Boundedness of operator-valued commutators involving martingale paraproducts

Abstract

Let $1<p<\infty$. We show the boundedness of operator-valued commutators $[{\pi }_a,M_b]$ on the noncommutative $L_p(L_{\infty }(\mathbb{R})\otimes \mathcal{M})$ for any von Neumann algebra $\mathcal{M}$, where ${\pi }_a$ is the $d$-adic martingale paraproduct with symbol $a\in BM{O}^d(\mathbb{R})$ and $M_b$ is the noncommutative left multiplication operator with $b\in BM{O}_{\mathcal{M}}^d(\mathbb{R})$. Besides, we consider the extrapolation property of semicommutative $d$-adic martingale paraproducts in terms of the $BM{O}_{\mathcal{M}}^d(\mathbb{R})$ space.