Molecules and Calderón-Zygmund operators with noncommuting kernels on ${\mathrm{H}}_1^c$

We study the description of semicommutative Hardy spaces in terms of molecules. We use this characterization to obtain ${\mathrm{H}}_1^c-{\mathrm{H}}_1^c$ estimates for Calderón-Zygmund operators with kernels with values in a semifinite von Neumann algebra $\mathcal{M}$.