Maximal singular integral operators acting on noncommutative $L_p$-spaces

In this paper, we study the boundedness theory for maximal Calder\'on-Zygmund operators acting on noncommutative $L_p$-spaces. Our first result is a criterion for the weak type $(1,1)$ estimate of noncommutative maximal Calder\'on-Zygmund operators; as an application, we obtain the weak type $(1,1)$ estimates of operator-valued maximal singular integrals of convolution type under proper {regularity} conditions. These are the {\it first} noncommutative maximal inequalities for families of linear operators that can not be reduced to positive ones. For homogeneous singular integrals, the strong type $(p,p)$ ($1<p<\infty$) maximal estimates are shown to be true even for {rough} kernels. As a byproduct of the criterion, we obtain the noncommutative weak type $(1,1)$ estimate for Calder\'on-Zygmund operators with integral regularity condition that is slightly stronger than the H\"ormander condition; this evidences somewhat an affirmative answer to an open question in the noncommutative Calder\'on-Zygmund theory.