  In this paper we extend a recent Pisier's inequality for p-orthogonal sums in
non-commutative Lebesgue spaces. To that purpose, we generalize the notion of
p-orthogonality to the class of multi-indexed families of operators. This kind
of families appear naturally in certain non-commutative Khintchine type
inequalities associated with free groups. Other p-orthogonal families are given
by the homogeneous operator-valued polynomials in the Rademacher variables or
the multi-indexed martingale difference sequences. As in Pisier's result, our
tools are mainly combinatorial.
