Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum

Abstract

Let $\mathcal M$ be a semifinite von Neumann algebra and $T : \mathcal{M} \to \mathcal{M}$ be a positive $L_\infty-L_1$ contraction in the sense of Junge-Xu, of which the numerical range, as an operator on $L_2(\mathcal M),$ is contained in a closed polygon with vertices on the unit circle. Let $1<p<\infty.$ In this article, we prove that there exists a positive constant $C_p(T)$ such that \begin{equation}\label{abstract1stin} \Big\|\sup_{n \ge 0}\!^{+} T^n x \Big\|_p \le C_p(T)\, \|x\|_p \end{equation} for all $x \in L_p(\mathcal{M})$, extending some noncommutative maximal ergodic inequalities proved by Junge-Xu \cite{junge-Xu} and later generalized by Bekjan \cite{Bekjan2008}. In the commutative setting, the similar inequalities as above hold for arbitrary $L_\infty-L_1$ contractions with the same condition on the numerical range, yielding a vast generalisation of a classical maximal ergodic theorem of Stein \cite{Stein-ergodic-theorem} proved in 1960s. We further prove a variational inequality for contractively regular operators $T:L_p(Ω)\to L_p(Ω)$ whose peripheral spectrum is finite and satisfies a suitable resolvent estimate, extending earlier work of Le Merdy and Xu \cite{le-Merdy-Xu-q-variational-inequality}. Finally, we establish a noncommutative weak-type maximal inequality for convolution powers which was proved by Calderón and Below \cite{Bellow-Calderon} in the classical setting, complementing our strong type noncommutative $(p,p)$-maximal ergodic inequalities. Our method relies on several new polynomial identities, suitable square function estimates tailored to fit our setting and generalisation of Stein&#39;s method of embedding maximal function into analytic family of operators.