Noncommutative martingale deviation and Poincar\'e type inequalities with applications

Understand

We prove a deviation inequality for noncommutative martingales by extending Oliveira's argument for random matrices.

• By integration we obtain a Burkholder type inequality with satisfactory constant.
• Using continuous time, we establish noncommutative Poincar\'e type inequalities for "nice" semigroups with a positive curvature condition.
• These results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to Bobkov and G\"otze in the commutative case.