Non-commutative ergodic averages of balls and spheres over Euclidean spaces

In this paper, we establish a non-commutative analogue of Calderón's transference principle, which allows us to deduce the non-commutative maximal ergodic inequalities from the special case - operator-valued maximal inequalities. As applications, we deduce the non-commutative Stein-Calderón maximal ergodic inequality and the dimension-free estimates of the non-commutative Wiener maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener's pointwise ergodic theorem, following a somewhat standard way we construct a dense subset on which pointwise convergence holds. To show Jones' pointwise ergodic theorem, we use again the transference principle together with the Littlewood-Paley method, which is different from Jones' original variational method that is still unavailable in the non-commutative setting.