Kolmogorov's law of the iterated logarithm for noncommutative martingales

We prove Kolmogorov's law of the iterated logarithm for noncommutative martingales. The commutative case was due to Stout. The key ingredient is an exponential inequality proved recently by Junge and the author.Comment: Revise

Noncommutative Bennett and Rosenthal inequalities

In this paper we extend the Bernstein, Prohorov and Bennett inequalities to the noncommutative setting. In addition we provide an improved version of the noncommutative Rosenthal inequality, essentially due to Nagaev, Pinelis and Pinelis, Utev for commutative random variables. We also present new best constants in Rosenthal's inequality. Applying these results to random Fourier projections, we recover and elaborate on fundamental results from compressed sensing, due to Candes, Romberg and Tao.Comment: Published in at http://dx.doi.org/10.1214/12-AOP771 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Free monotone transport for infinite variables

We extend the free monotone transport theorem of Guionnet and Shlyakhtenko to the case of infinite variables. As a first application, we provide a criterion for when mixed $q$-Gaussian algebras are isomorphic to $L(\mathbb{F}_\infty)$; namely, when the structure array $Q$ of a mixed $q$-Gaussian algebra has uniformly small entries that decay sufficiently rapidly. Here a mixed $q$-Gaussian algebra with structure array $Q=(q_{ij})_{i,j\in\mathbb{N}}$ is the von Neumann algebra generated by $X_n^Q=l_n+l_n^*, n\in\mathbb{N}$ and $(l_n)$ are the Fock space representations of the commutation relation $l_i^*l_j-q_{ij}l_jl_i^*=\delta_{i=j}, i,j\in\mathbb{N}$, $-1<q_{ij}=q_{ji}<1$