Hyperplane conjecture for quotient spaces of LpL_p

We give a positive solution for the hyperplane conjecture of quotient spaces F of $L_p$, where 1. vol(B_F)^{\frac{n-1}{n}} \kl c_0 \pl p' \pl \sup_{H \p hyperplane} vol(B_F\cap H) \pl. This result is extended to Banach lattices which does not contain $\ell_1^n$'s uniformly. Our main tools are tensor products and minimal volume ratio with respect to $L_p$-sections

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

Noncommutative Burkholder/Rosenthal inequalities II: applications

We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<\infty$ 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 $L_p$ for $2<p<\infty$.Comment: To appear in Isreal J; Mat