  Calder\'on-Zygmund operators with noncommuting kernels may fail to be
Lp-bounded for $p \neq 2$, even for kernels with good size and smoothness
properties. Matrix-valued paraproducts, Fourier multipliers on group vNa's or
noncommutative martingale transforms are frameworks where we find such
difficulties. We obtain weak type estimates for perfect dyadic CZO's and
cancellative Haar shifts associated to noncommuting kernels in terms of a
row/column decomposition of the function. Arbitrary CZO's satisfy $H_1 \to L_1$
type estimates. In conjunction with $L_\infty \to BMO$, we get certain
row/column Lp estimates. Our approach also applies to noncommutative
paraproducts or martingale transforms with noncommuting symbols/coefficients.
Our results complement recent results of Junge, Mei, Parcet and
Randrianantoanina.
