A variance bound for a general function of independent noncommutative random variables.

The main purpose of this paper is to establish a noncommutative analogue of the Efron-Stein inequality, which bounds the variance of a general function of some independent random variables. Moreover, we state an operator version including random matrices, which extends a result of D. Paulin et al., [Ann. Probab.44(5) (2016), 3431–3473]. Further, we state a Steele type inequality in the framework of noncommutative probability spaces. [ABSTRACT FROM AUTHOR]