certain extensions of the main result to noncommutative martingale Hardy-Orlicz spaces.\N\NThe paper is written in a very clear and informative way. In particular the author introduces carefully all the relevant interpolation theoretic and operator algebraic notions appearing in the main results.\N\NFor part~I see [\textit{N. Randrianantoanina}, Trans. Am. Math. Soc. 376, No.~3, 2089--2124 (2023; Zbl 1517.46048)].