Weak boundedness of Calder\'on-Zygmund operators on noncommutative $L_1$-spaces

In 2008, J. Parcet showed the $(1,1)$ weak-boundedness of Calder'on-Zygmund operators acting on functions taking values in a von Neumann algebra. We propose a simplified version of his proof using the same tools : Cuculescu’s projections and a pseudo-localisation theorem. This will unable us to recover the $L_p$-boundedness of Calder'on-Zygmund operators with Hilbert valued kernels acting on operator valued functions for $1<p<\mathrm{\infty }$ and an $L_p$-pseudo-localisation result of P. Hyt"onen.