Dimension-free maximal inequalities for noncommutative spherical means over cyclic groups

In this paper, we establish dimension-free $L_p$-estimates for operator-valued maximal spherical means over cyclic groups ${\mathbb{Z}}_{m+1}^d$ for all $p>1$ and $m\ge 1$. The key ingredient is a noncommutative extension of the spectral technique developed by Nevo and Stein. As an application, we obtain a noncommutative spherical maximal inequality for automorphism actions on von Neumann algebras, along with several concrete examples.