Mapping properties of operator-valued Bergman projections

In this paper, we study the boundedness theory for Bergman projection in the operator-valued setting. More precisely, let \mathbb {D} be the open unit disk in the complex plane \mathbb {C} and \mathcal {M} be a semifinite von Neumann algebra. We prove that \begin{equation*} \|P(f)\|_{L_{1,\infty }(\mathcal {N})}\leq C \|f\|_{L_1(\mathcal {N})}, \end{equation*} where \mathcal {N}=L_{\infty }(\mathbb {D})\bar {\otimes }\mathcal {M} and P denotes the Bergman projection. Consequently, P is bounded on L_{p}(\mathcal {N}) with 1