Convergence of noncommutative spherical averages for actions of free groups

Abstract: In this article, we extend the Bufetov pointwise ergodic theorem for spherical averages of even radius for free group actions on noncommutative $L\log L$-space. Indeed, we extend it to more general Orlicz space $L^\Phi(M, \tau)$ (noncommutative/classical), where $M$ is the semifinite von Neumann algebra with faithful normal semifinite trace $\tau$ and $\Phi: [0, \infty ) $ is a Orlicz function such that $ [0, \infty ) \ni t \rightarrow \left({\Phi(t)}\right)^{ \frac{1}{p}}$ is convex for some $p >1$. To establish this convergence we follow similar approach as Bufetov and Anantharaman-Delaroche. Thus, additionally we obtain Rota theorem on the same noncommutative Orlicz space by extending the earlier work of Anantharaman-Delaroche. Anantharaman-Delaroche proved Rota theorem for noncommutative $L^p$-spaces for $p >1$, and mentioned as ``interesting open problem'' to extend it to noncommutative $L\log L$-space as classical case. In the end we also look at the convergence of averages of spherical averages associated to free group and free semigroup actions on noncommutative spaces.