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