The classical Freedman inequality, as a martingale extension of the Bernstein inequal- ity, gives an upper bound for the tail probabilities of a supermartingale whose difference sequence is bounded above. In this paper, by employing a result of Lieb–Araki concerning the concavity of a certain map and construction of special projections corre- sponding to the event of the tail probabilities, we establish some Freedman inequalities for martingales in the setting of noncommutative probability spaces. As an application, among other things, we provide a noncommutative Bernstein-type inequality.

Keywords