Noncommutative Burkholder/Rosenthal inequalities II: applications

We show norm estimates for the sum of independent random variables in noncommutative Lp-spaces for 1 < p <∞ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications, we derive an equivalence for the p-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative Lp for 2 < p <∞. 0 Introduction and preliminaries This paper is a continuation of our previous work [JX1] on the investigation of noncommutative martingale inequalities. The classical theory of martingale inequalities has a long tradition in probability. It is well-known today that the applications of the works of Burkholder and his collaborators range from classical harmonic analysis to stochastic differential equations and the geometry of Banach spaces. When proving the estimates for the conditioned (or little) square function (cf. [Bu, BuG]), Burkholder was aware of Rosenthal’s result [Ro] on sums of independent random variables. Here we proceed differently and prove the noncommutative Rosenthal inequality along the same line as the noncommutative Burkholder inequality from [JX1]. This slightly modified proof yields a better constant. The main intention of this paper is to illustrate the usefulness of the conditioned square function by several examples. For many applications it is important to consider generalized notions of independence. This will allow us to explore applications towards random matrices and symmetric subspaces of noncommutative Lp-spaces. Our estimates on random matrices are motivated by the following noncommutative Khintchine inequality of Lust-Piquard [LP]. Let (εij) be an independent Rademacher family on a probability space (Ω, μ) and let (eij) be the canonical matrix units of B(l2). Then for any 2 ≤ p < ∞ there exists a positive constant cp, depending only on p, such that for scalar coefficients (aij)