  We establish an Azuma type inequality under a Lipshitz condition for
martingales in the framework of noncommutative probability spaces and apply it
to deduce a noncommutative Heoffding inequality as well as a noncommutative
McDiarmid type inequality. We also provide a noncommutative Azuma inequality
for noncommutative supermartingales in which instead of a fixed upper bound for
the variance we assume that the variance is bounded above by a linear function
of variables. We then employ it to deduce a noncommutative Bernstein inequality
and an inequality involving $L_p$-norm of the sum of a martingale difference.
