Pointwise convergence of noncommutative discrete spherical ergodic averages

Abstract

In this paper, we establish spherical maximal inequalities associated with commuting trace-preserving automorphisms on von Neumann algebras, extending Magyar-Stein-Wainger's discrete spherical maximal inequalities to the noncommutative setting. On a noncommutative probability space, we further prove that, under a strong ergodicity assumption, the corresponding discrete spherical averages converge bilaterally almost uniformly to $\tau (x)1$ for every $x\in L_p(\mathcal{M})$, $\frac{d}{d-2}<p\le \infty$, when $d\ge 5$.

arXiv comment24 pages