Abstract
We study two inequalities which are concerned with $${\mathbf {T}}$$ -martingales. More precisely, we establish the noncommutative Azuma inequality for $${\mathbf {T}}$$ -martingales and noncommutative $${\mathbf {T}}$$ -martingale deviation inequality using the Golden–Thompson inequality for a positive module operator. The inequalities in this paper are considered as generalizations of the inequalities for a conditional expectation and a state. As an application, we study some convergence results which are concerned with the weak law of large numbers for weighted sums in a noncommutative probability space.
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