A Non-commutative Individual Ergodic Theorem Along Sparse Random Subsequences

Let $(M,\tau )$ be a semifinite von Neumann algebra, let $J$ be a trace-preserving Jordan isomorphism, and let $(n_k{)}_{k\ge 1}$ be a random increasing sequence of integers obtained by selecting each integer $n\ge 1$ independently with probability $n^{-\alpha }$, where $0<\alpha <\frac{1}{2}$. We show that, almost surely, for every $x\in L^1(M)$, $\frac{1}{m}{\sum }_{k=1}^m{J}^{n_k}(x)$ converges bilaterally almost uniformly. This extends LaVictoire’s classical random $L^1$ ergodic theorem to the non-commutative setting.