  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.
