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 $(X_n{)}_{n\ge 1}$ with $\mathbb{P}(X_n=1)=n^{-\hat{I}\pm }$, 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.