An Extrapolation of Operator Valued Dyadic Paraproducts

We consider the dyadic paraproducts ${\pi }_{\text{f}}$ on $\text{T}$ associated with an $\text{M}$-valued function $\text{f}\mathrm{.}$ Here $\text{T}$ is the unit circle and $\text{M}$ is a tracial von Neumann algebra. We prove that their boundedness on $L^p(\text{T},L^p(\text{M}))$ for some $1<p<\mathrm{\infty }$ implies their boundedness on $L^p(\text{T},L^p(\text{M}))$ for all $1<p<\mathrm{\infty }$ provided $\text{f}$ is in an operator-valued BMO space. We also consider a modified version of dyadic paraproducts and their boundedness on $L^p(\T,L^p(\M)).