2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69598In 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.To appear in Indiana Univ. Math. J. 14 pagesOperator AlgebrasCombinatorics46L52; 05A18text