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 (2016), no. 5, 3431–3473]. Further, we state a Steele type inequality in the framework of noncommutative probability spaces.