MULTI-INDEXED p-ORTHOGONAL SUMS IN NON-COMMUTATIVE LEBESGUE SPACES

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. Introduction Let M be a von Neumann algebra equipped with a faithful, normal trace τ satisfying τ(1) = 1 and let us consider the associated non-commutative Lebesgue space Lp(τ) for an even integer p. Let Γ be the product set {1, 2, . . . , n} d and let f = (fγ)γ∈Γ be a family of operators in Lp(τ) indexed by Γ. We shall say that f is p-orthogonal with d indices if τ ( f h(1)fh(2)f ∗ h(3)fh(4) · · · f ∗ h(p−1)fh(p) ) = 0 whenever the function h : {1, 2, . . . , p} → Γ has an injective projection. In other words, whenever the coordinate function πk ◦ h : {1, 2, . . . , p} → {1, 2, . . . , n} is an injective function for some 1 ≤ k ≤ d. Of course, as it is to be expected, the product above can be replaced by fh(1)f ∗ h(2) · · · fh(p−1)f ∗ h(p), with no consequences in the forthcoming results. The case of one index d = 1 was already considered by Pisier in [6]. The main result in [6] is the following inequality, which holds for any p-orthogonal family f1, f2, . . . , fn with one index ∥∥ n ∑ k=1 fk ∥∥ Lp(τ) ≤ 3π 2 p max {∥∥∥ ( n ∑ k=1 f kfk ∥∥ Lp(τ) , ∥∥ ( n ∑ k=1 fkf ∗ k ∥∥ Lp(τ) } . Some natural examples of 1-indexed p-orthogonal sequences of operators are the (non-commutative) martingale difference sequences, the operators associated to a p-dissociate subset of any discrete group (via the left regular representation) or a free circular family in Voiculescu’s sense [10]. In particular, several relevant inequalities in Harmonic Analysis such as the Littlewood-Paley inequalities, the (non-commutative) Burkholder-Gundy inequalities [8], or the (non-commutative) Khintchine inequalities [3, 4] appear as particular cases. Moreover, it turns out that the combinatorial techniques applied in [6] led to the sharp order of growth Partially supported by the Project BFM 2001/0189, Spain. 2000 Mathematics Subject Classification: Primary 46L52. Secondary 05A18.