  The non-commutative Stein inequality asks whether there exists a constant
$C_{p,q}$ depending only on $p, q$ such that \begin{equation*} \left\|
\left(\sum_{n} |\mathcal{E}_{n} (x_n) |^{q}\right)^{\frac{1}{q}} \right\|_p
\leq C_{p,q} \left\| \left(\sum_{n} | x_n |^q \right)^{\frac{1}{q}}\right
\|_p\qquad \qquad (S_{p,q}), \end{equation*} for (positive) sequences $(x_n)$
in $L_p(\mathcal{M})$. The validity of $(S_{p,2})$ for $1 < p < \infty$ and
$(S_{p,1})$ for $1 \leq p < \infty$ are known. In this paper, we verify (i)
$(S_{p,\infty})$ for $1 < p \leq \infty$; (ii) $(S_{p,p})$ for $1 \leq p <
\infty$; (iii) $(S_{p,q})$ for $1 \leq q \leq 2$ and $q<p<\infty$. We also
present some applications.
