Pointwise Convergence for Random Ergodic Averages in Non-commutative $L^p$-spaces

Abstract

Let $M$ be a semifinite von Neumann algebra and $T$ a positive contraction on both $L^1(M)$ and $L^\infty(M)$. We consider ergodic averages along a random sparse subsequence determined by independent Bernoulli variables $(X_n)_{n\geq 1}$ with $\mathbb{P}(X_n = 1) = n^{-\alpha}$, and set $W_N = \sum_{n=1}^N \mathbb{E}[X_n]$. We prove that, almost surely, the averages $\frac{1}{W_N} \sum_{n=1}^N X_n\, T^n(x)$ converge bilaterally almost uniformly to the ergodic projection for all $1 < p < \infty$. This extends a theorem of Bourgain to the non-commutative setting.