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

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