  We give a systematic study on the Hardy spaces of functions with values in
the non-commutative $L^p$-spaces associated with a semifinite von Neumann
algebra ${\cal}M.$ This is motivated by the works on matrix valued Harmonic
Analysis (operator weighted norm inequalities, operator Hilbert transform), and
on the other hand, by the recent development on the non-commutative martingale
inequalities. Our non-commutative Hardy spaces are defined by the
non-commutative Lusin integral function.
